Problems 1-3 concern the following race. Consider three objects at the top of an

Question

Problems 1-3 concern the following race. Consider three objects at the top of an incline with = 12 and a height of 1.3 m: a sphere (I = 2 5mr2 ), a cylindrical shell (I = mr2 ), and a solid cylinder (I = 1 2mr2 ). Each object has a mass of 2 kg and a radius of 9 cm, and each is released from rest, then allowed to roll without slipping down the incline to the bottom.

1. What type(s) of energy do the three objects possess at the top of the incline? Find the total energy at the top of the incline.

2. At the bottom of the incline, each object will have both translational and rotational kinetic energy. a) What are the final velocities of each of the three objects? b) How much of the total energy is translational kinetic energy, and how much is rotational kinetic?

3. Now that you know the final velocity of each, solve for the time that each object will take each object to reach the bottom of the incline? Which one comes in first place? Second place? Third place?

Explanation / Answer

(1)
All the three object possese Potenital Energy at the top of the incline.

Total Energy = m*g*H
E = 2.0 * 9.8 * 1.3 J
E =  25.48 J

(2)
(a)
At the bottom of the incline,
Total Energy = translational l kinetic energy + rotational kinetic energy.
25.48 = 1/2*mv^2 + 1/2 * I * ^2

For sphere,
I = 2/5 * *r^2
= v/r ( roll without slipping )
25.48 = 1/2*mv^2 + 1/2 * 2/5 * m*r^2 * v^2/r^2
25.48 = 1/2*mv^2 + 1/5 *m*v^2
25.48 = 0.7 * 2.0 * v^2
v =  4.26 m/s

For cylindrical shell,
I = m*r^2
25.48 = 1/2*mv^2 + 1/2 *m*r^2 * v^2/r^2
25.48 = m*v^2
25.48 = 2.0 * v^2
v = 3.57 m/s

For Solid cylinder,
I = 1/2 m*r^2
25.48 = 1/2*mv^2 + 1/2 *1/2 * m*r^2 * v^2/r^2
25.48 = 1/2 * m*v^2 + 1/4 * m*v^2
25.48 = 0.75 * 2.0 * v^2
v = 4.12 m/s

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