Problem 6 - Master Resource Book in JEE Main Physics

Question

A bob of mass \(m\) accelerates uniformly from rest to \(v_{1}\) in time \(t_{1}\). As a function of \(t\), the instantaneous power delivered to the body is (a) \(\frac{m v_{1} f}{t_{2}}\) (b) \(\frac{m v_{1}}{t_{1}}\) (c) \(\frac{m v_{1} t^{2}}{t_{1}}\) (d) \(\frac{m v_{1}^{2} t}{t_{1}^{2}}\)

Step-by-Step Solution

Answer

The correct answer is (d) \(\frac{m v_{1}^{2} t}{t_{1}^{2}}\).

Step 1: Understanding Instantaneous Power

The power delivered to a body at any instant is given by the formula for power: \( P(t) = F(t) \cdot v(t) \), where \( F(t) \) is the instantaneous force and \( v(t) \) is the instantaneous velocity at time \( t \).

Step 2: Using Newton's Second Law

From Newton's second law, the force is given by \( F(t) = m \cdot a(t) \), where \( m \) is the mass of the bob, and \( a(t) \) is the acceleration. Because the bob accelerates uniformly, \( a(t) = \frac{v_{1}}{t_{1}} \).

Step 3: Expression for Instantaneous Speed

Since the motion is uniformly accelerated from rest, the instantaneous speed \( v(t) \) is given by \( v(t) = a(t) \cdot t = \frac{v_{1}}{t_{1}} \cdot t \).

Step 4: Calculating Instantaneous Power

Substitute \( F(t) \) and \( v(t) \) into the power formula: \( P(t) = F(t) \cdot v(t) = m \cdot a(t) \cdot v(t) = m \cdot \frac{v_{1}}{t_{1}} \cdot \frac{v_{1}}{t_{1}} \cdot t = \frac{m v_{1}^{2} t}{t_{1}^{2}} \).

Step 5: Select Correct Option

Compare the expression \( \frac{m v_{1}^{2} t}{t_{1}^{2}} \) with the given options. Option (d) matches the derived expression.

Key Concepts

Newton's Second Law

Newton's Second Law is a fundamental principle of physics. It provides a link between the net force applied to an object and its acceleration. This is described mathematically by the equation: \[ F = m \cdot a \]where:

• \( F \) is the net force applied to the object,
• \( m \) is the mass of the object, and
• \( a \) is the acceleration caused by the force.

So, when a bob accelerates uniformly, the force causing this motion can be determined using this law. With constant acceleration, the force is constant too, and acts in the direction of acceleration. This relationship is key to understanding how forces cause changes in motion and how to apply this understanding in solving problems related to forces and motion, like in the given exercise.

Uniform Acceleration

Uniform acceleration occurs when an object's velocity changes at a constant rate. This means the magnitude of acceleration remains constant over time. When studying motion under uniform acceleration, several equations help relate different quantities like velocity, time, distance, and acceleration.In the context of the exercise, the bob starts from rest and uniformly accelerates to a velocity \( v_1 \) over a time \( t_1 \). Thus, the acceleration \( a \) can be calculated as:\[ a = \frac{v_1}{t_1} \]This equation comes from the basic definition of acceleration as the change in velocity over time. Uniform acceleration simplifies motion analysis since the relationships remain linear over time. It's a useful concept, especially when coupled with Newton's Second Law, as it allows clear mathematical characterization of motion-related problems.

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific point in time. Unlike average velocity, which considers the total displacement divided by total time, instantaneous velocity provides the speed and direction of an object at a precise moment. In uniformly accelerated motion, calculating instantaneous velocity becomes straightforward.For an object that starts from rest and accelerates uniformly, like our bob, the instantaneous velocity \( v(t) \) at any time \( t \) is given by:\[ v(t) = a \cdot t \]Substituted with the expression for acceleration, it becomes:\[ v(t) = \frac{v_1}{t_1} \cdot t \]This formula underlines that the instantaneous velocity increases linearly with time in uniformly accelerated motion. Understanding this is crucial for solving problems where precise velocity at any given time is needed, such as in power calculations.

Power Formula

In physics, power is the rate at which work is done or energy is transferred over time. The instantaneous power delivered to a moving object can be calculated using the formula:\[ P(t) = F(t) \cdot v(t) \]Where \( F(t) \) represents the instantaneous force on the object at time \( t \), and \( v(t) \) is its instantaneous velocity.Using Newton's Second Law and the equation for instantaneous velocity, we substitute:\[ F(t) = m \cdot a(t) \]\[ v(t) = \frac{v_1}{t_1} \cdot t \]Thus, the expression for power becomes:\[ P(t) = m \cdot \left(\frac{v_1}{t_1}\right)^2 \cdot t \]This shows how power varies with time for an object undergoing uniform acceleration. It's a clear example of how force, velocity, and subsequently power, are interlinked in dynamics, especially illustrating the direct correlation between instantaneous velocity and instantaneous power.