At a typical bowling alley the distance from the line where...

At a typical bowling alley the distance from the line where the ball is released (foul line) to the first pin is $18.29 \mathrm{~m}$. Estimate the time it takes the ball to reach the pins after you release it if it rolls without slipping and has a constant translational speed. Assume that the mass of the ball is $5.44 \mathrm{~kg}$ and its diameter $21.6 \mathrm{~cm}$. (a) Use your estimate to calculate the rotation rate of the ball, in rev/s. (b) What is its total kinetic energy in joules and what fraction of the total is its rotational kinetic energy? Ignore the finger holes and treat the bowling ball as a uniform sphere.

Key Concepts

Rolling Without Slipping

This concept links translational motion with rotational motion by stating that the linear velocity at the center of mass equals the angular speed times the radius of the object. It guarantees that the point of contact with the surface is momentarily at rest, enabling the use of the relation v = ?r to connect the two types of motion.

Kinematics of Constant Velocity Motion

This concept involves the basic kinematic relationship between distance, speed, and time. When an object moves with a constant speed, the time taken to cover a given distance can simply be calculated by dividing the distance by the speed.

Moment of Inertia

The moment of inertia quantifies an object's resistance to rotational acceleration about an axis. For uniform shapes, standard formulas (like I = (2/5)mR² for a sphere) are used to calculate this value, which is essential for determining the rotational kinetic energy of rolling objects.

Rotational Kinetic Energy

This is the energy stored as a result of the rotation of an object, expressed by the formula K_rot = (1/2)I?². It represents the part of the total kinetic energy due to the spinning motion of the object and is directly influenced by both the moment of inertia and the angular velocity.

Translational Kinetic Energy

Translational kinetic energy is the energy associated with the linear movement of an object and is calculated using the formula K_trans = (1/2)mv². In rolling motion, this energy, together with the rotational kinetic energy, constitutes the total kinetic energy of the object.

Rotation Rate Conversion

This concept involves converting the angular speed of an object from radians per second to revolutions per second, typically by dividing the angular speed by 2?. This conversion makes it easier to interpret and compare rotational speeds in more familiar terms.

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