Problem 5 - Master Resource Book in JEE Main Physics

Question

A ball of mass \(0.2 \mathrm{~kg}\) is thrown vertically upwards by applying a force by hand. If the hand moves \(0.2 \mathrm{~m}\) while applying the force and the ball goes upto \(2 \mathrm{~m}\) height further. Find the magnitude of force (Consider \(g=10 \mathrm{~m} / \mathrm{s}^{2}\) ) (a) \(22 \mathrm{~N}\) (b) \(4 \mathrm{~N}\) (c) \(16 \mathrm{~N}\) (d) \(20 \mathrm{~N}\)

Step-by-Step Solution

Answer

(d) 20 N

Step 1: Calculate Work Done by Gravitational Force

The ball gains potential energy when it reaches a height of 2 meters. The work done by the gravitational force (\( W_g \)) is given by the change in potential energy:\[ W_g = mgh = 0.2 \times 10 \times 2 = 4 \, \text{Joules} \]

Step 2: Calculate Initial Kinetic Energy Required

The work done by the hand's force must provide enough energy to overcome the gravitational force and reach 2 meters. Thus, the initial kinetic energy (\( KE_i \)) should equal 4 Joules.

Step 3: Work-Energy Principle for the Force

When the force is applied, the work done by the force (\( W_f \)) is used to give the ball kinetic energy equal to the potential energy gained plus any kinetic energy just as it leaves the hand. Since it just reaches the top, the kinetic energy at the top is 0, so:\[ W_f = KE_i = 4 \, \text{Joules} \]

Step 4: Calculate Force using Work Done Formula

The work done by the force can also be calculated as force times distance moved in the direction of the force. Thus,\[ W_f = F \times d \]Given that\[ d = 0.2 \text{ meters}, \]\[ F \times 0.2 = 4 \]Solving for the force \( F \):\[ F = \frac{4}{0.2} = 20 \, \text{Newtons} \]

Step 5: Verify the Answer

The calculated magnitude of the force is 20 N, which matches option (d). Thus, the solution is consistent, accurate, and complete.

Key Concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. When you throw a ball upwards, it starts moving with a certain speed. This motion gives the ball its kinetic energy. The formula to calculate kinetic energy (KE) is given by:

• \[KE = \frac{1}{2}mv^2\]

where:

• \(m\) is the mass of the object
• \(v\) is the velocity of the object

It's important to note that the kinetic energy depends on both mass and velocity. When you throw the ball, the work done by your hand provides this kinetic energy. As it ascends, this kinetic energy gets converted into potential energy, slowing the ball down until it stops momentarily at the highest point.

Potential Energy

Potential energy is the energy stored in an object due to its position relative to a reference point, often the ground. When a ball is thrown upwards, it gains height and, consequently, potential energy. The potential energy (PE) is calculated using the formula:

• \[PE = mgh\]

where:

• \(m\) is the mass of the object
• \(g\) is the acceleration due to gravity
• \(h\) is the height above the reference point

As the ball rises, it gains height, and thus, its potential energy increases. At the peak of its trajectory, all the initial kinetic energy has been converted into potential energy. Understanding this conversion is essential in the work-energy principle, as showcased in this physics problem.

Gravitational Force

Gravitational force is the attraction force that the Earth exerts on objects, pulling them towards its center. This force is responsible for the downward acceleration of freely falling objects. In physics problems such as the exercise given, we assume a constant gravitational acceleration (\(g\)) at Earth's surface, typically approximated as \(10 \, ext{m/s}^2\) for simplicity.
The gravitational force (\(F_g\)) acting on an object is computed by:

• \[F_g = mg\]

where \(m\) is the mass of the object. In our exercise, the ball's ascent is resisted by gravitational force, which requires a greater upward force from the hand to achieve a net positive velocity upward. Calculating the gravitational potential energy change helps determine how much work the applied force must do to lift the ball against gravity.

Physics Problem Solving

Physics problem solving involves using logical strategies to understand and solve physics questions, integrating various principles like work-energy, forces, and motion. In problems like the ball throwing exercise, following a structured approach can simplify the solution process:
1. **Identify the Types of Energy:** Recognize kinetic and potential energy transformations during motion.
2. **Apply the Work-Energy Principle:** Use this principle to relate work done by forces to energy changes.
3. **Calculate Each Force and Energy Component:** Quantify gravitational and external forces (such as the hand's applied force) and the associated work done.
4. **Verify Results:** Always cross-check against given options to ensure the solution's consistency.
This methodical process helps break down complex problems into manageable steps, aiding comprehension and accuracy in solving physics exercises.