Concept #1: Energy of Rolling Motion (Surface vs Air)

Practice: : A 150-g baseball, 3.85 cm in radius, leaves the pitcher’s hand with 30 m/s horizontal and 20 rad/s clockwise. Calculate the ball’s linear, rotational, and total kinetic energy.

Example #1: Ratio of energies of cylinder on surface

Practice: A hollow sphere of mass M and radius R rolls without slipping on a horizontal surface with angular speed W. Calculate the ratio of its linear kinetic energy to its total kinetic energy.

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?

An object rolls without slipping, carrying a kinetic energy of (5/6)mv2. What is the moment of inertia of the object?

Consider a solid sphere of radius R and mas M rolling wihtout slipping. At any instant during the motion, which form of kinetic energy is larger, translational or rotational?
A) Rotational kinetic energy is larger.
B) Translational kinetic is larger
C) Both are equal.
D) You need to know the speed of the sphere to tell.
E) You need to know the acceleration of the sphere to tell.

A disk (I = 1⁄2 MR2) and a hoop (I = MR2) of the same mass and radius are released at the same time at the top of an inclined plane and allowed to roll without slipping to the bottom.
a) The hoop always wins.
b) The disk always wins
c) The hoop wins as often as the disk
d) Both reach the bottom at the same time.
e) Insuffient information to answer.

A cylinder with mass m, radius a, and moment of inertia with respect to the center of mass ICM = 1/2 ma2, rolls without slipping around a loop with radius R as shown in the figure.
a) What is the minimum speed of the center of mass at point C (vC) for the cylinder to move around the loop without falling off?
b) What is the total kinetic energy of the cylinder at point C in the case of minimum speed?
c) What is the minimum speed of the center of mass necessary at point B (vB) for the cylinder to move around the loop without falling off at the top (point C)?
Write your results in terms of a, R, m, and g. Check the units/dimensions for each answer.

A spool floating in space has radius r mass m and moment of inertia about its center I = β mr2. The spool is unwound by a constant force F. If initially the spool is motionless, at some later time what is the ratio of translational kinetic energy to rotational kinetic energy?
1. Ktrans / Krot = (1 + β)2
2. Ktrans / Krot = β2
3. Ktrans / Krot = 1 / β
4. Ktrans / Krot = 1 − β
5. Ktrans / Krot = (1 + β) / β
6. Ktrans / Krot = 1 / β 2
7. Ktrans / Krot = (1 − β)2
8. Ktrans / Krot = 1 + β
9. Ktrans / Krot = β
10. Ktrans / Krot = (1 − β) / β

Consider a solid sphere of radius R and mass M rolling without slipping. At any instant during the motion, which form of kinetic energy is larger, translational or rotational? A) Rotational kinetic energy is larger.B) Translational kinetic energy is larger.C) Both are equal.D) You need to know the speed of the sphere to tell.E) You need to know the acceleration of the sphere to tell.