Question

The bob of simple pendulum (mass \(\mathrm{m}\) and length 1 ) dropped from a horizontal position strike a block of the same mass elastically placed on a horizontal frictionless table. The K.E. of the block will be (A) \(2 \mathrm{mg} 1\) (B) \(\mathrm{mg} 1 / 2\) (C) \(\mathrm{mg} 1\) (D) zero

Short answer

The kinetic energy of the block after the collision is \(2mg1\).

Step-by-step solution

- Identify the Initial and Final Conditions

Initially, the pendulum is in a horizontal position with only potential energy due to its height and the block is at rest on the frictionless table. After collision, the pendulum and the block have kinetic energy from the transferred potential energy of the pendulum.

- Calculate the Initial Potential Energy of the Pendulum

When the pendulum is horizontally positioned, the entire length of the pendulum (1) is lifted against gravity. The potential energy (PE) at this point can be calculated using the formula: \(PE = mgh\) where m = mass of the bob g = acceleration due to gravity h = height of the bob In this case, h equals the entire length of the pendulum, which is 1. Therefore, \(PE = mg * 1 = mg\)

- Apply the Principle of Energy Conservation

According to the principle of energy conservation, the total mechanical energy before collision equals the total mechanical energy after collision. Initially, the pendulum has potential energy (PE) and the block has a kinetic energy of zero. After the collision, the pendulum has some kinetic energy (KE1) and the block has some kinetic energy (KE2). To find the total mechanical energy after collision, we can equate the initial mechanical energy (PE) with the final mechanical energy (KE1 + KE2): \(PE = KE1 + KE2\) Since we are interested in finding the kinetic energy of the block (KE2), we can rewrite this equation as: \(mg = KE1 + KE2\)

- Apply the Principle of Elastic Collisions

In elastic collisions, both momentum and kinetic energies are conserved. We can use the momentum conservation principle to find the relative velocities of the pendulum and the block after collision. Let the velocity of mass m1 (pendulum) be \(v1_f\) and the velocity of mass m2 (block) be \(v2_f\) after collision. The initial velocities are 0. The equation for conservation of momentum is: \(m1 * v1_i + m2 * v2_i = m1 * v1_f + m2 * v2_f\) As both masses are equal and initial velocities are zero, we get: \(m * 0 + m * 0 = m * v1_f + m * v2_f\) Solving for \(v2_f\): \(v2_f = -v1_f\)

- Calculate the Kinetic Energy of Mass m2 (Block)

Now we have the relationship between the final velocities of the two masses, we can use it to find the kinetic energy of the block (m2). From step 4, we have: \(v2_f = -v1_f\) The kinetic energy of a mass is given by: \(KE = \frac{1}{2}mv^2\) Substitute the expression for \(v2_f\) from step 4: \(KE2 = \frac{1}{2}m(-v1_f)^2\) \(KE2 = \frac{1}{2}mv1_f^2\) Now we can substitute this expression for KE2 in the energy conservation equation from step 3: \(mg = KE1 + \frac{1}{2}mv1_f^2\) We also know that: \(KE1 = \frac{1}{2}mv1_f^2\) So the equation becomes: \(mg = 2 * \frac{1}{2}mv1_f^2\) Divide both sides by m to obtain the Kinetic Energy of the block (KE2): \(KE2 = 2g * 1\) \(KE2 = 2mg\) Hence, the kinetic energy of the block after the collision is \(2mg\). So, the correct answer is: (A) \(2mg1\)

Key concepts

Elastic Collision

An elastic collision is a type of collision where total kinetic energy and total momentum are conserved. In our case, when the pendulum bob collides with the block on the table, both objects interact but do not lose any kinetic energy to deformation or heat. This conservation of kinetic energy ensures that the motion remains predictable.

• Both objects retain their physical integrity.
• No energy is lost to the surroundings.

Elasticity is ideal because it simplifies calculations. We can equate the sum of kinetic energies before and after the collision. In practical scenarios, especially small, hard objects, collisions approach this ideal.

Kinetic Energy

Kinetic energy refers to the energy an object has due to its motion. It is given by the formula: \( KE = \frac{1}{2}mv^2 \), where \( m \) is mass and \( v \) is velocity.

• This energy is zero if the object is stationary.
• It increases with the square of the velocity.

In the exercise, after the pendulum bob falls and hits the block, the energy it had from motion is partly transferred to the block. This transfer is crucial for calculating how the moving pendulum influences the block's speed and resultant energy.

Potential Energy

Potential energy is the stored energy an object has due to its position or state. The pendulum stores potential energy when it is at its highest point because work has been done against gravity to get it there.

• It's calculated by \( PE = mgh \), with \( h \) being the height above the reference point.
• This energy is highest when the pendulum is fully horizontal, preparing to swing down.

Understanding how potential energy converts to kinetic energy helps explain the pendulum’s energy distribution during its swing. As the pendulum swings down, its potential energy decreases while kinetic energy increases, showing an energy transformation.

Conservation of Momentum

The conservation of momentum principle states that in an isolated system, the total momentum remains constant unless acted upon by an external force. In the context of the pendulum and block system, it ensures:

• The combined momentum of both before the collision equals that after.
• Since the system starts with no motion, total momentum during the interaction remains zero.

This principle is mathematically expressed as:\[ m1 \cdot v1_i + m2 \cdot v2_i = m1 \cdot v1_f + m2 \cdot v2_f \]
In our exercise, this implies that any momentum the pendulum had pre-collision is perfectly transferred to the block, thus maintaining system equilibrium.

Mechanical Energy Conservation

Mechanical energy conservation combines both kinetic and potential energy. In a system void of non-conservative forces like friction, the total mechanical energy stays constant.

• Initially, the energy is purely potential when the pendulum is at its peak.
• All potential energy converts to kinetic energy just before collision.
• Post-collision, this kinetic energy is distributed between the pendulum and the block.

Using \( PE + KE = \text{constant} \), one can solve for different states of motion within a closed system. In the exercise, right after impact, the energy from the pendulum manifests as kinetic energy, distributed according to the conservation rules.