Question

A particle is revolving in a circle of radius R with initial speed \(u\). It starts retarding with constant retardation \(\frac{\mathrm{v}^{2}}{4 \pi \mathrm{R}}\). The number of revolutions it makes in time \(\frac{8 \pi \mathrm{R}}{\mathrm{u}}\) is : (1) 3 (2) 4 (3) 2 (4) none of these

Short answer

The number of revolutions is 4.

Step-by-step solution

Identify the given values

The initial speed of the particle is given as \(u\). The particle is retarding with a constant retardation \( \frac{v^2}{4 \pi R} \). The time given is \( \frac{8 \pi R}{u} \).

Write the equation of motion

Using the equation of motion under retardation, we have: \[ v = u + at \]Here, \(a\) is the retardation and is given as \( \frac{v^2}{4 \pi R} \). We need to solve for \(v = 0\) (where the particle stops).

Solve the retardation equation

We need to solve for the time when the particle stops \( v = u - \frac{v^2}{4 \pi R} t \). Since \( v = 0 \) at the time it stops,\(0 = u - \frac{(u^2)}{4 \pi R} t\)Thus, the stopping time \( t_{stop} \) is:\[ t_{stop} = \frac{4 \pi R}{u} \]

Compare the given time with the stopping time

The given time is \( \frac{8 \pi R}{u} \), which is twice the stopping time:\[ t = 2 \times t_{stop} = 2 \times \frac{4 \pi R}{u} = \frac{8 \pi R}{u} \]

Calculate the number of revolutions

Each revolution takes time \( T \) where \( T = \frac{2 \pi R}{u} \). The total number of revolutions is given by the total time divided by the time for one revolution: \[ \text{Number of revolutions} = \frac{\frac{8 \pi R}{u}}{ \frac{2 \pi R}{u} } = 4 \]

Key concepts

Circular Motion

Circular motion involves an object moving in a circular path with a fixed radius. Think about how planets orbit the sun or a car traveling around a circular track.
The main elements to understand are:

• Radius (R): The distance from the center of the circle to any point on its edge.
• Speed (u): How fast the object is moving along the circular path.

In uniform circular motion, speed is constant, and the only factor acting on the object is centripetal force, which keeps it moving in a circle.However, in our exercise, the particle experiences retardation, meaning it slows down over time.

Constant Retardation

In this exercise, the particle does not maintain constant speed; it experiences constant retardation. Retardation, also known as negative acceleration, is when an object slows down over time.
The retardation here is given as \(\frac{v^2}{4 \pi R}\).
This means the particle's speed decreases consistently as it travels along the circle.
It is important to understand that, unlike uniform circular motion, retardation introduces a change in the object's velocity.

• The object's initial speed is u.
• Over time, this speed decreases due to the constant retardation force acting on it.

The slowing down process continues until the particle eventually comes to a stop.

Equations of Motion

To study the particle’s motion, we use the equations of motion. These equations allow us to calculate various dynamics of the moving particle. Here, we use the equation:
\[v = u + at\]
Where:

• v: Final velocity
• u: Initial velocity
• a: Acceleration (or retardation in this case)
• t: Time

Given the constant retardation ( \(-\frac{v^2}{4 \pi R})\), our goal is to find when the particle stops, meaning final velocity \(v\) is zero.
By substituting the values and solving, we determined that the time to stop ( \( t_{stop} = \frac{4 \pi R}{u})\). Given the total time in the problem, we then compared it and found that the total time was twice the stopping time.

Number of Revolutions

Finally, we need to figure out how many times the particle goes around the circle before stopping. If we know how long one full revolution takes, we can divide the total time by this period.
Using these step-by-step calculations:

• One revolution time (T) is calculated as \(\frac{2 \pi R}{u}\).


• Total time given is \( \frac{8 \pi R}{u}\).

By dividing the total time by the time for one revolution, we find:
\[\text{Number of revolutions} = \frac{\frac{8 \pi R}{u}}{\frac{2 \pi R}{u}} = 4\]
Thus, the particle makes four revolutions before it stops.