Ch 11: Rotational Inertia & EnergySee 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: 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 thin-walled hollow sphere with mass 5.0 kg and radius 0.20 m is rolling without slipping at the base of an incline that slopes upward at 37° above the horizontal. At the base of the incline the translational speed of the center of mass of the sphere is v = 12.0 m/s. If the sphere rolls without slipping as it travels up the incline, what is the maximum vertical height that it reaches before it starts to roll back down?
A solid disk is released from rest and rolls without slipping down an inclined plane that makes an angle of 25.0° with the horizontal. What is the speed of the disk after it has rolled 3.00 m, measured along the inclined plane? A) 4.07 m/s B) 6.29 m/s C) 3.53 m/s D) 5.71 m/s E) 2.04 m/s
A thin-walled hollow cylinder (I = MR2), with mass M = 3.00 kg and radius R = 0.200 m, is rolling without slipping at the bottom of a hill. At the bottom of the hill the center of mass of the cylinder has translational velocity 16.0 m/s. The cylinder then rolls without slipping to the top of a hill. The top of the hill is a vertical height of 6.00 m above the bottom of the hill. What is the translational velocity of the center of mass of the cylinder when the cylinder reaches the top of the hill?
A solid cylinder with moment of inertia I = 1/2 MR2, a hollow cylinder with moment of inertia I = MR2 and a solid sphere with moment of inertia I = 2/5 MR2 all have a uniform density, the same mass and the same radius. They are placed at the top of an inclined plane and allowed to roll down the inclined plane without slipping. Rank them in order of total kinetic energy at the bottom of the incline, from higher to lower. [a] sphere(1st), hollow cylinder(2nd), solid cylinder(3rd) [b] sphere(1st), solid cylinder(2nd), hollow cylinder(3rd) [c] hollow cylinder(1st), sphere(2nd), solid cylinder(3rd) [d] hollow cylinder(1st), solid cylinder(2nd), sphere(3rd) [e] they all have the same kinetic energy