Question

current attempt in progress
an object is rolling, so its motion involves both rotation and translation. which one of the following statements must be true concerning this situation?
the translational kinetic energy may be equal to zero joules.
the gravitational potential energy must be changing as the object rolls.
the total mechanical energy is equal to the sum of the translational kinetic energy and the gravitational potential energy of the object.
the rotational kinetic energy must be constant as the object rolls.
the total mechanical energy is equal to the sum of the translational and rotational kinetic energies plus the gravitational potential energy of the object.

Explanation:

• Translational kinetic energy \(K_{trans}=\frac{1}{2}mv^{2}\). If the object is rolling, \(v

eq0\) (otherwise it would not be in translational motion related to rolling), so translational kinetic energy cannot be zero.

• Gravitational potential energy \(U = mgh\). If the object is rolling on a horizontal surface (\(h\) is constant), \(U\) does not change.
• Total mechanical energy \(E=K_{trans}+K_{rot}+U\). The first option that mentions only \(K_{trans}\) and \(U\) is incorrect as rotational kinetic energy \(K_{rot}=\frac{1}{2}I\omega^{2}\) (where \(I\) is the moment of inertia and \(\omega\) is the angular velocity) is also part of the mechanical energy for a rolling object.
• Rotational kinetic energy depends on \(\omega\). If the object is accelerating or decelerating while rolling (e.g., rolling down an incline where \(\omega\) changes), \(K_{rot}\) changes.
• For a rolling object, by the definition of total mechanical energy (sum of translational kinetic energy \(K_{trans}=\frac{1}{2}mv^{2}\), rotational kinetic energy \(K_{rot}=\frac{1}{2}I\omega^{2}\) and gravitational potential energy \(U = mgh\)), \(E=K_{trans}+K_{rot}+U\)

Answer:

The total mechanical energy is equal to the sum of the translational and rotational kinetic energies plus the gravitational potential energy of the object.