Question

Give a common, everyday example for each of the following: (a) an object that has zero rotational kinetic energy but nonzero translational (linear) kinetic energy, (b) an object that has zero translational (linear) kinetic energy but nonzero rotational kinetic energy, (c) an object that has nonzero rotational and translational kinetic energies.

Short answer

(a) A sliding car; (b) A spinning top; (c) A rolling soccer ball.

Step-by-step solution

Understand Rotational and Translational Kinetic Energy

Translational kinetic energy is the energy due to the motion of an object moving through space, calculated by the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity. Rotational kinetic energy, on the other hand, is the energy an object has due to its rotation, given by \( KE_{rot} = \frac{1}{2}I\omega^2 \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

Identify an Object with Zero Rotational and Nonzero Translational Kinetic Energy

A car moving in a straight line without spinning or rotating is a common example. Here, the car has translational kinetic energy because it is moving linearly along a road, but it has zero rotational kinetic energy if none of its parts (like the wheels spinning) are counted, for example if it's imagined to simply slide.

Identify an Object with Zero Translational and Nonzero Rotational Kinetic Energy

A spinning top that is fixed in one place on the ground is a classic example. The top doesn't move linearly, so it has zero translational kinetic energy. However, it is spinning in place, which means it has rotational kinetic energy.

Identify an Object with Both Rotational and Translational Kinetic Energy

A rolling soccer ball combines both types of kinetic energy. As it rolls across a field, its motion includes both translation, as it moves from one location to another, and rotation, as it spins around its axis. Therefore, it possesses both translational and rotational kinetic energy.

Key concepts

Translational Kinetic Energy

Translational kinetic energy is the energy that an object possesses due to its linear motion through space. It is calculated with the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) stands for mass and \( v \) denotes velocity. Simply put, any object moving in a straight line has translational kinetic energy. A common example is a car driving down a straight road. The car's motion along the road is linear, hence it has translational kinetic energy. This is separate from any spinning or rotational movement. If the car does not rotate around itself, no rotational kinetic energy is present; this emphasizes the difference between these two types of motion-based energies.

When looking at examples of translational kinetic energy in everyday life, consider:

• A rolling ball moving in a straight path on a flat surface.
• A train traveling along a track.
• A person running in a straight line.

Under these circumstances, the objects or people are moving in a straight line, showcasing translational kinetic energy.

Rotational Kinetic Energy

Rotational kinetic energy arises from an object's rotation around its axis. It is determined using the formula \( KE_{rot} = \frac{1}{2}I\omega^2 \), where \( I \) is the moment of inertia, and \( \omega \) represents the angular velocity. A spinning top is a perfect example of an object exhibiting rotational kinetic energy. As it spins around, it retains energy without moving linearly, thus possessing no translational kinetic energy.

The rotational kinetic energy is significant in scenarios such as:

• The spinning wheel of a bicycle when the bike is stationary.
• A ceiling fan spinning overhead.
• The blades of a wind turbine rotating as they generate power.

Elements like the moment of inertia and angular speed of these objects define how much energy they possess due to rotation.

Moment of Inertia

Moment of inertia is a crucial concept when it comes to understanding rotational motion. It measures how difficult it is to change an object's rotational velocity. Analogous to mass in translational motion, the moment of inertia depends on both the mass of the object and its distribution relative to the axis around which it rotates. The formula typically used for a simplistic model is \( I = \sum m_ir_i^2 \), summarizing mass units \( m_i \) multiplied by the square of their distances \( r_i \) from the axis.

Moment of inertia can vary greatly depending on the shape and axis:

• A solid cylinder rotating about its central axis has a different moment of inertia than a hollow cylinder of the same mass.
• The distribution of weight in a figure skater's body affects her moment of inertia when she pulls in her arms to spin faster.
• A door rotating around its hinges has a different inertia compared to a merry-go-round with distributed seats.

These scenarios illustrate how the positioning of mass and the axis of rotation impact the calculation and effect of the moment of inertia in rotational dynamics.

Angular Velocity

Angular velocity refers to the rate of change of angular position of a rotating object, generally measured in radians per second (rad/s). If you imagine an object spinning around an axis, angular velocity denotes how fast it rotates. This is a key factor in calculating rotational kinetic energy, as it directly affects how energy is distributed because the faster an object spins, the more kinetic energy it contains. For instance, when a disc spins faster, its angular velocity increases, thereby increasing its rotational kinetic energy.

Scenarios often involving considerations of angular velocity include:

• A Ferris wheel turning slowly on a fairground, illustrating low angular velocity.
• The rapid spin of a washing machine drum during the spin cycle showing high angular velocity.
• Changes in a figure skater's speed when she tucks her arms in, which alters her angular velocity.

Understanding angular velocity is vital when assessing the dynamics of any rotating system, as it helps explain both speed and energy relations in rotational motions.