Question

Motion on a Circle An object is traveling on a circle with a radius of 5 centimeters. If in 20 seconds a central angle of \(\frac{1}{3}\) radian is swept out, what is the angular speed of the object? What is its linear speed?

Short answer

Angular speed: \( \frac{1}{60} \) radians per second. Linear speed: \( \frac{1}{12} \) cm per second.

Step-by-step solution

Determine the Angular Speed

To find the angular speed, use the formula \(\text{angular speed} = \frac{\theta}{t}\), where \( \theta \) is the central angle in radians and \( t \) is the time in seconds. Here, \( \theta = \frac{1}{3} \) radians and \( t = 20 \) seconds. Substitute these values into the formula: \( \text{angular speed} = \frac{\frac{1}{3}}{20} = \frac{1}{60} \) radians per second.

Determine the Linear Speed

The linear speed can be found using the formula \( v = r \times \text{angular speed} \), where \( r \) is the radius and \( \text{angular speed} \) is what we found in the previous step. Here, \( r = 5 \) cm and \( \text{angular speed} = \frac{1}{60} \) radians per second. Substitute these values into the formula: \( v = 5 \times \frac{1}{60} = \frac{5}{60} = \frac{1}{12} \) cm per second.

Key concepts

Linear Speed

Linear speed refers to how fast an object is moving along a path. In the context of circular motion, it's the rate at which the object covers distance along the circle's circumference. Imagine a car driving around a race track. Linear speed would be the car’s speedometer reading - showing how fast it's traveling along the track. For linear speed, we use the formula: \( v = r \times \text{angular speed} \), where v is linear speed, r is the radius of the circle, and angular speed is how fast the angle is changing.
In our exercise, the radius (r) is 5 cm and we've calculated the angular speed to be \( \frac{1}{60} \) radians per second. So, to find the linear speed: \( v = 5 \times \frac{1}{60} = \frac{5}{60} = \frac{1}{12} \) cm per second.
Understanding linear speed can help visualize how fast an object is moving along its circular path, which is essential in various real-world applications like designing gears and wheels.

Circular Motion

Circular motion happens when an object moves along a circular path. It can be uniform (constant speed) or non-uniform (changing speed). In our problem, the object is moving at a constant angular speed. Imagine tying a ball to a string and swinging it around in a circle. The ball’s motion represents circular motion.
This type of motion is governed by certain principles:

• Radius (r): The distance from the center of the circle to the object’s path.
• Central angle (\theta): The angle formed at the center between two points on the circle.
• Angular speed (\text{angular speed}): How quickly the central angle changes over time.

In our exercise, the ball swings around a 5 cm radius circle, sweeping out a central angle of \( \frac{1}{3} \) radians in 20 seconds.
Circular motion is pivotal in understanding systems like satellites, rotation of celestial bodies, and machinery parts.

Radian Measurement

Radian is a unit of angular measure used in many areas of mathematics. It's the standard unit for measuring angles in circular motion because of its natural relationship with the radius and circumference of a circle. One radian is the angle created when the arc length is equal to the circle's radius.
Using radians simplifies many formulas in physics and engineering, including the calculation of angular speed. The formula for angular speed is: \( \text{angular speed} = \frac{\theta}{t} \), where \( \theta \) is the central angle in radians and t is the time in seconds.
For our exercise, \( \theta \) is \( \frac{1}{3} \) radians, and \(\frac{\theta}{t}\text{time}\). By using radian measures, we can easily find the relationship between the angle and the radius, helping us solve problems in circular motion with ease.