Question

An 8.7-cm-diameter, 400 g solid sphere is released from rest at the top of a 1.8-m-long, 16° incline. It rolls, without slipping, to the bottom. What is the sphere's final velocity at the bottom of the incline?

A) 2.3 m/s
B) 4.6 m/s
C) 6.9 m/s
D) 9.2 m/s

Answer 1

To find the sphere's final velocity at the bottom of an incline, conservation of energy principles are applied considering both translational and rotational kinetic energy. By setting up an equation equating the initial potential energy to the sum of kinetic energies and solving for the final velocity, we can obtain the sphere's final kinetic state.

Step-by-step explanation:

The question relates to the physics concept of conservation of energy for a rolling object down an incline. The student's task is to find the sphere's final velocity at the bottom of a 1.8-meter long, 16° incline. To solve this problem, energy conservation principles are employed, considering both translational and rotational kinetic energies as well as gravitational potential energy.

Using the energy conservation law:

• Initial potential energy = Final translational kinetic energy + Final rotational kinetic energy
• mgh = ½ mv² + ½ Iω²

Where:

• m is the mass of the system
• g is the acceleration due to gravity (9.8 m/s²)
• h is the height of the incline
• v is the final velocity
• I is the moment of inertia of a sphere (½ mR² for a solid sphere)
• ω is the angular velocity

Since the sphere rolls without slipping, v = Rω. This relationship allows us to eliminate the angular velocity variable. Inserting the given data into our equations and solving gives us the sphere's final velocity at the bottom of the incline.