Question

A golf ball with mass \(45.90 \mathrm{~g}\) and diameter \(42.60 \mathrm{~mm}\) is struck such that it moves with a speed of \(51.85 \mathrm{~m} / \mathrm{s}\) and rotates with a frequency of \(2857 \mathrm{rpm} .\) What is the kinetic energy of the golf ball?

Short answer

Answer: The total kinetic energy of the golf ball is approximately 63.0161 J.

Step-by-step solution

1. Convert mass and diameter to standard units

To solve this problem, the mass should be converted from grams to kilograms, and the diameter should be converted from millimeters to meters. Mass: \(45.90 \mathrm{~g} = 0.0459 \mathrm{~kg}\) Diameter: \(42.60 \mathrm{~mm} = 0.0426 \mathrm{~m}\)

2. Calculate the radius of the golf ball

Using the diameter, we can calculate the radius of the golf ball. Radius: \(r = \frac{d}{2} = \frac{0.0426 \mathrm{~m}}{2} = 0.0213 \mathrm{~m}\)

3. Calculate the moment of inertia

Assume the golf ball is a solid sphere. The moment of inertia for a solid sphere is given by the formula: \(I = \frac{2}{5}mr^2\) Plug in the values for mass and radius: \(I = \frac{2}{5}(0.0459 \mathrm{~kg})(0.0213 \mathrm{~m})^2 = 3.22678 \times 10^{-5} \mathrm{kg\cdot m^2}\)

4. Remove the RPM unit and convert it to radians per second

To work with the rotational frequency in the kinetic energy formula, we need to convert it from revolutions per minute (rpm) to radians per second. There are 2π radians in one revolution and 60 seconds in one minute: \(\omega = 2857 \mathrm{rpm} \times \frac{2\pi \mathrm{rad}}{1 \mathrm{rev}} \times \frac{1 \mathrm{min}}{60 \mathrm{s}} = 299.054 \mathrm{rad/s}\)

5. Calculate the translational kinetic energy

The translational kinetic energy (K.E.) can be calculated using the following formula: \(K.E. = \frac{1}{2}mv^2\) Plug in the values for mass and speed: \(K.E. = \frac{1}{2}(0.0459 \mathrm{~kg})(51.85 \mathrm{m/s})^2 = 61.5719 \mathrm{J}\)

6. Calculate the rotational kinetic energy

The rotational kinetic energy can be calculated using the following formula: \(K.E. = \frac{1}{2}I\omega^2\) Plug in the values for the moment of inertia and the angular velocity: \(K.E. = \frac{1}{2}(3.22678 \times 10^{-5} \mathrm{kg\cdot m^2})(299.054\mathrm{rad/s})^2 = 1.44519 \mathrm{J}\)

7. Calculate the total kinetic energy

Add both the translational and rotational kinetic energies to get the total kinetic energy: \(K.E._{total} = K.E._{translational} + K.E._{rotational} = 61.5719 \mathrm{J} + 1.44519 \mathrm{J} \approx 63.0161 \mathrm{J}\) Hence, the total kinetic energy of the golf ball is approximately \(63.0161 \mathrm{J}\).

Key concepts

Moment of Inertia

The concept of moment of inertia is central to understanding rotational motion, much like how mass is central to translational motion. It quantifies an object's resistance to changes in its rotation, which depends not only on the mass of the object but also on how that mass is distributed relative to the axis of rotation.

For a solid sphere like a golf ball, the moment of inertia is given by the formula:
\[\begin{equation}I = \frac{2}{5}mr^2\tag{1}\right).\right)Where \(m\) is the mass and \(r\) is the radius of the sphere. It is crucial to use the correct formula for the moment of inertia because it varies with the object's shape. For example, a solid cylinder would have a different equation from a hollow sphere. The correct calculation ensures an accurate assessment of the rotational kinetic energy.

Rotational Kinetic Energy

Rotational kinetic energy is the energy possessed by a rotating object and is part of an object's total kinetic energy. It is given by the formula:
\[\begin{equation}K.E. = \frac{1}{2}I\tilde{\theta}^2\tag{2}\right).\right)The symbol \(I\) represents the moment of inertia, and \(\tilde{\theta}\) is the angular velocity, or how fast the object rotates. This angular speed, often given in revolutions per minute (rpm), should be converted to radians per second to work within the standard units of the formula. Understanding rotational kinetic energy is vital, as it allows us to calculate the energy an object has due to its rotation. This calculation is particularly important for objects like the golf ball in our exercise that has both rotational and translational motion.

Translational Kinetic Energy

Translational kinetic energy refers to the energy that an object has due to its motion from one location to another. It is the more commonly understood form of kinetic energy and is mathematically expressed as:
\[\begin{equation}K.E. = \frac{1}{2}mv^2\tag{3}\right).\right)Here, \(m\) is the mass and \(v\) is the velocity. An object may have translational kinetic energy, rotational kinetic energy, or both, depending on its movement. For example, a rolling ball has both translational kinetic energy (because it's moving across the ground) and rotational kinetic energy (because it's spinning). In the case of the golf ball from the exercise, it's crucial to calculate both types of kinetic energy to determine the total energy it possesses after being struck.