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Objects rolling down a slope

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Wize University Physics Textbook (Master) > Angular / Rotational Motion

Rotational Kinetic Energy

3 Activities

This is a classic problem - professors love to ask questions about it!

Consider any round thing you can think of rolling down a hill. Here, we will use four objects: a ring (I=MR2I=MR^2), a disk (I=12MR2I=\frac12MR^2), a solid sphere (I=25MR2I=\frac25MR^2), and a hollow sphere (I=23MR2I=\frac23MR^2).

You release them from rest at the same time from height h and they all roll without slipping (that is, there is enough static friction to allow for this). The angle of the incline is θ\theta.

The static friction for this case is f=mgsin⁡θ1+mR2If=\frac{mg\sin\theta}{1+\frac{mR^2}{I}}.






a) Which one gets to the bottom of the slope first? In what order do the four objects finish?
More Rotational Kinetic Energy Questions:
You have two spheres of same mass M and radius R, one solid and one hollow, and an incline on which they can roll without slipping. Which one has a higher angular velocity while rolling? Show your work.
Isolid sphere=25MR2;       Ihollow sphere=23MR2I_{solid~sphere}=\frac{2}{5}MR^2; ~~~~~~~I_{hollow~sphere}=\frac{2}{3}MR^2
(PS: If you are wondering how these two spheres can have both the same mass and radius: it is because the one with less material would have a higher density. Perhaps one is made of dense steel and the other is made of a water-equivalent material.)
A stone with a mass of 9.5kg is hung vertically from the end of a rope that is wound around a pulley on a boat. The moment of inertia of the pulley is 0.47kg m2 and the radius is 0.25m. The stone is released from rest and falls toward the water, causing the pulley to rotate. After the stone has fallen 2.5 m, what is the angular velocity of the pulley? (Enter your answer in radians per second.)
Kinetic Energy: Rotating Body
A thin rod with mass of M and length of L is hinged to the wall at one end. The rod is held horizontally and then released. What is the speed of the tip of the rod as it hits the wall? Assume the hinge is frictionless. The axis of rotation for a thin rod about one of its ends is 13ML2\frac{1}{3}ML^2.
Hint: You should take the CM into consideration for problems of this kind.
A marble starts from rest and rolls without slipping on the loop-the-loop track in the figure. Find the minimum starting height h from which the ball will remain on the track throughout the loop. Assume the marble's radius is small compared with R. The moment of inertia for a solid sphere is 25Mr2\frac25Mr^2.
Find your answer by using R = 70 cm (radius of the loop) and r = 1.0 cm (radius of the marble).
A uniform solid disk and a uniform hoop are placed side by side at the top of an incline of height h.
If they are released from rest and roll without slipping, which object reaches the bottom first?
Verify your answer by calculating their speeds when they reach the bottom in terms of h.
A disk of mass 5Kg and radius 0.20 m rolls smoothly on a horizontal floor. If the kinetic energy of rolling of the disk is 70J at a certain instant, find the speed of the center of mass of disk.
Icom(disk)=12MR2I_{com} (disk) = \dfrac{1}{2} MR^2
Give answer is m/s.
A spherical ball with a mass of 1kg and radius of 10cm is rolling without slipping down the track shown in the picture starting at rest.
a) If the radius of the loop in the track is R=2m, find the linear speed of the ball on top of the loop:
A kid with mass m and velocity v runs toward a playground merry-go-round, which is initially at rest, and jumps on at its edge. The kid and the merry-go-round (mass M, radius R, and I = cMR2) then spin together with a constant angular velocity ωf.
If kid's initial velocity is tangent to the circular merry-go-round, what is ωf?
What is the energy loss?
You have two spheres of same mass M and radius R, one solid and one hollow, and an incline on which they can roll without slipping. How can you identify both of them on the incline?
An object of mass m and speed v0 strikes a rigid uniform rod of length l and mass M that is hanging by a frictionless pivot from the ceiling. Immediately after striking the rod, the object continues forward with half its initial speed.
a) Find the angular speed immediately after the collision
b) For what value of v0 will the rod just touch the ceiling?