A uniform sphere with mass 28.0 kg and radius 0.380 m is rotating at constant angular velocity about a stationary axis that lies along a diameter of the sphere. If the kinetic energy of the sphere is 236 J, what is the tangential velocity of a point on the rim of the sphere?

Verified step by step guidance

1. Identify the formula for the kinetic energy of a rotating object, which is given by $K = \frac{1}{2} I \omega^2$, where $I$ is the moment of inertia and $\omega$ is the angular velocity.
2. Calculate the moment of inertia for a uniform sphere rotating about a diameter using the formula $I = \frac{2}{5} m r^2$, where $m$ is the mass of the sphere and $r$ is its radius.
3. Rearrange the kinetic energy formula to solve for the angular velocity $\omega$. Substitute the expression for $I$ from step 2 into this rearranged formula to express $\omega$ in terms of known quantities.
4. Use the relationship between tangential velocity $v$ and angular velocity $\omega$ at the rim of the sphere, which is $v = r \omega$, where $r$ is the radius of the sphere.
5. Substitute the value of $\omega$ from step 3 into the formula from step 4 to find the tangential velocity $v$ of a point on the rim of the sphere.

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, expressed in radians per second. For a rotating object, it relates to the linear velocity of points on the object’s surface through the equation v = ωr, where v is the tangential velocity, ω is the angular velocity, and r is the radius. Understanding angular velocity is crucial for determining the motion of points on a rotating sphere.

Kinetic Energy of Rotation

The kinetic energy of a rotating object is given by the formula KE = 0.5 I ω², where I is the moment of inertia and ω is the angular velocity. For a uniform sphere, the moment of inertia is I = (2/5)mr², where m is the mass and r is the radius. This concept is essential for relating the sphere's kinetic energy to its rotational motion and ultimately finding the tangential velocity.

Tangential Velocity

Tangential velocity is the linear speed of a point on the circumference of a rotating object, calculated as v = ωr. It represents how fast a point moves along its circular path and is directly proportional to both the angular velocity and the radius of the object. In this problem, finding the tangential velocity of a point on the sphere's rim requires understanding its relationship with angular velocity and the sphere's radius.