Ch 08: Conservation of EnergyWorksheetSee all chapters
All Chapters
Ch 01: Units & Vectors
Ch 02: 1D Motion (Kinematics)
Ch 03: 2D Motion (Projectile Motion)
Ch 04: Intro to Forces (Dynamics)
Ch 05: Friction, Inclines, Systems
Ch 06: Centripetal Forces & Gravitation
Ch 07: Work & Energy
Ch 08: Conservation of Energy
Ch 09: Momentum & Impulse
Ch 10: Rotational Kinematics
Ch 11: Rotational Inertia & Energy
Ch 12: Torque & Rotational Dynamics
Ch 13: Rotational Equilibrium
Ch 14: Angular Momentum
Ch 15: Periodic Motion (NEW)
Ch 15: Periodic Motion (Oscillations)
Ch 16: Waves & Sound
Ch 17: Fluid Mechanics
Ch 18: Heat and Temperature
Ch 19: Kinetic Theory of Ideal Gasses
Ch 20: The First Law of Thermodynamics
Ch 21: The Second Law of Thermodynamics
Ch 22: Electric Force & Field; Gauss' Law
Ch 23: Electric Potential
Ch 24: Capacitors & Dielectrics
Ch 25: Resistors & DC Circuits
Ch 26: Magnetic Fields and Forces
Ch 27: Sources of Magnetic Field
Ch 28: Induction and Inductance
Ch 29: Alternating Current
Ch 30: Electromagnetic Waves
Ch 31: Geometric Optics
Ch 32: Wave Optics
Ch 34: Special Relativity
Ch 35: Particle-Wave Duality
Ch 36: Atomic Structure
Ch 37: Nuclear Physics
Ch 38: Quantum Mechanics

Example #1: Energy in Horizontal Springs

Practice: A 4-kg block moving on a frictionless, horizontal surface with 20 m/s strikes a massless, horizontal spring of force constant 600 N/m. Calculate the maximum distance that the block will compress the spring by.  

EXTRA: What maximum acceleration will the block experience? (hint: this happens when it temporarily stops).

Example #2: Springs in Rough Surfaces

Practice: A 4-kg block moving on a flat surface strikes a massless, horizontal spring of force constant 600 N/m with a 20 m/s. The block-surface coefficient of friction is 0.5. Calculate the maximum compression that the spring will experience.

Example #3: Energy in Vertical Springs

Example #4: Energy in Vertical Springs

Additional Problems
A 205 kg block is released at a 6.3 m height as shown. The track is frictionless. The block travels down the track, hits a spring of force constant k = 1285 N/m . The acceleration of gravity is 9.8 m/s2. Determine the compression of the spring x from its equilibrium position before coming to rest momentarily. A. 3.39911 B. 2.46691 C. 3.05595 D. 2.80797 E. 3.4477 F. 3.57838 G. 2.50776 H. 3.66395 I. 3.72911 J. 4.43837
A 1.86-kg block is held in place against the spring by an 81-N horizontal external force (see the figure). The external force is removed, and the block is projected with a velocity v1 = 1.2 m/s upon separation from the spring. The block descends a ramp and has a velocity v2 = 1.9 m/s at the bottom. The track is frictionless between points A and B. The block enters a rough section at B, extending to E. The coefficient of kinetic friction over this section is 0.28. The velocity of the block is v3 = 1.4 m/s at C. The block moves on to D, where it stops. The height h of the ramp is closest to _____. A. 15 cm B. 17 cm C. 11 cm D. 18 cm E. 7.3 cm
It takes 4.36 J of work to stretch a Hooke’s-law spring 7.54 cm from its unstressed length. How much the extra work is required to stretch it an additional 4.8 cm? 1. 27.5923 2. 13.8816 3. 14.269 4. 3.54616 5. 6.58426 6. 5.18114 7. 15.988 8. 17.1564 9. 18.2072 10. 7.31815
On a horizontal surface, a crate with mass 50.0 kg is placed against a spring that stores 360 J of energy. The spring is released and the crate slides 5.60 m before coming to rest.What is the speed of the crate when it is 2.00 m from its initial position?
A horizontal spring with spring constant 130 N/m is compressed 18 cm and used to launch a 2.6 kg box across a frictionless, horizontal surface. After the box travels some distance, the surface becomes rough. The coefficient of kinetic friction of the box on the surface is 0.15.Use work and energy to find how far the box slides across the rough surface before stopping.
Consider the track shown in the figure. The section AB is one quadrant of a circle of radius 2.0 m and is frictionless. B to C is a horizontal span 3.1 m long with a coefficient of kinetic friction exttip{mu_{ m k}}{mu} = 0.24. The section CD under the spring is frictionless. A block of mass 1.0 kg is released from rest at A. After sliding on the track, it compresses the spring by 0.30 m . Determine the velocity of the block at point B.Determine the thermal energy produced as the block slides from B to C.Determine the velocity of the block at point C.Determine the stiffness constant k for the spring.
A 0.620 kg wood block is firmly attached to a very light horizontal spring ( exttip{k}{k} = 180 N/m ) as shown in the figure. This block-spring system, when compressed 5.1 cm and released, stretches out 2.3 cm beyond the equilibrium position before stopping and turning back.What is the coefficient of kinetic friction between the block and the table?
It takes 26 N of force to compress a very light spring 18 cm. The spring is placed parallel to a table and its ends are firmly attached to the wall and to a 240-g wood block . The coefficient of kinetic friction between the block and the table is 0.36. The spring is compressed a second time to 18 cm and then released from rest.How far beyond its equilibrium position will it stretch on its first cycle?
A 2.1 kg block slides along a horizontal surface with a coefficient of kinetic friction exttip{mu _{ m k}}{mu_k} = 0.31. The block has a speed exttip{v}{v} = 1.4 m/s when it strikes a massless spring head-on.If the spring has force constant exttip{k}{k} = 120 N/m , how far is the spring compressed?What minimum value of the coefficient of static friction, S, will assure that the spring remains compressed at the maximum compressed position?If S is less than this, what is the speed of the block when it detaches from the decompressing spring? [Hint: Detachment occurs when the spring reaches its natural length (x = 0).]Explain why the detachment occurs when the spring reaches its natural length (x = 0).
A 2.55 kg block on a horizontal floor is attached to a horizontal spring that is initially compressed 0.0360 m . The spring has force constant 855 N/m . The coefficient of kinetic friction between the floor and the block is 0.41 . The block and spring are released from rest and the block slides along the floor.What is the speed of the block when it has moved a distance of 0.0160 m from its initial position? (At this point the spring is compressed 0.0200 m .)
A block with mass 0.50 kg is forced against a horizontal spring of negligible mass, compressing the spring a distance of 0.20 m (the figure ). When released, the block moves on a horizontal tabletop for 1.00 m before coming to rest. The spring constant exttip{k}{k} is 100 N/m.What is the coefficient of kinetic friction k between the block and the tabletop?
A 13.0 kg stone slides down a snow-covered hill (the figure ), leaving point exttip{A}{A} with a speed of 12.0 m/s . There is no friction on the hill between points exttip{A}{A} and exttip{B}{B}, but there is friction on the level ground at the bottom of the hill, between exttip{B}{B} and the wall. After entering the rough horizontal region, the stone travels 100 m and then runs into a very long, light spring with force constant 2.30 N/m . The coefficients of kinetic and static friction between the stone and the horizontal ground are 0.20 and 0.80, respectively.What is the speed of the stone when it reaches point exttip{B}{B}?How far will the stone compress the spring?Will the stone move again after it has been stopped by the spring?
A box of mass 2.00 kg is placed against a compressed spring. The spring is released and the box slides along a horizontal surface. The spring initially had 82.0 J of potential energy stored in it. The coefficient of kinetic friction between the box and the surface is μk = 0.400. What is the speed of the box after it has traveled a distance of 3.00 m from its initial position? (At this point the box is no longer in contact with the spring and no potential energy is left in the spring.)
A box of mass 3.0 kg is placed against a compressed spring. The spring is released and the box slides along a horizontal surface. The spring initially had 76.0 J of potential energy stored in it and after the box leaves the spring then there is no energy stored in the spring. After the box has traveled a distance of 2.0 m from its initial position its speed is 6.0 m/s. What is the coefficient of kinetic friction μk between the box and the surface?
A 7-kg mass slides to the right on a surface having a coefficient of friction 0.57 as shown in the figure. The mass has a speed of 6 m/s when contact is made with a spring that has a spring constant 150 N/m. The mass comes to rest after the spring has been compressed a distance d. The mass is then forced toward the left by the spring and continues to move in that direction beyond the unstretched position. Finally, the mass comes to rest a distance D to the left of the unstretched spring. The acceleration of gravity is 9.8 m/s2. Find the compressed distance d. 1. 1.20704 2. 0.851612 3. 1.34481 4. 1.22121 5. 1.17285 6. 0.771988 7. 1.14619 8. 1.31355 9. 1.06142 10. 1.00331
You are an industrial engineer with a shipping company. As part of the package-handling system, a small box with mass 1.60 kg is placed against a light spring that is compressed 0.280 m. The spring has force constant 46.0 N/m . The spring and box are released from rest, and the box travels along a horizontal surface for which the coefficient of kinetic friction with the box is k = 0.300. When the box has traveled 0.280 m and the spring has reached its equilibrium length, the box loses contact with the spring.What is the speed of the box at the instant when it leaves the spring?What is the maximum speed of the box during its motion?
A 2.00-kg package is released on a 53.1 53.1 s = 0.40 incline, 4.00 m from a long spring with force constant 120 N/m that is attached at the bottom of the incline . The coefficients of friction between the package and the incline are k = 0.20 and mu_{k} ;=; 0.20. The mass of the spring is negligible.What is the speed of the package just before it reaches the spring?What is the maximum compression of the spring?The package rebounds back up the incline. How close does it get to its initial position?
A box of mass exttip{m}{m} is pushed against a spring of negligible mass and force constant exttip{k}{k}, compressing it a distance exttip{x}{x}. The box is then released and travels up a ramp that is at an angle exttip{alpha }{alpha} above the horizontal. The coefficient of kinetic friction between the box and the ramp is exttip{mu _{ m k}}{mu_k}, where k < 1. The box is still moving up the ramp after traveling a distance s > | x | along the ramp.Calculate the angle exttip{alpha }{alpha} for which the speed of the box after traveling distance exttip{s}{s} is a minimum.
A small rock block with mass 0.400 kg is placed against a compressed spring at the bottom of a 37.0° incline. The compressed spring has 50.0 J of elastic potential energy stored in it. The spring is released and the block moves a distance of 12.0 m along the incline before momentarily coming to rest. How much work does the friction force do on the block during the motion? How much work does the friction force do on the block during the motion?
A 19 kg box slides 4.0 m down the frictionless ramp shown in the figure, then collides with a spring whose spring constant is 200 N/m .What is the maximum compression of the spring?At what compression of the spring does the box have its maximum speed?
You are designing a delivery ramp for crates containing exercise equipment. The 1670-N crates will move at 1.8 m/s at the top of a ramp that slopes downward at 22.0N. The ramp exerts a 515-N kinetic friction force on each crate, and the maximum static friction force also has this value. Each crate will compress a spring at the bottom of the ramp and will come to rest after traveling a total distance of 5.0 m m along the ramp. Once stopped, a crate must not rebound back up the ramp.Calculate the largest force constant of the spring that will be needed to meet the design criteria.
The spring shown in the figure is compressed 51 cm and used to launch a 100 kg physics student. The track is frictionless until it starts up the incline. The students coefficient of kinetic friction on the 30 30 (Page 231) incline is 0.16 . You may want to review (Page 231) . For help with math skills, you may want to review: Mathematical Expressions Involving Squares For general problem-solving tips and strategies for this topic, you may want to view a Video Tutor Solution of Pulling a box across the floor.What is the students speed just after losing contact with the spring?How far up the incline does the student go?
A freight company uses a compressed spring to shoot 1.90 kg packages up a 1.00-m -high frictionless ramp into a truck, as the figure shows. The spring constant is 371 N/m and the spring is compressed 33.0 cm .What is the speed of the package when it reaches the truck?A careless worker spills his soda on the ramp. This creates a 50.0-cm-long sticky spot with a coefficient of kinetic friction 0.300. Will the next package make it into the truck?
A 56 g ice cube can slide without friction up and down a 3030 slope. The ice cube is pressed against a spring at the bottom of the slope, compressing the spring 10 cm. The spring constant is 26 { m N/m} . When the ice cube is released, what total distance will it travel up the slope before reversing direction?The ice cube is replaced by a 56 g plastic cube whose coefficient of kinetic friction is 0.20. How far will the plastic cube travel up the slope?
A small block with mass 0.500 kg is placed against a compressed spring at the bottom of a ramp (point A). The ramp is inclined at 37.0° above the horizontal. Initially the potential energy stored in the spring is 90.0 J. The spring is released with the block initially at rest and the block slides up the ramp to point B. As the block travels from A to B, the work done by friction has magnitude 40.0 J. What is the kinetic energy of the block when it reaches point B?
The spring in the figure has a spring constant of 1300 N/m . It is compressed 16.0 cm , then launches a 200 g block. The horizontal surface is frictionless, but the blocks coefficient of kinetic friction on the incline is 0.210. What distance exttip{d}{d} does the block sail through the air?
A small glider is placed against a compressed spring at the bottom of an air track that slopes upward at an angle of 41.0 above the horizontal. The glider has mass 8.00×10−2 kg . The spring has 640 N/m and negligible mass. When the spring is released, the glider travels a maximum distance of 1.60 m along the air track before sliding back down. Before reaching this maximum distance, the glider loses contact with the spring.What distance was the spring originally compressed?When the glider has traveled along the air track 0.600 m from its initial position against the compressed spring, is it still in contact with the spring?What is the kinetic energy of the glider at this point?
A 55 g ice cube can slide up and down a frictionless 30 slope. At the bottom, a spring with spring constant 30 cm is compressed 10 { m cm} and used to launch the ice cube up the slope.How high does it go above its starting point?
A spring (80 N/m ) has an equilibrium length of 1.00 m. The spring is compressed to a length of 0.50 m and a mass of 2.1 kg is placed at its free end on a frictionless slope which makes an angle of 41 with respect to the horizontal. The spring is then released. If the mass is not attached to the spring, how far up the slope from the compressed point will the mass move before coming to rest?If the mass is attached to the spring, how far up the slope from the compressed point will the mass move before coming to rest?Now the incline has a coefficient of kinetic friction k. If the block, attached to the spring, is observed to stop just as it reaches the springs equilibrium position, what is the coefficient of friction k?
A 2.00-kg block is pushed against a spring with negligible mass and force constant k = 400 N/m, compressing it 0.220 m. When the block is released, it moves along a frictionless, horizontal surface and then up a frictionless incline with slope 37.0 .What is the speed of the block as it slides along the horizontal surface after having left the spring?How far does the block travel up the incline before starting to slide back down?
The Great Sandini is a 60-kg circus performer who is shot from a cannon (actually a spring gun). You dont find many men of his caliber, so you help him design a new gun. This new gun has a very large spring with a very small mass and a force constant of 1300 N/m that he will compress with a force of 6500 N . The inside of the gun barrel is coated with Teflon, so the average friction force will be only 41 N during the distance of 5.0 m that he moves in the barrel.At what speed will he emerge from the end of the barrel, a distance 1.6 m above his initial rest position?
A wooden block with mass 1.55 kg is placed against a compressed spring at the bottom of a slope inclined at an angle of 31.0 (point A). When the spring is released, it projects the block up the incline. At point B, a distance of 6.05 m up the incline from A, the block is moving up the incline at a speed of 6.30 m/s and is no longer in contact with the spring. The coefficient of kinetic friction between the block and incline is exttip{mu_k}{mu_k} = 0.50. The mass of the spring is negligible.Calculate the amount of potential energy that was initially stored in the spring.
An elevator cable breaks when a 950 kg elevator is 23.0 m above the top of a huge spring ( exttip{k}{k} = 8.00×104 N/m ) at the bottom of the shaft.Calculate the work done by gravity on the elevator before it hits the spring.Calculate the speed of the elevator just before striking the spring.Calculate the amount the spring compresses (note that here work is done by both the spring and gravity).
An experimental apparatus with mass m is placed on a vertical spring of negligible mass and pushed down until the spring is compressed a distance exttip{x}{x}. The apparatus is then released and reaches its maximum height at a distance h above the point where it is released. The apparatus is not attached to the spring, and at its maximum height it is no longer in contact with the spring. The maximum magnitude of acceleration the apparatus can have without being damaged is exttip{a}{a}, where a > g.What should the force constant of the spring be?What distance exttip{x}{x} must the spring be compressed initially?
An 80.0-kg man jumps from a height of 2.50 m onto a platform mounted on springs. As the springs compress, he pushes the platform down a maximum distance of 0.240 m below its initial position, and then it rebounds. The platform and springs have negligible mass.What is the mans speed at the instant he depresses the platform 0.120 m?If the man just steps gently onto the platform, what maximum distance would he push it down?
A spring of negligible mass has force constant exttip{k}{k} = 860 N/m .How far must the spring be compressed for 160 J of potential energy to be stored in it?You place the spring vertically with one end on the floor. You then lay a 1.60-kg book on top of the spring and release the book from rest. Find the maximum distance the spring will be compressed.
A 1100 kg safe is 2.1 m above a heavy-duty spring when the rope holding the safe breaks. The safe hits the spring and compresses it 52 cm .What is the spring constant of the spring?
A 60 kg bungee jumper leaps from a bridge. She is tied to a bungee cord that is 13 m long when unstretched, and falls a total of 32 m .Calculate the spring constant k of the bungee cord assuming Hookes law applies.Calculate the maximum acceleration she experiences.
A 71 kg trampoline artist jumps vertically upward from the top of a platform with a speed of 4.3 m/s . How fast is he going as he lands on the trampoline, 2.0 m below?If the trampoline behaves like a spring of spring constant 6.1×104 N/m , what is the distance he depress it?
A vertical spring (ignore its mass), whose spring constant is 880 N/m , is attached to a table and is compressed down by 0.170 m .What upward speed can it give to a 0.370 kg ball when released?How high above its original position (spring compressed) will the ball fly?
An ingenious bricklayer builds a device for shooting bricks up to the top of the wall where he is working. He places a brick on a vertical compressed spring with force constant k = 450 N/m and negligible mass. When the spring is released, the brick is propelled upward.If the brick has mass 1.80 kg and is to reach a maximum height of 3.6 m above its initial position on the compressed spring, what distance must the bricklayer compress the spring initially?
An engineer is designing a spring to be placed at the bottom of an elevator shaft.If the elevator cable breaks when the elevator is at a height h above the top of the spring, calculate the value that the spring constant k should have so that passengers undergo an acceleration of no more than 7.0 g when brought to rest. Let M be the total mass of the elevator and passengers.
When a 63 kg cheerleader stands on a vertical spring, the spring compresses by 5.8 cm. when a second cheerleader stands on the shoulders of the first, the spring compresses an additional 4.8 cm.What is the mass of the second cheerleader?
An ideal spring of negligible mass is 13.00 cm long when nothing is attached to it. When you hang a 3.15-kg weight from it, you measure its length to be 14.40 cm .If you wanted to store 10.0 J of potential energy in this spring, what would be its total length? Assume that it continues to obey Hookes law.
A slingshot will shoot a 10-g pebble 22.0 m straight up.How much potential energy is stored in the slingshots rubber band?With the same potential energy stored in the rubber band, how high can the slingshot shoot a 25-g pebble?What physical effects did you ignore in solving this problem?
A piece of cheese with a mass of 1.21 kg is placed on a vertical spring of negligible mass and a force constant exttip{k}{k} = 2100 N/m that is compressed by a distance of 15.1 cm .When the spring is released, how high does the cheese rise from this initial position? (The cheese and the spring are not attached.)
In a "worst-case" design scenario, a 2000-kg elevator with broken cables is falling at 4.00 m/s when it first contacts a cushioning spring at the bottom of the shaft. The spring is supposed to stop the elevator, compressing 2.00 m as it does so. During the motion a safety clamp applies a constant 17000-N frictional force to the elevator.What is the speed of the elevator after it has moved downward 1.00 m from the point where it first contacts a spring?When the elevator is 1.00 m below point where it first contacts a spring, what is its acceleration?
If you stand on a bathroom scale, the spring inside the scale compresses 0.55 mm , and it tells you your weight is 760 N .Now if you jump on the scale from a height of 1.3 m , what does the scale read at its peak?
A bungee cord is 30.0 m long and, when stretched a distance exttip{x}{x}, it exerts a restoring force of magnitude exttip{kx}{kx}. Your father-in-law (mass 96.0 kg ) stands on a platform 45.0 m above the ground, and one end of the cord is tied securely to his ankle and the other end to the platform. You have promised him that when he steps off the platform he will fall a maximum distance of only 41.0 m before the cord stops him. You had several bungee cords to select from, and you tested them by stretching them out, tying one end to a tree, and pulling on the other end with a force of 410 N .When you do this, what distance will the bungee cord that you should select have stretched?
A 3.05 kg fish is attached to the lower end of a vertical spring that has negligible mass and force constant 880 N/m . The spring initially is neither stretched nor compressed. The fish is released from rest.What is its speed after it has descended 0.0510 m from its initial position?What is the maximum speed of the fish as it descends?
A basket of negligible weight hangs from a vertical spring scale of force constant 1500 N/m .If you suddenly put a 3.00 kg adobe brick in the basket, find the maximum distance that the spring will stretch.If, instead, you release the brick from 1.00 m above the basket, by how much will the spring stretch at its maximum elongation?
A spring of negligible mass has force constant exttip{k}{k} = 1700 N/m. You may want to review (Pages 212 - 217).For related problemsolving tips and strategies, you may want to view a Video Tutor Solution of Motion with gravitational, elastic, and friction forces. How far must the spring be compressed for an amount 3.40 J of potential energy to be stored in it?You place the spring vertically with one end on the floor. You then drop a book of mass 1.40 kg onto it from a height of 0.800 m above the top of the spring. Find the maximum distance the spring will be compressed.
In Figure 7.16 in the textbook a glider with mass m=0.200 kg sits on a frictionless horizontal air track, connected to a spring with force constant k=5.00 N/m. The glider is released from rest with the spring stretched 0.100 m.What is the displacement x of the glider from its equilibrium position when its speed is 0.20 m/s? (You should get more than one answer.)Explain why you get more than one answer.
A machine part of mass m is attached to a horizontal ideal spring of force constant exttip{k}{k} that is attached to the edge of a friction-free horizontal surface. The part is pushed against the spring, compressing it a distance x0, and then released from rest.Find the maximum speed.Find the maximum acceleration of the machine part.Where in the motion do the maxima in part A occurs?What will be the maximum extension of the spring?Describe the subsequent motion of this machine part. Will it ever stop permanently?Where in the motion do the maxima in part B occurs?
A spring gun shoots out a plastic ball at speed v0 . The spring is then compressed twice the distance it was on the first shot.By what factor is the balls speed increased?
You have been hired to design a spring-launched roller coaster that will carry two passengers per car. The car goes up a 12-m-high hill, then descends 18 m to the tracks lowest point. Youve determined that the spring can be compressed a maximum of 2.3 m and that a loaded car will have a maximum mass of 430 kg . For safety reasons, the spring constant should be 13 % larger than the minimum needed for the car to just make it over the top.What spring constant should you specify?What is the maximum speed of a 350 kg car if the spring is compressed the full amount?
A block with mass 2.0 kg is on a horizontal frictionless surface and is placed against a compressed spring that has force constant 128 N/m. The spring is released and the block moves along the surface away from the spring. If the spring was initially compressed 0.20 m, what is the speed of the block after it has left the spring?  A) l.6 m/s  B) 2.4 m/s  C) 3.2 m/s  D) 12.8 m/s  E) 19.2 m/s  F) 25.6 m/s  G) none of the above answers
A spring with force constant 720 N/m has one end attached to a wall. A 5.00 kg block on the floor is pushed against the spring, compressing it 5.00 cm. The block is released from rest and moves away from the wall. Friction between the block and the floor can be neglected. What is the speed of the block after it leaves the spring?  (a) 0.300 m/s (b) 0.600 m/s (c) 0.720 m/s (d) 0.960 m/s (e) 1.20 m/s (f) none of the above answers
A 2.60 kg mass is pushed against a horizontal spring of force constant 26.0 N/cm on a frictionless air table. The spring is attached to the tabletop, and the mass is not attached to the spring in any way. When the spring has been compressed enough to store 13.0 J of potential energy in it, the mass is suddenly released from rest.Find the greatest speed the mass reaches.What is the greatest acceleration of the mass?When does this occur?When does it occur?
The mass exttip{m}{m} = 5.5 kg resting on a frictionless horizontal table is connected to a horizontal spring with stiffness constant exttip{k}{k} = 180 N/m . The mass is pulled a distance to the right so that the spring is stretched a distance exttip{x_{ m 0}}{x_0} = 1.6 m initially, and then the mass is released from rest.Determine the total energy of the system.Determine the kinetic energy when x = 1 2x0.Determine the maximum kinetic energy.Determine the maximum speed.Determine the maximum acceleration.At what position it occurs?At what position it occurs?
A 0.500-kg block, attached to a spring with length 0.60 m and force constant 40.0 N/m, is at rest with the back of the block at point exttip{A}{A} on a frictionless, horizontal air table (the figure ). The mass of the spring is negligible. You move the block to the right along the surface by pulling with a constant 20.0-N horizontal force.What is the blocks speed when the back of the block reaches point exttip{B}{B}, which is 0.25 m to the right of point exttip{A}{A}?When the back of the block reaches point exttip{B}{B}, you let go of the block. In the subsequent motion, how close does the block get to the wall where the left end of the spring is attached?
You are asked to design a spring that will give a 1160 kg satellite a speed of 2.85 m/s relative to an orbiting space shuttle. Your spring is to give the satellite a maximum acceleration of 5.00g. The springs mass, the recoil kinetic energy of the shuttle, and changes in gravitational potential energy will all be negligible.You may want to review (Pages 217 - 221). For related problemsolving tips and strategies, you may want to view a Video Tutor Solution of Motion with elastic potential energy.What must the force constant of the spring be?What distance must the spring be compressed?
A force of 531 N keeps a certain spring stretched a distance of 0.600 m .What is the potential energy of the spring when it is stretched 0.600 m ?What is its potential energy when it is compressed 8.00 cm ?
How far must you stretch a spring with exttip{k}{k} = 1100 N/m to store 210 J of energy?
A stretched spring stores 3.3 J of energy.How much energy will be stored if the spring is stretched three times as far?
Tendons are strong elastic fibers that attach muscles to bones. To a reasonable approximation, they obey Hookes law. In laboratory tests on a particular tendon, it was found that, when a 251 g object was hung from it, the tendon stretched 1.23 cm .Find the force constant of this tendon in N/m.Because of its thickness, the maximum tension this tendon can support without rupturing is 141 N . By how much can the tendon stretch without rupturing?How much energy is stored in it at that point?
A spring stores potential energy U0 when it is compressed a distance x0 from its uncompressed length.In terms of U0, how much energy does it store when it is compressed twice as much?In terms of x0, how much must it be compressed from its uncompressed length to store twice as much energy?In terms of U0, how much energy does it store when it is compressed half as much?In terms of x0, how much must it be compressed from its uncompressed length to store half as much energy?
Rank in order, from most to least, the elastic potential energy ( Us )a to ( Us )d stored in the springs of .
A spring is compressed 7.0 cm.How far must you compress a spring with twice the spring constant to store the same amount of energy?
A block sliding along a horizontal frictionless surface with speed exttip{v}{v} collides with a spring and compresses it by 2.2 cm .What will be the compression if the same block collides with the spring at a speed of 4v?
A 13 kg runaway grocery cart runs into a spring with spring constant 250 N/m and compresses it by 61 cm .What was the speed of the cart just before it hit the spring?
As a 1.6×104 kg jet plane lands on an aircraft carrier, its tail hook snags a cable to slow it down. The cable is attached to a spring with spring constant 6.1×104 N/m .If the spring stretches 31 m to stop the plane, what was the planes landing speed?
You are designing an amusement park ride. A cart with two riders moves horizontally with speed exttip{v}{v} = 5.60 m/s . You assume that the total mass of cart plus riders is 300 kg. The cart hits a light spring that is attached to a wall, momentarily comes to rest as the spring is compressed, and then regains speed as it moves back in the opposite direction. For the ride to be thrilling but safe, the maximum acceleration of the cart during this motion should be 3.00g. Ignore friction.What is the required force constant of the spring?What is the maximum distance the spring will be compressed?
A small rock block with mass 0.400 kg is placed against a compressed spring at the bottom of a 37.0° incline. The compressed spring has 50.0 J of elastic potential energy stored in it. The spring is released and the block moves a distance of 12.0 m along the incline before momentarily coming to rest. How much work does the friction force do on the block during the motion? What is the coefficient of kinetic frinction μk between the block and the incline?
In a physics lab experiment, a compressed spring launches a 31 g metal ball at a 35° angle. Compressing the spring 18 cm causes the ball to hit the floor 1.8 m below the point at which it leaves the spring after traveling 5.6 m horizontally. What is the spring constant?
Its your birthday, and to celebrate you're going to make your first bungee jump. You stand on a bridge 120 m above a raging river and attach a 33-m-long bungee cord to your harness. A bungee cord, for practical purposes, is just a long spring, and this cord has a spring constant of 43 N/m . Assume that your mass is 76 kg . After a long hesitation, you dive off the bridge. How far are you above the water when the cord reaches its maximum elongation?
A spring has a spring constant k of 82.0 N/m. How much must this spring be compressed to store 40.0 J of potential energy?
A student places her 410 g physics book on a frictionless table. She pushes the book against a spring, compressing the spring by 7.00 cm, then releases the book. What is the books speed as it slides away? The spring constant is 1650 N/m.
A 1300 kg car moving on a horizontal surface has speed exttip{v}{v} = 80 km/h when it strikes a horizontal coiled spring and is brought to rest in a distance of 2.3 m. What is the spring stiffness constant of the spring?
A 1.50-kg object is held 1.20 m above a relaxed massless, vertical spring with a force constant of 320 N/m. The object is dropped onto the spring.(a) How far does the object compress the spring?(b) Repeat part (a), but this time assume a constant air-resistance force of 0.700 N acts on the object during its motion.(c) How far does the object compress the spring if the same experiment is performed on the Moon, where g = 1.63 m/s2 and air resistance is neglected?
What should be the spring constant k of a spring designed to bring a 1300 kg car to rest from a speed of 90 km/h so that the occupants undergo a maximum acceleration of 5.5 g?
When an object at the end of a spring compresses that spring by some distance, the object gains a potential energy of (1/2)kd2, where k is known as the “force constant”. A 15 kg mass compresses a spring with a force constant of 200 N/m by 15 cm, and is then released, launching it across a frictionless floor. How much kinetic energy does the mass gain from the spring? If that mass crosses 15 cm of the floor and then slides up a 40o incline, how far up the ramp will the mass get before it comes to a stop and begins to slide down the ramp?
A 0.160 kg block of ice is placed against a horizontal, compressed spring mounted on a horizontal tabletop that is 1.30 m above the floor. The spring has force constant 2050 N/m and is initially compressed 0.045 m. The mass of the spring is negligible. The spring is released, and the block slides along the table, goes off the edge, and travels to the floor. If there is negligible friction between the block of ice and the tabletop, what is the speed of the block of ice when it reaches the floor?
A box with mass 5.00 kg is placed against a compressed spring. The spring is released and the box slides 4.00 m along a horizontal surface before coming to rest. (At this point the box is no longer in contact with the spring and there is no potential energy left stored in the spring.) The coefficient of kinetic friction between the block and the surface is 0.300. How much energy was initially stored in the compressed spring?
A 10.0-kg block is released from rest at point A in Figure P8.63. The track is frictionless except for the portion between points B and C, which has a length of 6.00 m. The block travels down the track, hits a spring of force constant 2 250 N/m, and compresses the spring 0.300 m from its equilibrium position before coming to rest momentarily. Determine the coefficient of kinetic friction between the block and the rough surface between points B and C.
A block of mass m = 2.00 kg is attached to a spring of force constant k = 500 N/m as shown in Figure P8.15. The block is pulled to a position xi = 5.00 cm to the right of equilibrium and released from rest. Find the speed the block has as it passes through equilibrium if(a) the horizontal surface is frictionless and(b) the coefficient of friction between block and surface is µk = 0.350.
A toy cannon uses a spring to project a 5.30-g soft rubber ball. The spring is originally compressed by 5.00 cm and has a force constant of 8.00 N/m. When the cannon is fired, the ball moves 15.0 cm through the horizontal barrel of the cannon, and the barrel exerts a constant friction force of 0.032 0 N on the ball.(a) With what speed does the projectile leave the barrel of the cannon?(b) At what point does the ball have maximum speed?(c) What is this maximum speed?
A massless spring of constant k = 78.4 N/m is fixed on the left side of a level track. A block of mass m = 0.50 kg is pressed against the spring and compresses it a distance d, as in Figure P7.76. The block (initially at rest) is then released and travels toward a circular loop-the-loop of radius R = 1.5 m. The entire track and the loop-the-loop are frictionless, except for the section of track between points A and B. Given that the coefficient of kinetic friction between the block and the track along AB is μk = 0.30, and that the length of AB is 2.5 m, determine the minimum compression d of the spring that enables the block to just make it through the loop-the-loop at point C. Hint: The force exerted by the track on the block will be zero if the block barely makes it through the loop-the-loop.
A 1.00-kg object slides to the right on a surface having a coefficient of kinetic friction 0.250 (Fig. P8.62a). The object has a speed of vi = 3.00 m/s when it makes contact with a light spring (Fig. P8.62b) that has a force constant of 50.0  N/m. The object comes to rest after the spring has been compressed a distance d (Fig. P8.62c). The object is then forced toward the left by the spring (Fig. P8.62d) and continues to move in that direction beyond the spring’s unstretched position. Finally, the object comes to rest a distance D to the left of the unstretched spring (Fig. P8.62e). Find(a) the distance of compression d,(b) the speed v at the unstretched position when the object is moving to the left (Fig. P8.62d), and(c) the distance D where the object comes to rest.
A 200-g block is pressed against a spring of force constant 1.40 kN/m until the block compresses the spring 10.0  cm. The spring rests at the bottom of a ramp inclined at 60.0o to the horizontal. Using energy considerations, determine how far up the incline the block moves from its initial position before it stops(a) if the ramp exerts no friction force on the block and(b) if the coefficient of kinetic friction is 0.400.
An inclined plane of angle θ = 20.0° has a spring of force constant k = 500 N/m fastened securely at the bottom so that the spring is parallel to the surface as shown in Figure P7.63. A block of mass m = 2.50 kg is placed on the plane at a distance d = 0.300 m from the spring. From this position, the block is projected downward toward the spring with speed v = 0.750 m/s. By what distance is the spring compressed when the block momentarily comes to rest?
An inclined plane of angle θ has a spring of force constant k fastened securely at the bottom so that the spring is parallel to the surface. A block of mass m is placed on the plane at a distance d from the spring. From this position, the block is projected downward toward the spring with speed v as shown in Figure P7.63. By what distance is the spring compressed when the block momentarily comes to rest?
A block of mass M rests on a table. It is fastened to the lower end of a light, vertical spring. The upper end of the spring is fastened to a block of mass m. The upper block is pushed down by an additional force 3mg, so the spring compression is 4mg/k. In this configuration, the upper block is released from rest. The spring lifts the lower block off the table. In terms of m, what is the greatest possible value for M?
A daredevil plans to bungee jump from a balloon 65.0 m above the ground. He will use a uniform elastic cord, tied to a harness around his body, to stop his fall at a point 10.0 m above the ground. Model his body as a particle and the cord as having negligible mass and obeying Hooke’s law. In a preliminary test he finds that when hanging at rest from a 5.00-m length of the cord, his body weight stretches it by 1.50 m. He will drop from rest at the point where the top end of a longer section of the cord is attached to the stationary balloon.(a) What length of cord should he use?(b) What maximum acceleration will he experience?
Hooke’s law describes a certain light spring of unstretched length 35.0 cm. When one end is attached to the top of a doorframe and a 7.50-kg object is hung from the other end, the length of the spring is 41.5 cm. (a) Find its spring constant. (b) The load and the spring are taken down. Two people pull in opposite directions on the ends of the spring, each with a force of 190 N. Find the length of the spring in this situation.
A child’s pogo stick (Fig. P8.61) stores energy in a spring with a force constant of 2.50 x  104 N/m. At position A (xA = - 0.100  m), the spring compression is a maximum and the child is momentarily at rest. At position B (xB = 0), the spring is relaxed and the child is moving upward. At position C, the child is again momentarily at rest at the top of the jump. The combined mass of child and pogo stick is 25.0 kg. Although the boy must lean forward to remain balanced, the angle is small, so let’s assume the pogo stick is vertical. Also assume the boy does not bend his legs during the motion.(a) Calculate the total energy of the child–stick–Earth system, taking both gravitational and elastic potential energies as zero for x = 0.(b) Determine xC.(c) Calculate the speed of the child at x = 0.(d) Determine the value of x for which the kinetic energy of the system is a maximum.(e) Calculate the child’s maximum upward speed.
A 6 000-kg freight car rolls along rails with negligible friction. The car is brought to rest by a combination of two coiled springs as illustrated in Figure P7.59. Both springs are described by Hooke’s law and have spring constants k1 = 1 600 N/m and k2 = 3 400 N/m. After the first spring compresses a distance of 30.0 cm, the second spring acts with the first to increase the force as additional compression occurs as shown in the graph. The car comes to rest 50.0 cm after first contacting the two-spring system. Find the car’s initial speed.