Problem 53 - Master Resource Book in JEE Main Physics

Question

A body of mass \(M\) is moving with a uniform speed of \(10 \mathrm{~m} / \mathrm{s}\) on frictionless surface under the influence of two forces \(F_{1}\) and \(F_{2}\). The net power of the system is [MP PET 2010] (a) \(10 F F_{12} M\) (b) \(10\left(F_{i}+F_{2}\right) M\) (c) \(\left(F_{1}+F_{2}\right) M\) (d) zero

Step-by-Step Solution

Answer

The net power of the system is zero (option d).

Step 1: Understand the Concept of Power

Power is the rate at which work is done or energy is transferred. Mathematically, it is defined as the dot product of force and velocity, i.e., \( P = \mathbf{F} \cdot \mathbf{v} \).

Step 2: Analyze the Problem Setup

The body is moving on a frictionless surface, meaning there is no resisting force. Given that two forces \( F_1 \) and \( F_2 \) act on the body, the net force must equal the mass times the acceleration.

Step 3: Calculate the Net Force

Since the object is moving with a 'uniform speed' of 10 m/s, it implies there is no acceleration. Thus, the net force acting on it, \( F_{net} = F_1 + F_2 = 0 \).

Step 4: Determine the Net Power

Substitute the net force into the power formula \( P = \mathbf{F}_{net} \cdot \mathbf{v} \). Since \( F_{net} = 0 \), the power \( P = 0 \cdot 10 = 0 \).

Step 5: Choose the Correct Option

Based on our calculations, the correct answer is option (d) zero, since the net power of a system on a frictionless surface with no net acceleration is zero.

Key Concepts

Power Calculation

Power refers to how quickly work is done or energy is transferred over time. In physics, it's an important concept to understand how systems operate.
When we talk about power, we use the formula:

• \( P = \mathbf{F} \cdot \mathbf{v} \)
  This denotes power \( P \) as the dot product of force \( \mathbf{F} \) and velocity \( \mathbf{v} \).

This means for a body moving at a constant speed, the power exerted by the forces acting upon it can be calculated by multiplying the net force by its velocity.
However, if the net force is zero, as in the given exercise on a frictionless surface, then the power becomes zero too.
This is because no work is needed to maintain a constant velocity when there's no opposing force or change in motion.

Net Force

The net force is a crucial concept in understanding motion. It represents the total force acting on an object after taking into account all individual forces.
Mathematically, it's the vector sum of all forces involved.

• If the net force is zero, as in this exercise where forces \( F_1 \) and \( F_2 \) cancel each other out, the object moves at a constant velocity.
• This is because acceleration, which results from net force (using \( F = ma \)), is absent.

Thus, a zero net force implies uniform motion, meaning the object maintains a constant speed and direction.
In the context of our exercise, since the speed is constant and the net force comes out to zero, it reinforces the idea that the object is moving with uniform velocity without any acceleration.

Frictionless Surface

A frictionless surface is a theoretical concept used to simplify problems in physics.
In real scenarios, surfaces with truly zero friction don't exist, but in physics problems, considering such surfaces helps us focus on fundamental forces without resistance.

• On a frictionless surface, no opposing force acts against motion.
• This means once an object starts moving, it will continue to move indefinitely with the same velocity, unless another force acts on it.

In our exercise, the body moves on a frictionless surface, meaning forces \( F_1 \) and \( F_2 \) are responsible for any change, but they balance out, maintaining uniform motion.
This setup shows how the absence of friction helps to keep the problem simple, emphasizing the role of applied forces alone in dictating motion.

Velocity

Velocity is a vector quantity that describes both the speed and direction of an object's motion. It's important to understand how it differs from speed, which is a scalar and measures only magnitude.

• In the exercise, the body's velocity is given as 10 m/s, indicating its speed along a particular direction.
• Uniform velocity means the speed and direction remain constant over time.

In scenarios involving no net force, like the one with the frictionless surface, velocity plays a key role as it remains constant.
This reinforces Newton's first law of motion, that a body in motion will stay in uniform motion unless acted upon by a net external force.
Understanding velocity helps distinguish between situations where forces result in acceleration versus those where motion remains unchanged.