An 8.0-cm-diameter, 400 g solid sphere is released from rest at the top of a 2.1-m-long, 25 incline. It rolls, without slipping, to the bottom..(a)What is the sphere’s angular velocity at the bottom of the incline?

Verified step by step guidance

1. Identify the given values: radius of the sphere (R) = 8.0 cm = 0.08 m, mass of the sphere (m) = 400 g = 0.4 kg, length of the incline (L) = 2.1 m, and angle of the incline (\( \theta \)) = 25 degrees.
2. Calculate the moment of inertia (I) for a solid sphere, which is given by the formula \( I = \frac{2}{5} mR^2 \).
3. Use the conservation of energy principle. The initial potential energy at the top of the incline will be equal to the sum of translational kinetic energy and rotational kinetic energy at the bottom. The potential energy (PE) at the top is given by \( PE = mgh \), where \( h = L \sin(\theta) \) is the vertical height of the incline.
4. Set up the energy conservation equation: \( mgh = \frac{1}{2} mv^2 + \frac{1}{2} I \omega^2 \), where \( v \) is the linear velocity and \( \omega \) is the angular velocity at the bottom. Remember that for rolling without slipping, \( v = R\omega \).
5. Solve the energy conservation equation for \( \omega \) (angular velocity) by substituting \( v = R\omega \) and the expression for I, and then isolate \( \omega \) to find its value at the bottom of the incline.

Key Concepts

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. In this scenario, the potential energy of the sphere at the top of the incline is converted into kinetic energy as it rolls down. This includes both translational kinetic energy and rotational kinetic energy, which must be accounted for to find the angular velocity.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation. For a solid sphere, the moment of inertia is given by the formula I = (2/5)mr², where m is the mass and r is the radius. This concept is crucial for determining how the sphere's mass and shape affect its rotational motion as it rolls down the incline.

Rolling Motion

Rolling motion occurs when an object rotates about an axis while simultaneously translating along a surface. For the sphere in this problem, rolling without slipping means that the point of contact with the incline does not slide. This relationship between linear velocity and angular velocity is described by the equation v = rω, where v is the linear velocity, r is the radius, and ω is the angular velocity.