Problem 17 - Master Resource Book in JEE Main Physics

Question

An object of mass \(10 \mathrm{~kg}\) is whirled round a horizontal circle of radius \(4 \mathrm{~m}\) by a revolving string inclined \(30^{\circ}\) to the vertical. If the uniform speed of the object is \(5 \mathrm{~ms}^{-1}\), the tension in the string (approximately) is (a) \(720 \mathrm{~N}\) (b) \(960 \mathrm{~N}\) (c) \(114 \mathrm{~N}\) (d) \(125 \mathrm{~N}\)

Step-by-Step Solution

Answer

The tension in the string is approximately \(125 \mathrm{~N}\) (option d).

Step 1: Identify Given Values

We need to identify the values given in the problem:- Mass of the object, \( m = 10 \mathrm{~kg} \).- Radius of the circle, \( r = 4 \mathrm{~m} \).- Angle of the string with the vertical, \( \theta = 30^{\circ} \).- Speed of the object, \( v = 5 \mathrm{~ms}^{-1} \).

Step 2: Understand the Forces Involved

The forces acting on the object are its weight (\( mg \)), the tension in the string (\( T \)), and the centripetal force required to keep it moving in a circle. The tension can be resolved into two components: 1. Vertical component: \( T\cos{\theta} \) that supports the weight of the object.2. Horizontal component: \( T\sin{\theta} \) that provides centripetal force.

Step 3: Apply the Vertical Forces Equilibrium

The vertical component of tension balances the weight of the object:\[T\cos{\theta} = mg\]Substituting the known values:\[T\cos{30^{\circ}} = 10 \times 9.81\]\[T\frac{\sqrt{3}}{2} = 98.1\]

Step 4: Solve for Tension Using Centripetal Force

The horizontal component provides the centripetal force needed:\[T\sin{\theta} = \frac{mv^2}{r}\]Substituting the known values:\[T\sin{30^{\circ}} = \frac{10 \times 5^2}{4}\]\[T\frac{1}{2} = \frac{250}{4}\]\[T \times 0.5 = 62.5\]

Step 5: Solve for T using Equations

From Step 3 and 4, we have the following equations:1. \( T\frac{\sqrt{3}}{2} = 98.1 \)2. \( T \times 0.5 = 62.5 \)Solve these simultaneously to find \( T \):From the second equation, \( T = 125 \mathrm{~N} \).Check if the tension value satisfies both equations, which it does.

Key Concepts

Centripetal Force

Centripetal force is a crucial concept in understanding circular motion. When an object moves in a circle, it is constantly changing direction. This change in direction requires a net force to act on the object, pulling it towards the center of the circle. This net force is known as centripetal force.

For an object of mass \( m \) moving at a velocity \( v \) in a circle of radius \( r \), the formula for centripetal force is given by:
\[ F_c = \frac{mv^2}{r} \]

In the exercise, the horizontal component of the tension in the string provides the necessary centripetal force to keep the object moving in a circle. Therefore, the tension \( T \) multiplied by \( \sin\theta \) equals the centripetal force. This understanding helps us solve for the tension and ensure the object continues its circular path.

Forces in Equilibrium

Forces in equilibrium occur when all the forces acting on an object balance each other out, resulting in no net force and no change in motion. In the context of circular motion, equilibrium can be observed in the vertical direction.

The vertical component of the tension in the string must balance the weight of the object to maintain equilibrium. The weight of an object of mass \( m \) is given by \( mg \), where \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).

The formula for vertical equilibrium with tension is:
\[ T\cos\theta = mg \]

This equation indicates that the tension times the cosine of the angle with the vertical equals the weight of the object. Solving this equation provides us with one component of the forces acting in equilibrium in the vertical plane.

Mechanics

Mechanics is the branch of physics that deals with the motion of objects and the forces that cause these movements. It encompasses a wide range of principles, among which centripetal force and equilibrium are crucial in analyzing motion in a circular path.

In this exercise, we apply mechanics to understand how a force (tension in the string) can control the motion of an object in a circle. The mechanics involve resolving the tension into horizontal and vertical components, then using those components to balance the forces acting on the object.

Key steps in using mechanics for this exercise include:

• Identifying the given values such as mass, radius, angle, and speed.
• Understanding the forces involved, namely tension, weight, and centripetal force.
• Applying mathematical relationships to solve for unknowns, such as tension in this case.

By breaking down the problem using basic mechanics principles, we can solve complex problems involving motion and forces more effectively.