Download Practice problems (Rotational Motion)

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Rotational Motion
1.) A runner of mass m = 36 kg and running at 3.6 m/s runs as shown and jumps on the rim of a playground
merry-go-round which has a moment of inertia of 360 kgm2 and a radius of 2 meters. Assuming the merry-go-round
is initially at rest, what is its final angular velocity to three decimal places?
2.) A disk of radius 2.06 cm and mass 1 kg is pulled by a string wrapped around its circumference with a constant
force of 0.51 newtons. What is its angular acceleration to three decimal places? What is the angular velocity of the
disk to three decimal places after it has been turned through 0.77 of a revolution?
3.)A CD disk is reputed to accelerate from 0 to 378 rpm in 1.4 seconds. What is its acceleration to the nearest
radian/sec2 ? How many radians does it turn through (to the nearest radian)?
4.)A 96 kg disk, with a radius of 1.03 meters, is rotating at 308 rpm. What is its angular momentum, to the
nearest tenth of a Joule*sec?
5.) A 1-kg mass hangs by a string from a disk with radius 10.3 cm which has a rotational inertia of 5 10-5 kgm2.
After it falls a distance of 0.7 meters, how fast is it going to the nearest hundredth of a m/s?. What is the angular
velocity of the disk to the nearest tenth of a radian per second?
6.) Three masses 5 kg, 8 kg, 5 kg, are held together at the corners of a triangle by light rods. The (x,y)
coordinates of which are (-3,-2), (0,3),(3,-2) 1. Calculate the rotational inertia about the (a) x-axis, (b) y-axis, and
(c) z-axis. 2. What would be the kinetic energy of the system if it rotates about the z-axis at 10 rpm? 3. What
would be its angular momentum?
7.)A man of mass Mm = 80 kg sits on the edge of a uniform rectangular crate of width 1 m and height 2 m. The
crate has a mass of 120 kg. A person tries to tilt the crate by pulling with a rope attached to a corner of the crate at
an angle of 30◦ , as shown below. What force does the person have to apply to the rope? Assume the crate will tilt
without slipping.
8.) A rope is wrapped around a pulley that is free to rotate about an axis through the center. In case (a) you pull
on the rope with a constant force F = 10N . In case (b) you hang a block which has a weight W = mg = 10N on the
end of the rope. In each case, calculate the angular acceleration of the pulley. The radius of the pulley is r = 0.2 m
and the moment of Inertia is 0.1 kg/m2
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9.) A solid ball (I = 52 M R2 ) rolls down a hill without slipping. Assuming it starts from rest at point A, what is
the speed (v) and angular velocity (w) of the ball at point B?
10.) A solid cylinder of radius R = 2 cm and mass M = 100 g rolls without slipping down an inclined plane. If it
starts from rest at an elevation h = 25 cm, what is its translational speed when it reaches the base of the incline?
Repeat the calculation for a solid cylinder with R = 4 cm and M = 100 g. Repeat again for R = 2 cm and M = 200
g. Hint: Use conservation of energy and the fact that w = v/R.
11.) Assume that the rotational inertia of a round object can be written as I = kMR2, where k is a shape factor.
Derive a general expression for the speed of the object in terms of k and the initial height, h.
12.) Pick two objects that have the same shape but different radii and/or different masses. Place them side-by-side
on the incline and release them from rest at the same time.
a) How does the speed depend on the radius? On the mass?.
b) Now compare the rolling speeds of objects of different shapes (cylindrical shell, solid cylinder, spherical shell, solid
sphere).
c) What are your results? Are your experimental observations consistent with the general equation for the speed that
you derived above?
d) Explain.How would you design a rolling object that was faster than any of the ones that you have used here?