Problem 2 - A Complete Resource Book in Physics for JEE Main

Question

A body of mass \(2 \mathrm{~kg}\) moves vertically downwards with an acceleration \(a=19.6 \mathrm{~m} / \mathrm{s}^{2}\). The force acting on the body simultaneously with the force of gravity is \(\left(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\right.\), neglect air resistance) (A) \(19.6 \mathrm{~N}\) (B) \(19.2 \mathrm{~N}\) (C) \(59.2 \mathrm{~N}\) (D) \(58.8 \mathrm{~N}\)

Step-by-Step Solution

Answer

The force acting simultaneously with the force of gravity is \(19.6\,\text{N}\).

Step 1: Recall Newton's second law of motion

Newton's second law of motion states that the force acting on an object is equal to the product of its mass and acceleration. Mathematically, the force can be expressed as: \(F = m \times a\) Where \(F\) is the force, \(m\) is the mass, and \(a\) is the acceleration.

Step 2: Calculate the force due to gravity

To find the force due to gravity, we use the formula: \(F_g = m \times g\) Where \(F_g\) is the force due to gravity, \(m\) is the mass, and \(g\) is the acceleration due to gravity. Plugging in the given values: \(F_g = 2\,\text{kg} \times 9.8\,\frac{\text{m}}{\text{s}^2}\) \(F_g = 19.6\,\text{N}\)

Step 3: Calculate the total force acting on the body

Now, we need to find the total force acting on the body. We will use Newton's second law of motion formula: \(F = m \times a\) Where \(F\) is the force, \(m\) is the mass, and \(a\) is the acceleration. Plugging in the given values: \(F = 2\,\text{kg} \times 19.6\,\frac{\text{m}}{\text{s}^2}\) \(F = 39.2\,\text{N}\)

Step 4: Calculate the force acting simultaneously with gravity

To find the force acting simultaneously with gravity, we will subtract the force due to gravity from the total force acting on the body: \(F_{simultaneous} = F - F_g\) \(F_{simultaneous} = 39.2\,\text{N} - 19.6\,\text{N}\) \(F_{simultaneous} = 19.6\,\text{N}\) Therefore, the force acting simultaneously with the force of gravity is \(19.6\,\text{N}\). The correct answer is (A) \(19.6\,\text{N}\).

Key Concepts

Newton's Second Law of Motion

Understanding Newton's Second Law of Motion is crucial for solving many problems in physics, particularly those involving forces and motion. Put simply, this law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The direction of the acceleration is the same as the direction of the net force.

The mathematical expression for Newton's Second Law is \( F = m \times a \), where \( F \) represents the net force applied to the object, \( m \) is the mass of the object, and \( a \) is the acceleration that the object experiences. To illustrate, if a body of mass 2kg experiences a net force that causes an acceleration of \( 19.6 \mathrm{~m} / \mathrm{s}^{2} \), according to Newton's Second Law, the force can be calculated simply by multiplying the mass by the acceleration.

Force due to Gravity

Our Earth exerts a gravitational force on objects, pulling them towards its center. This force, known as the force due to gravity, is what gives objects weight. It is expressed with the formula \( F_g = m \times g \), where \( F_g \) is the gravitational force, \( m \) is the mass of an object, and \( g \) stands for the acceleration due to gravity. On Earth, \( g \) is approximately \( 9.8 \mathrm{~m} / \mathrm{s}^{2} \).

For a body with a mass of 2 kg, the force due to gravity can be calculated by multiplying the mass (\text{2kg}) by the acceleration due to gravity (\text{9.8m/s}^2), resulting in a gravitational force of \( 19.6 \text{N} \). This force acts on the body regardless of any other forces that may be present and is a key consideration when analyzing the motion of objects.

Equations of Motion

The equations of motion form the foundation of classical mechanics and describe how the velocity of a body changes under the influence of a force. There are three primary equations of motion that relate displacement, velocity, acceleration, and time. These are used to determine the position and velocity of an object at any point in time, given its initial position and velocity, and the forces acting upon it. The three equations are:

• The first equation \( v = u + at \) relates initial velocity \( u \) with final velocity \( v \) over time \( t \) with acceleration \( a \).
• The second equation \( s = ut + \frac{1}{2}at^2 \) gives the displacement \( s \) of an object initially traveling with velocity \( u \) and accelerated by \( a \) over a time \( t \).
• The third equation \( v^2 = u^2 + 2as \) relates the squares of the final and initial velocities with displacement and acceleration.

These equations provide powerful tools for predicting the future state of moving objects and are widely used not just in physics, but also in engineering, navigation, and various scientific analyses.