Ch 06: Centripetal Forces & GravitationWorksheetSee 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

Concept #1: Intro to Centripetal Forces

Practice: For the situation above, suppose the string breaks if its tension exceeds 50 N. Calculate the maximum speed that the object can attain without breaking the string.

Example #1: Intro to Centripetal Forces

Example #2: Vertical Centripetal Forces

Example #3: Vertical Centripetal Forces

Practice: A pendulum is made from a light, 2 m-long rope and a 5-kg small object. When you release the object from rest as of a certain height, it swings from side to side, attaining a maximum speed of 10 m/s. At the object’s lowest point: 

(a) Draw a Free Body Diagram.
(b) Find the magnitude of its acceleration.
(c) Find the tension on the rope.

Concept #2: Flat Curve

Concept #3: Banked Curve

Practice: You are designing a highway curve to allow cars to turn, without any banking, at a maximum speed of 50 m/s. The average coefficient of friction between cars and asphalt, for dry roads, is roughly 0.7. What radius would this curve have to have, for this to be possible? For the radius you just found, how much would you have to bank the same curve, in order to attain the same maximum speed, but at the absence of friction?

Additional Problems
A car travels around a curve banked at a 15° incline, with a radius of curvature of 10 m. If a 1000 kg car had a coefficient of static friction between the road and its tires of 0.5, what is the maximum speed that the car can round the curve at?  
A ball of mass m = 1.30 kg is attached to a massless rope and swung in a horizontal circle of radius r = 1.70 m, as shown in the figure. The rope makes a constant angle θ with the vertical as the ball swings around the circle, and the ball makes one revolution around the circle in a time Δt = 3.00 s. Calculate the centripetal acceleration of the ball.
A ball of mass m = 1.30 kg is attached to a massless rope and swung in a horizontal circle of radius r = 1.70 m, as shown in the figure. The rope makes a constant angle θ with the vertical as the ball swings around the circle, and the ball makes one revolution around the circle in a time Δt = 3.00 s. Make a diagram showing all the forces acting on the ball and the net force acting on the ball. With the help of your diagram, write down equations for the horizontal and vertical forces acting on the ball and use these to determine the angle θ.
A ball of mass m = 1.30 kg is attached to a massless rope and swung in a horizontal circle of radius r = 1.70 m, as shown in the figure. The rope makes a constant angle θ with the vertical as the ball swings around the circle, and the ball makes one revolution around the circle in a time Δt = 3.00 s. Find the tension in the rope.
A ball on a string is swung in a horizontal circle on a frictionless table. The tension in the string is To when the ball completes a rotation in 1 s. You then increase the speed so that the ball completes a rotation in 0.5 s. What is the new tension in the string? A) 1/4 To B) 1/2 To C) To D) 2 To E) 4 To
You tie a light rope to a pail of water and you swing the pail in a vertical circle of radius 0.600 m. What minimum speed must the pail have at the highest point of its circular path if no water is to spill from it? (a) 1.98 m/s (b) 2.21 m/s (c) 2.42 m/s (d) 3.92 m/s (e) 4.00 m/s (f) 5.88 m/s (g) none of the above answers
Suppose a highway curve is properly banked to eliminate friction for a speed of 45 mph. If your tires were bald and you wanted to avoid sliding on the road, you would have to drive A) somewhat above 45 mph. B) somewhat below 45 mph. C) at exactly 45 mph.
A passenger on a ferris wheel moves in a vertical circle of radius R with constant speed v. Assuming that the seat remains upright during the motion, derive an expression for the magnitude of the upward force the seat exerts on the passenger at the bottom of the circle if the passenger's mass is m. 
A passenger on a ferris wheel moves in a vertical circle of radius R with constant speed v. What are the magnitudes of the vertical forces on the passenger at the bottom of the circle if the passenger's mass is 60 kg, the radius of the circle is 16 m, and the wheel makes one revolution in 10 s?
A car travels around a flat curve. If the radius of curvature is 150 m, the car has a mass of 1500 kg, and the car wants to be able to round the curve at 25 m/s, what is the minimum coefficient of static friction between the tires and the ground required?
A 600-kg car is going around a banked curve with a radius of 110 m at a speed of 24.5 m/s. What is the appropriate banking angle so that the car stays on its path without the assistance of friction?[A] 29.1°[B] 13.5°[C] 33.8°[D] 56.2°[E] 60.9°
A passenger on a Ferris wheel moves in a vertical circle of radius R with constant speed v. What are the magnitudes of the vertical forces on the passenger at the bottom of the circle if the passenger's mass is 95 kg, the radius of the circle is 17 m and the wheel makes on revolution in 12 s?
A ball of mass m = 1.30 kg is attached to a massless rope and swung in a horizontal circle of radius r = 1.70 m, as shown in the figure. The rope makes a constant angle θ with the vertical as the ball swings around the circle, and the ball makes one revolution around the circle in a time Δt = 3.00 s.Determine the ball's speed as it moves around the circle.
A small rock is tied to a light string set it motion in a horizontal circle with constant speed. The string is 8.00 m long and makes a constant angle of 36.9° with the vertical direction. The tension in the string is 6.20 N.What is the radius r of the circular path of the rock?
A small rock with mass 2.00 kg slides on the inside of a circular track that has radius 0.600 m. When the block is at highest point of its path (point A), its speed is 3.00 m/s. What is the downward normal force that the track exerts on the block when it is at point A during its motion?A) 10.4 NB) 15.0 NC) 19.6 ND) 30.0 NE) 49.6 NF) None of the above answers
A new park attraction consists of a vertical wheel of radius  R, spinning at constant angular velocity, with cars for passengers. The passengers feel weightless at the top of the wheel. Assume that the typical mass of a passenger is m. a) What is the magnitude of the force exerted on a passenger by the car at the top of the wheel? b) What is the speed of the car? c) What is the magnitude of the force exerted on a passenger by the car at the bottom of the wheel? d) What is the magnitude of the force exerted on a passenger by the car when they are 90° from the bottom, going up? Write your results in terms of R, m, and g. Check the units/dimensions for each answer.
A small block is sliding on the inside of a circular track that has radius  R = 2.0 m. Point A is at the top of the block's path. What is the minimum speed   v that the block must have at point A so the block continues in its circular path an doesn't fall from the track at point A.A) 3.1 m/sB) 4.4 m/sC) 5.4 m/sD) 6.3 m/sF) None of the above answers