Answer:
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
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
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
