Ch 11: Rotational Inertia & 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 18: Electric Forces and Electric Fields
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: Conservation of Energy in Rolling Motion

Practice: A solid sphere of mass M = 10 kg and radius R = 2 is rolling without slipping with speed V = 5 m/s on a flat surface when it reaches the bottom of an inclined plane that makes an angle of Θ = 37° with the horizontal. The plane has just enough friction to cause the sphere to roll without slipping while going up. What maximum height will the sphere attain?

Example #1: Sphere on rough and smooth hills

Practice: You may remember that the lowest speed that an object may have at the top of a loop-the-loop of radius R, so that it completes the loop without falling, is √gR . Calculate the lowest speed that a solid sphere must have at the bottom of a loop-the-loop, so that it reaches the top with enough speed to complete the loop. Assume the sphere rolls without slipping.

Additional Problems
A spring is compressed against a 1 kg disk with radius 5 cm such that as the spring expands, its force is applied to the rim of the disk, allowing it to roll without slipping. If the spring has a force constant of 10 N/m, and it's compressed to 10 cm, how fast does the disk roll once the spring has expanded fully?
A wheel with a 10 cm radius rolls down an incline with an angular acceleration of 35 rad/s2. If the incline angle is 30°, and the wheel starts at a height of 0.5 m, what will its linear velocity be at the bottom of the hill?
A thin-walled hollow sphere with radius R = 0.050 m is released from rest at the top of an incline, a vertical distance of 2.0 m above the bottom of the incline. The moment of inertia of the sphere about the rotation axis through its center is (2/3) mR2. There is sufficient friction for the sphere to roll without slipping. What is the angular velocity of rotation of the sphere when it gets to the bottom of the incline?
Calculate the translational speed of a cylinder when it reaches the foot of an incline 7.30 m high. Assume it starts from rest and rolls without slipping.
A sphere of radius exttip{r_0}{r_0} = 23.0 cm and mass m = 1.20 kg starts from rest and rolls without slipping down a 36.0 degree incline that is 13.0 m long.a) Calculate its translational speed when it reaches the bottom.b) Calculate its rotational speed when it reaches the bottom.c) What is the ratio of translational to rotational kinetic energy at the bottom?d) Does your answer in part A depend on mass or radius of the ball? Part B? Part C?
A string is wrapped several times around the rim of a small hoop with radius 8.00 cm and mass 0.180 kg. The free end of the string is held in place and the hoop is released from rest (the figure). After the hoop has descended 75.0 cm , calculatea) the angular speed of the rotating hoop andb) the speed of its center
A size-5 soccer ball of diameter 22.6 cm and mass 426 g rolls up a hill without slipping, reaching a maximum height of 6.50 m above the base of the hill. We can model this ball as a thin-walled hollow sphere.a) At what rate was it rotating at the base of the hill?b) How much rotational kinetic energy did it then have?
A basketball (which can be closely modeled as a hollow spherical shell) rolls down a mountainside into a valley and then up the opposite side, starting from rest at a height H0 above the bottom. In the figure, the rough part of the terrain prevents slipping while the smooth part has no friction.a) How high, in terms of H0, will it go up the other side?b) Why doesn't the ball return to height H0? Has it lost any of its original potential energy?
A hollow sphere is rolling along a horizontal floor at 6.00 m/s when it comes to a 33.0 degree incline. How far up the incline does it roll before reversing direction?