Sand drops vertically from a stationary hopper at the rate of 2 kg/s onto a horizontal
conveyer belt moving with a constant speed of 2 m/s. The height from which sand
falls is negligible and the sand comes to rest with respect to the belt at the instant it
comes in contact with the belt. Neglect all horizontal forces except that between the
sand and the belt. The power (in W) delivered by the motor moving the belt is

To determine the power delivered by the motor moving the conveyor belt in this scenario, we need to focus on how the kinetic energy of the falling sand is transferred to the belt. Since the sand is falling vertically and comes to rest upon contact with the belt, we can use the principles of momentum and power to find the solution.

Understanding Momentum Transfer

The sand is falling at a rate of 2 kg/s, which means that every second, 2 kilograms of sand is added to the conveyor belt. The belt is moving horizontally at a constant speed of 2 m/s. When the sand lands on the belt, it must acquire the same horizontal velocity as the belt, which is 2 m/s.

Calculating Horizontal Momentum

The horizontal momentum gained by the sand when it lands on the belt can be calculated using the formula:

• Momentum (p) = mass (m) × velocity (v)

Here, the mass flow rate of the sand is 2 kg/s, and the velocity of the belt is 2 m/s. Therefore, the horizontal momentum imparted to the sand each second is:

p = 2 kg/s × 2 m/s = 4 kg·m/s

Power Calculation

Power is the rate at which work is done or energy is transferred, and it can also be defined in terms of momentum as:

• Power (P) = force (F) × velocity (v)

In this case, the force exerted on the sand as it comes to rest with respect to the belt is equal to the change in momentum per unit time (which is the mass flow rate times the velocity of the belt). Thus, the power delivered by the motor can be calculated as:

P = (mass flow rate) × (velocity) × (velocity of the belt)

Plugging in our values:

P = 2 kg/s × 2 m/s = 4 W

Final Result

Hence, the power delivered by the motor moving the belt is 4 watts. This power is necessary to keep the belt moving at a constant speed while accommodating the new sand that comes to rest on it.

This example illustrates how momentum transfer and power calculations can be applied practically in scenarios involving moving objects and forces acting on them. By understanding the dynamics at play, we can better appreciate the energy requirements in mechanical systems.