Concern the following race. Consider three objects at the top of an incline with

Question

Concern the following race. Consider three objects at the top of an incline with theta = 12 degree and a height of 1.3 m: a sphere (I = 2/3 mr^3), a cylindrical shell (I = mr^2), and a solid cylinder (I = 1/2 mr^2). 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. What type(s) of energy do the three objects posses at the top of the incline? Find the total energy at the top of the incline. At the bottom of the incline, each object will have both translational and rotational kinetic energy. What are the final velocities of each of the objects? How much of the total energy is translational kinetic energy, and how much is rotational kinetic? 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.

The total energy at the top of the incline is

P.E = m* g * h = 2 * 9.8 * 1.3

P.E = 25.48 J ( for each object)

2.

the moment of inertia for each object can be written as

I = c * m * r^2

a)

for sphere c = 2/5

cylidrical shell c = 1

solid cylinder c = 1/2

the speed at the bottom of incline is

v = sqrt ((2* g * h) / ( 1 + c))

for sphere v = sqrt ((2* 9.8 * 1.3) / ( 1 + 2/5))

v = 4.26 m/s

for cylindrical shell c= 1

v = sqrt ((2* 9.8 * 1.3) / ( 1 + 1))

v = 3.57 m/s

for solid cylinder c = 1/2

v = sqrt ((2* 9.8 * 1.3) / ( 1 + 1/2))

v = 4.12 m/s

b)

for sphere

translation K.E = 1/2 * m * v^2

translation K.E = 1/2 * 2 * 4.26^2

translation K.E = 18.14 J

rotatinal K.E = P.E - translation K.E

rotational K.E = 25.48 - 18.14 = 7.33 J

for cylindrical shell

translation K.E = 1/2 * m * v^2

translation K.E = 1/2 * 2 * 3.57^2

translation K.E = 12.74 J

rotatinal K.E = P.E - translation K.E

rotational K.E = 25.48 - 12.74 = 12.73 J

for solid cylinder

translation K.E = 1/2 * m * v^2

translation K.E = 1/2 * 2 * 4.12^2

translation K.E = 16.97 J

rotatinal K.E = P.E - translation K.E

rotational K.E = 25.48 - 16.97 = 8.5 J

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