Problem 361 - JEE Physics

Question

A body of mass \(0.05 \mathrm{~kg}\) is falling with acceleration \(9.4 \mathrm{~ms}^{-2}\). The force exerted by air opposite to motion is \(\mathrm{N}\) \(\left(g=9.8 \mathrm{~ms}^{-2}\right)\) (A) \(0.02\) (B) \(0.20\) (C) \(0.030\) (D) Zero

Step-by-Step Solution

Answer

The force exerted by air opposite to motion is \(0.02 \mathrm{ ~N}\), which corresponds to option (A).

Step 1: Identify the accelerations (gravitational and net)

The body is falling under the influence of gravity (g = 9.8 m/s²) and is accelerated downward. The net acceleration (a_net) of the body is given as 9.4 m/s². Notice that the net acceleration is less than the gravitational acceleration, this is because of the air resistance acting on the body.

Step 2: Write down Newton's second law equation

According to Newton's second law, the net force acting on an object is equal to the product of its mass and acceleration. Mathematically, it can be represented as: \(F_{net} = m \cdot a_{net}\) where \(F_{net}\) is the net force, m is the mass of the body and \(a_{net}\) is the net acceleration.

Step 3: Calculate the gravitational force acting on the body

As the body is falling downward, we have a gravitational force acting on it which can be calculated as: \(F_g = m \cdot g\) where \(F_g\) is the gravitational force, m is the mass of the body, and g = 9.8 m/s² Substituting the given values, we get: \(F_g = 0.05 \mathrm{~kg} \cdot 9.8 \mathrm{~ms}^{-2} = 0.49 \mathrm{ ~N}\)

Step 4: Calculate the net force acting on the body

Now, use the Newton's second law equation to calculate the net force acting on the body: \(F_{net} = m \cdot a_{net}\) Substituting the given values, we get: \(F_{net} = 0.05 \mathrm{~kg} \cdot 9.4 \mathrm{~ms}^{-2} = 0.47 \mathrm{ ~N}\)

Step 5: Determine the force exerted by air opposite to motion

Now that we have the gravitational force and the net force acting on the body, we can determine the force exerted by air opposite to motion, \(F_{air}\), using the equation: \(F_{air} = F_g - F_{net}\) Substituting the calculated values, we get: \(F_{air} = 0.49 \mathrm{ ~N} - 0.47 \mathrm{ ~N} = 0.02 \mathrm{ ~N}\) So, the force exerted by air opposite to motion is 0.02 N, which corresponds to option (A).

Key Concepts

Gravitational Force

Gravitational force is a fundamental concept in physics. It is the force that attracts any two objects with mass towards each other. This force is what makes apples fall from trees and keeps planets orbiting the sun. For any object near the surface of the Earth, the gravitational acceleration is usually taken as \(g = 9.8 \, \text{ms}^{-2}\).
The formula to calculate this force acting on an object is: - \(F_g = m \cdot g\) - where \(F_g\) is the gravitational force, \(m\) is the mass of the object, and \(g\) is the gravitational acceleration.
In the exercise, the gravitational force on a 0.05 kg object is calculated as \(0.49 \, \text{N}\). This force causes the object to accelerate downwards.

Net Force

Net force is the overall force acting on an object when all individual forces are considered. According to Newton's Second Law of Motion, the net force \(F_{net}\) is related to mass \(m\) and net acceleration \(a_{net}\) by the equation:- \(F_{net} = m \cdot a_{net}\)Understanding net force is critical because it tells us how the motion of an object will change.
A net force causes a change in the object's velocity, leading to acceleration, deceleration, or a change in direction.
In our example, the net force is less than the gravitational force because air resistance also acts on the falling object. This results in a net force of \(0.47 \, \text{N}\).

Air Resistance

When an object moves through the air, it experiences a force opposing its motion. This force is known as air resistance or drag force. Air resistance acts in the opposite direction to the motion of the object and increases with speed.
For falling objects, air resistance reduces the net acceleration below gravitational acceleration.
Mathematically, air resistance \(F_{air}\) can be determined by:- \(F_{air} = F_g - F_{net}\) - where \(F_g\) is the gravitational force and \(F_{net}\) is the net force.
In the exercise, air resistance was calculated to be \(0.02 \, \text{N}\), which is the difference between gravitational force and the net force.

Acceleration

Acceleration is the rate at which an object's velocity changes with time. It can be influenced by forces such as gravity and air resistance. In physics, acceleration is not only about changing speed, but also about changes in direction.
The key to understanding problems involving acceleration is Newton's Second Law:- \(F_{net} = m \cdot a_{net}\) - where \(F_{net}\) represents the sum of all forces, and \(a_{net}\) is the resulting acceleration.
In the given exercise, the object's acceleration is given as \(9.4 \, \text{ms}^{-2}\), slightly less than \(g = 9.8 \, \text{ms}^{-2}\). This indicates the effect of air resistance reducing the acceleration.