Calculating Velocity in Circular Motion

The question asks us to find the velocity of a body moving in a circular path given the radius and the time taken for one complete revolution. This scenario describes uniform circular motion, where the speed is constant, but the velocity changes direction continuously. The magnitude of the velocity is the speed of the body.

Understanding Velocity and Circular Path

Velocity is defined as the rate of change of displacement. In simpler terms, for motion at a constant speed, it's the distance traveled divided by the time taken.

In this problem:

• The path is a circle with radius 'r'.
• The body completes one full revolution, meaning it covers the entire circumference of the circle.
• The time taken for one full revolution is given as 't' seconds. This time is also known as the time period (T). So, T = t.

Distance Traveled in One Revolution

The distance covered by the body in one full circle is equal to the circumference of the circle.

The formula for the circumference of a circle with radius 'r' is:

Circumference \(C = 2\pi r\)

Calculating Speed (Magnitude of Velocity)

Since the body travels the distance \(2\pi r\) in time 't', the speed (magnitude of velocity) can be calculated using the formula:

Speed \(v = \frac{\text{Distance}}{\text{Time}}\)

Substituting the distance \(2\pi r\) and time 't':

\(v = \frac{2\pi r}{t}\)

This equation gives the magnitude of the velocity, which is often referred to as the speed in the context of uniform circular motion.

Analyzing the Options

Let's look at the given options:

• Option 1: \(v = \frac{t}{{2\pi r}}\) - This has time in the numerator and distance in the denominator, which is incorrect for velocity/speed.
• Option 2: \(v = \frac{{2\pi r}}{t}\) - This represents distance divided by time, which is the correct formula for speed in this scenario.
• Option 3: \(v = \frac{{2\pi {r^2}}}{t}\) - This includes \(r^2\), which is related to the area of a circle, not the circumference. It is incorrect.
• Option 4: \(v = \frac{{\pi r}}{{2t}}\) - This is not the correct formula for the circumference divided by time.

Comparing our derived formula with the options, Option 2 matches our result.

Summary of Calculation

Therefore, the velocity 'v' (magnitude or speed) for a body taking 't' seconds to go once around a circular path of radius 'r' is given by \(v = \frac{{2\pi r}}{t}\).

Revision Table: Circular Motion Velocity

Additional Information: Related Circular Motion Concepts

Understanding velocity in circular motion is fundamental. Here are a few related concepts that are important:

• Angular Velocity (\(\omega\)): This is the rate of change of angular displacement. It is measured in radians per second. The relationship between linear speed (v) and angular velocity (\(\omega\)) is \(v = r\omega\). Since the angular displacement for one revolution is \(2\pi\) radians and the time is 't', the angular velocity is \(\omega = \frac{2\pi}{t}\). Substituting this into \(v = r\omega\) gives \(v = r \left(\frac{2\pi}{t}\right) = \frac{2\pi r}{t}\), which confirms our velocity formula.
• Frequency (f): This is the number of revolutions per unit time. It is the reciprocal of the time period (T). \(f = \frac{1}{T} = \frac{1}{t}\). The velocity formula can also be written as \(v = 2\pi r f\).
• Centripetal Acceleration (\(a_c\)): Although the speed is constant in uniform circular motion, the velocity is constantly changing direction. This change in velocity means there is acceleration. This acceleration is always directed towards the center of the circle and is called centripetal acceleration. Its magnitude is given by \(a_c = \frac{v^2}{r}\) or \(a_c = r\omega^2\).
• Centripetal Force (\(F_c\)): According to Newton's second law, an acceleration is caused by a force. The force responsible for centripetal acceleration is called centripetal force. It is also directed towards the center and is given by \(F_c = m a_c = \frac{m v^2}{r}\) or \(F_c = mr\omega^2\), where 'm' is the mass of the body.