Problem 74 - Master Resource Book in JEE Main Physics

Question

An engine of power \(7500 \mathrm{~W}\) makes a train move on a horizontal surface with constant velocity of \(20 \mathrm{~ms}^{-1} .\) The force involved in the problem is (a) \(375 \mathrm{~N}\) (b) \(400 \mathrm{~N}\) (c) \(500 \mathrm{~N}\) (d) \(600 \mathrm{~N}\)

Step-by-Step Solution

Answer

The force involved is 375 N.

Step 1: Understanding the Problem

We need to find the force exerted by the engine on the train as it moves with a constant velocity on a horizontal surface. We are given the power of the engine as 7500 W and the velocity as 20 m/s.

Step 2: Using the Power Formula

The formula relating power, force, and velocity is given by: \[ P = F \times v \] where \( P \) is the power in watts, \( F \) is the force in newtons, and \( v \) is the velocity in meters per second. We can rearrange this formula to solve for the force \( F \) as:\[ F = \frac{P}{v} \]

Step 3: Substitute the Known Values

Substitute the given values into the formula:\[ F = \frac{7500}{20} \]

Step 4: Calculate the Force

Perform the division to find the force:\[ F = 375 \text{ N} \] This means the force involved is 375 N.

Key Concepts

Power and Force Relationship

The power and force relationship is a fundamental concept in physics, especially when understanding how machines work. Power signifies the rate at which work is done, and it relates directly to force and velocity. In the problem involving a train on a horizontal surface, we learn that power (P) connects to force (F) by the equation:

• \( P = F \times v \)

Breaking it down:

• Power (\( P \)): It is the energy transfer per unit time, measured in watts (W).
• Force (\( F \)): Causes the train to move and is measured in newtons (N).
• Velocity (\( v \)): Constant speed of the object, in meters per second (m/s).

By rearranging the formula to \( F = \frac{P}{v} \), it becomes possible to deduce the force when the power and velocity are given. Here, knowing how 7500 W of power pushes the train at 20 m/s, simplifies to find out the force exerted is 375 N. This relationship clarifies that with constant velocity, an incremental increase in power implies an increase in force.

Physics Formulas

Physics relies on several essential formulas that relate different physical quantities. They serve as tools to interpret and calculate unseen forces and motions around us. Let's highlight one crucial equation from the original exercise:

• Power Equation: \( P = F \times v \)

This formula is a simple yet powerful descriptor of how mechanical work is transferred in a system. Power dictates how much force is being applied over a distance in a given time period.
To solve practical problems, understanding and using these formulas is vital:

• Identify the known quantities – these are parameters like power, speed, force.
• Choose the appropriate formula – like the power equation, so it fits the situations described.
• Re-arrange the equation – solve for the unknown, by using algebraic manipulation.

For example, rearranging the power formula to solve for force helps find the precise force maintaining constant velocity in our exercise.

Newton's Laws of Motion

Newton's Laws of Motion underpin almost every principle of classical mechanics, and are crucial when solving dynamics problems. The scenario with the train involves concepts from these laws:

• First Law (Inertia): An object stays at rest or moves at a constant velocity unless acted upon by a net external force.

This law explains why the train's constant velocity remains unchanged as the force exerted by the engine matches any opposing forces like friction or air resistance.

• Second Law (Force and Acceleration): The principle expressed by \( F = ma \) indicates how force relates to mass and acceleration.

Here, although the acceleration is zero due to constant speed, it helps to understand how force balances out to maintain stability.

• Third Law (Action and Reaction): For every action, there is an equal and opposite reaction.

This law isn't overtly visible here but it reminds us that forces come in pairs. The train engine pushes forward while the tracks push back with equal force.
These laws form the foundation of relationships discussed, guiding steps in determining forces in motion.