In this worksheet, we will practice calculating the motion of rolling objects by referring to the conservation of rotational kinetic energy and of angular momentum.

**Q3: **

A marble that is initially at rest rolls down a slope inclined at below the horizontal for a time interval of 3.0 s.

What is the magnitude of the acceleration of the marble downward along the incline? Answer to a precision of two significant figures.

What distance downward along the incline does the marble travel in 3.0 s?

**Q4: **

A rigid body with a cylindrical cross-section is released from the top of an incline that is angled at below the horizontal. The body rolls a 10.0 m distance downward along the incline and reaches the base of it after 2.6 s. Find the moment of inertia of the body in terms of its mass and radius .

**Q5: **

A hollow cylinder is given an initial speed of 5.0 m/s upward along an incline. The cylinder rolls up the incline and instantaneously stops when its vertical upward displacement from its initial position is 1.0 m. If a hollow sphere of the same mass and radius as the cylinder is given the same initial speed upward along the incline, what vertical upward displacement does the sphere have when it instantaneously stops?

**Q6: **

A solid cylinder rolls upward along an incline angled at above the horizontal. If the cylinder has a speed of 10 m/s at the base of the incline, what distance along the incline does the cylinder roll before coming to rest?

**Q7: **

A solid cylinder of radius 10.0 cm rolls with slipping down an incline that is angled below the horizontal. The coefficient of kinetic friction between the cylinder and the surface of the incline is 0.40.

What is the magnitude of the angular acceleration of the solid cylinder?

What is the magnitude of the linear acceleration of the solid cylinder?

**Q8: **

A hollow marble is rolling across the floor at a speed of 7.0 m/s when it starts up a plane inclined at to the horizontal.

How far along the plane does the marble travel before coming to a rest?

How much time elapses while the marble moves up the plane?

**Q9: **

A bowling ball rolls without slipping upward along a ramp. The initial speed of the bowling ball’s center of mass is 2.5 m/s. To reach the top of the ramp, the bowling ball is displaced vertically upward by 0.32 m. What is the speed of the ball’s center of mass when the ball reaches the top of the ramp?

**Q10: **

A marble is rolling across a horizontal floor at a speed of 6.4 m/s and starts rolling upward along a plane that is inclined at above the horizontal.

How far is the marble displaced along the plane before coming instantaneously to rest?

How much time elapses between the marble starting to move along the plane and coming instantaneously to rest?

**Q11: **

A bowling ball of radius 9.30 cm is rolled down a bowling lane with a speed of 8.80 m/s. The direction of the roll is to the left, as viewed by an observer, so the bowling ball starts to rotate counterclockwise when in contact with the floor. The coefficient of kinetic friction on the lane is 0.250.

What is the time required for the ball to come to the point where it is not slipping?

What is the distance to the point where the ball is rolling without slipping?

**Q12: **

A yo-yo can be thought of as a solid cylinder of mass and radius that has a light string wrapped around its circumference, as shown in the diagram. One end of the string is held fixed in space and the cylinder falls as the string unwinds without slipping. What is the magnitude of the linear acceleration of the cylinder?

**Q13: **

A solid cylinder is at rest. The cylinder then rolls downward along a slope that is inclined below the horizontal.

What is the magnitude of the acceleration of the cylinder down the slope?

How much has the cylinder been displaced along the direction of the slope after it has rolled for 5.5 s?

**Q14: **

A solid cylindrical wheel has a mass of 35 kg and radius . The wheel is pulled along a surface by a force , applied to the center of the wheel and acting at an angle above the horizontal, as shown in the diagram. If the wheel is to roll without slipping, what is the maximum value of ? The coefficient of static friction between the wheel and the surface is 0.33.

**Q15: **

A solid cylinder that is initially at rest rolls downward along a slope that is angled at below the horizontal. The cylinder does not slip while rolling. The cylinder’s mass is 1.5 kg and its radius is 0.15 m.

What is the magnitude of the linear acceleration of the cylinder parallel to the slope’s surface?

What is the magnitude of the cylinder’s angular acceleration as it rolls?

Assuming that the frictional contact between the cylinder and the slope is just sufficient to prevent the cylinder slipping when it rolls, what is the coefficient of static friction between the cylinder and the slope?

**Q16: **

A slope inclined at below the horizontal has a coefficient of static friction of 0.60 with two cylinders, one solid and one hollow. When the cylinders are placed on the slope, which of the following is likely to happen?

- ABoth cylinders slip at the time that they are rolling.
- BBoth cylinders roll without slipping.
- CThe solid cylinder slips and the hollow cylinder does not.
- DThe hollow cylinder slips and the solid cylinder does not.
- EBoth cylinders slip but one starts slipping before the other.

**Q17: **

A solid cylinder at rest is placed on a slope inclined at below the horizontal and starts to simultaneously roll and slip downward along the slope. The cylinder has a mass of 0.75 kg and a radius of 0.25 m. The coefficient of kinetic friction between the cylinder and the slope is 0.55.

What is the magnitude of the linear acceleration of the cylinder parallel to the slope’s surface?

What is the magnitude of the cylinder’s angular acceleration due to its rolling motion?