Question

True or False The angular speed \(\omega\) of an object traveling on a circle of radius \(r\) is the angle \(\theta\) (measured in radians) swept out, divided by the elapsed time \(t\).

Short answer

True

Step-by-step solution

Understand the given statement

The statement says that the angular speed \(\omega\) of an object moving in a circle with radius \(r\) is the angle \(\theta\) divided by the elapsed time \(t\).

Recall the definition of angular speed

Angular speed \(\omega\) is defined as the rate at which an angle is swept out by an object moving along a circular path. Mathematically, this is expressed as \(\omega = \frac{\theta}{t}\).

Compare the given statement with the definition

The given statement precisely mirrors the mathematical definition of angular speed: \(\omega = \frac{\theta}{t}\).

Conclusion

Since the statement is true based on the definition of angular speed, the final answer is True.

Key concepts

Circular Motion

Circular motion occurs when an object moves along the circumference of a circle. This motion can be uniform, where the speed remains constant, or non-uniform, where the speed varies. In uniform circular motion:

• The direction of the velocity changes continuously.
• The magnitude of the velocity remains constant.
• Acceleration is directed towards the center of the circle, called centripetal acceleration.

Key elements to understand circular motion include the radius of the circle, the speed of the object, and the period, which is the time it takes to complete one full revolution. The relationship between these elements helps us understand how objects move in circular paths. For example, in a Ferris wheel, the ride moves in circular motion providing both centripetal force and acceleration to the riders.

Angular Velocity

Angular velocity is a measure of how fast an object rotates or revolves relative to another point, typically the center of the circle. It is different from linear velocity, which measures the rate of change of displacement.
Angular velocity is defined by the angle \(\theta\) (in radians) swept out by the radius of the circle per unit time \(t\). Hence, the formula for angular velocity is:
\(\omega = \frac{\theta}{t}\)
Key points to note about angular velocity include:

• It is a vector quantity, having both magnitude and direction.
• The SI unit is radians per second (rad/s).
• It provides insight into how fast the angle changes as the object travels along the circular path.

For instance, if a car is taking a round turn, its angular velocity helps us understand how quickly it is turning around the curve.

Radians

Radians are the standard unit of angular measure used in many areas of mathematics and science. They provide a direct link between the angle and the arc length. One radian is defined as the angle formed when the radius of a circle is wrapped along the circumference.
Important aspects of radians include:

• There are \(2\pi = 6.2832\) radians in a full circle.
• The relationship between degrees and radians is given by \(1\pi\) radian equals \(180^{\circ}\).
• Angular measurements in radians are often more natural when dealing with periodic phenomena.

In practical terms, understanding radians allows us to calculate angular distances easily, tying together linear and angular measurements effectively. For example, the angle for half a circle (180 degrees) is \(\pi\) radians, which can be useful when solving problems involving circular movements.