Problem 36 - Master Resource Book in JEE Main Physics

Question

A body of mass \(5 \mathrm{~kg}\) is moving in a circle of radius \(1 \mathrm{~m}\) with angular velocity of \(2 \mathrm{rad} / \mathrm{s}\). The centripetal force is [Orissa JEE 2011] (a) \(10 \mathrm{~N}\) (b) \(20 \mathrm{~N}\) (c) \(30 \mathrm{~N}\) (d) \(40 \mathrm{~N}\)

Step-by-Step Solution

Answer

The centripetal force is 20 N, option (b).

Step 1: Understanding the problem

We have a body of mass \(5 \mathrm{~kg}\) moving in a circle of radius \(1 \mathrm{~m}\) with an angular velocity of \(2 \mathrm{rad/ s}\). We need to find the centripetal force acting on this body.

Step 2: Identify the formula for centripetal force

The centripetal force \(F_c\) for a body moving in a circle is given by the formula \(F_c = m \cdot r \cdot \omega^2\), where \(m\) is the mass, \(r\) is the radius, and \(\omega\) is the angular velocity.

Step 3: Substitute known values into the formula

We know that \(m = 5 \mathrm{~kg}\), \(r = 1 \mathrm{~m}\), and \(\omega = 2 \mathrm{rad/ s}\). Substituting these into the formula, we get: \[ F_c = 5 \cdot 1 \cdot (2)^2 \]

Step 4: Calculate the centripetal force

Perform the calculations: \[ F_c = 5 \cdot 1 \cdot 4 \]\[ F_c = 20 \mathrm{~N} \]

Step 5: Determine the correct option

Based on our calculation, the centripetal force is \(20 \mathrm{~N}\), which corresponds to option (b).

Key Concepts

Circular Motion

Circular motion occurs when an object moves along a curved path or a circle. It is a fascinating concept in physics because it illustrates how forces keep objects in motion along a circular path. The key to understanding circular motion lies in recognizing that even if the speed of an object remains constant, its velocity is not. Velocity, a vector quantity, considers both speed and direction. Therefore, in circular motion, since the direction is continuously changing, the velocity is changing too.
A critical force in circular motion is the centripetal force, which acts toward the center of the circle. It is this force that keeps the object from flying outwards due to inertia.
In everyday life, circular motion is all around us, from the spinning of the Earth to the wheels of a moving car. Understanding this helps us appreciate the direction of forces at play and the object's speed needed for successful circular motion.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates or revolves relative to a point or axis. It is vital in determining how fast an object moves around a circular path. The angular velocity is denoted by the symbol \(\omega\) and is typically expressed in radians per second (rad/s). This unit choice highlights the relationship with the angle the object moves through in a specified time.
It's crucial to understand that angular velocity differs from linear velocity. While linear velocity refers to how fast an object is moving along a path, angular velocity concerns itself with the rotational aspect. For instance, all points on the spinning wheel of a bicycle have the same angular velocity because they travel around the same axis at the same rate.
By understanding angular velocity, we can better comprehend the relationships and forces in systems involving rotation, such as gears, orbits, and even amusement park rides.

Mass and Radius in Physics

In physics, mass and radius are essential parameters that significantly influence the motion of an object, particularly in circular motion. Mass refers to the quantity of matter in an object, generally measured in kilograms. It plays a critical role in determining the object's inertia and its response to forces acting upon it. In our exercise, the mass is 5 kg, a key value used in calculating centripetal force.
The radius is the distance from the center of the circle to its perimeter. It's significant because it directly affects the path length an object travels during one rotation. In our context, a radius of 1 meter is crucial to determining the centripetal force when combined with mass and angular velocity.
Both these factors, mass and radius, particularly affect the centripetal force needed to keep an object moving in a circle. A larger mass means more force is needed, and a larger radius increases the distance, affecting the calculations for circular velocity and acceleration.