Suppose that the height of the incline is h = 14.7 m. Find the speed at the bottom for each of the following objects. 1. solid sphere 2. spherical shell 3. hoop 4. cylinder m/s

1. 14.4 m/s 2. 13.2 m/s 3. 12.0 m/s 4. 13.9 m/s

Explanation:

Assuming no friction present, the different objects roll without slipping, so there is a constant relationship between linear and angular velocity, as follows:

ω = v/r

If no friction exists, the change in total kinetic energy must be equal in magnitude to the change in the gravitational potential energy:

∆K = - ∆U

½ * m*v² + ½ * I * ω² = m*g*h

Simplifying and replacing the value of the angular velocity:

½ * v² + ½ I * (v/r) ² = g*h (1)

In order to answer the question, we just need to replace h by the value given, and I (moment of inertia) for the value for each different object, as follows:

Solid Sphere I = 2/5 * m * r²

Replacing in (1):

½ * v² + ½ (2/5 * m*r²) * (v/r) ² = g*h

Replacing by the value given for h, and solving for v:

v = √ (10/7*9.8 m/s2*14.7 m) = 14. 4 m/s

Spherical shell I=2/3*m*r²

Replacing in (1):

½ * v² + ½ (2/3 * m*r²) * (v/r) ² = g*h

Replacing by the value given for h, and solving for v:

v = √ (6/5*9.8 m/s2*14.7 m) = 13.2 m/s

Hoop I = m*r²

Replacing in (1):

½ * v² + ½ (m*r²) * (v/r) ² = g*h

Replacing by the value given for h, and solving for v:

v = √ (9.8 m/s2*14.7 m) = 12.0 m/s

Cylinder I = 1/2 * m * r²

Replacing in (1):

½ * v² + ½ (1/2 * m*r²) * (v/r) ² = g*h

Replacing by the value given for h, and solving for v:

v = 2*√ (1/3*9.8 m/s2*14.7 m) = 13.9 m/s