Problem 1 - Master Resource Book in JEE Main Physics

Question

The bob of a pendulum is released from a horizontal position \(A\) as shown in the figure. If the length of the pendulum is \(1.5 \mathrm{~m}\), then the speed with which the bob arrives at the lower most point \(B\). Given that it dissipated \(5 \%\) of its initial energy against air resistance? (a) \(6.0 \mathrm{~m} / \mathrm{s}\) (b) \(6.5 \mathrm{~m} / \mathrm{s}\) (c) \(4.5 \mathrm{~m} / \mathrm{s}\) (d) \(5.3 \mathrm{~m} / \mathrm{s}\)

Step-by-Step Solution

Answer

The speed of the bob is approximately \( 5.3 \ \mathrm{m/s} \), option (d).

Step 1: Understand the Problem

The pendulum is released from a horizontal position, which means it starts with gravitational potential energy and no kinetic energy at the start. We need to find the speed of the bob at the lowermost point after accounting for a 5% energy loss due to air resistance.

Step 2: Determine Initial Energy

At the horizontal position (point A), all energy is potential. The potential energy (PE) is given by \( PE = mgh \), where \( h = L = 1.5 \ \mathrm{m} \) is the vertical height from the lowermost point, so \( PE = mg \times 1.5 \).

Step 3: Calculate Energy at Point B

At the lowest point (point B), ideally, all the potential energy would convert to kinetic energy. However, due to 5% energy loss, only 95% of the initial energy becomes kinetic. Hence, \( KE = 0.95 \times mgh \).

Step 4: Relate Kinetic Energy to Speed

The kinetic energy at point B can be expressed as \( KE = \frac{1}{2}mv^2 \). Set this equal to the energy at point B to find \( v \): \( \frac{1}{2}mv^2 = 0.95 \times mgh \).

Step 5: Solve for Speed

Cancel \( m \) from both sides of \( \frac{1}{2}mv^2 = 0.95 \times m \times g \times 1.5 \), so \( v^2 = 2 \times 0.95 \times g \times 1.5 \). With \( g = 9.8 \ \mathrm{m/s^2} \), calculate \( v \):\[v^2 = 2 \times 0.95 \times 9.8 \times 1.5 \v = \sqrt{2 \times 0.95 \times 9.8 \times 1.5}\]

Step 6: Calculate and Choose Correct Option

Compute the numerical value: \[v = \sqrt{27.93} \approx 5.3 \ \mathrm{m/s}\]Hence, the speed of the bob at point B is approximately \( 5.3 \ \mathrm{m/s} \). The correct option is (d).

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is the energy stored due to an object's position in a gravitational field. It is calculated by the formula:

• Potential Energy (PE) = \( mgh \)
• Where
  • \( m \) is the mass of the object,
  • \( g \) is the acceleration due to gravity, approximately \( 9.8 \, \mathrm{m/s^2} \),
  • \( h \) is the height above the reference point.

In the context of a pendulum, at the highest point of its swing, all its energy is gravitational potential energy, because its height is maximized and its speed is zero. When the pendulum bob begins its descent from the highest position, it converts this potential energy into kinetic energy as it moves downward.

Kinetic Energy

Kinetic energy is the energy of motion. When an object is moving, it possesses kinetic energy, which is given by the formula:

• Kinetic Energy (KE) = \( \frac{1}{2}mv^2 \)
• Where
  • \( m \) is the mass of the object,
  • \( v \) is its velocity.

In our pendulum example, as the bob swings downwards, it speeds up, increasing its kinetic energy. At the lowest point of the swing, ideally, all the gravitational potential energy is converted into kinetic energy, making the speed and therefore the kinetic energy maximum. However, in reality, some energy is lost, as mentioned, due to air resistance.

Energy Dissipation

Energy dissipation involves the loss of energy from a system, usually in the form of thermal energy due to friction or air resistance. In the pendulum scenario, when the bob swings from its highest point, not all the initial potential energy becomes kinetic energy.
Instead, some percentage — 5% in this problem — is dissipated mostly due to air resistance.

• This means the energy that's available to convert into kinetic energy is only 95% of the initial potential energy.
• Energy dissipation highlights the natural tendency of real-world systems to lose energy to the surroundings, which complicates theoretical models that assume perfect energy conservation.

Taking this dissipation into account allows us to realistically calculate the actual speed of the pendulum bob at its lowest point.

Conservation of Energy

The principle of conservation of energy states that energy in a closed system remains constant over time. Energy cannot be created or destroyed but can only change forms. In the pendulum exercise:

• The total energy transitions between potential and kinetic forms as the pendulum swings.
• At the start, all energy is potential, and at the bottom, it is (ideally) all kinetic.
• Nonetheless, due to external factors like air resistance, not all energy is exchanged perfectly, hence some gets dissipated.

This principle helps us connect the dots between the start and end states of the pendulum bob, allowing calculations about the bob’s speed, knowing energy is conserved barring losses due to dissipative forces.