A loaded ore car has a mass of 900 kg and rolls on rails with negligible friction. It starts from rest and is pulled up a mine shaft by a cable connected to a winch. The shaft is inclined at 23.0° above the horizontal. The car accelerates uniformly to a speed of 2.20 m/s in 11.5 s and then continues at constant speed. What power must the winch motor provide when the car is moving at constant speed????

To determine the power that the winch motor must provide when the ore car is moving at a constant speed, we need to consider the forces acting on the car and how they relate to power. Since the car is moving at a constant speed, the net force acting on it is zero, meaning that the force exerted by the winch must equal the gravitational force component acting down the incline.

Understanding the Forces at Play

First, let's break down the forces acting on the ore car. The gravitational force acting on the car can be calculated using the formula:

• Weight (W) = mass (m) × gravitational acceleration (g)

Here, the mass of the car (m) is 900 kg, and the gravitational acceleration (g) is approximately 9.81 m/s². Therefore, the weight of the car is:

W = 900 kg × 9.81 m/s² = 8829 N

Calculating the Gravitational Force Component

Since the car is on an incline of 23.0°, we need to find the component of the weight acting parallel to the incline. This can be calculated using the sine function:

• Force parallel to incline (F_parallel) = W × sin(θ)

Substituting the values:

F_parallel = 8829 N × sin(23.0°)

Using a calculator, we find:

F_parallel ≈ 8829 N × 0.3907 ≈ 3446.6 N

Power Calculation

Power is defined as the rate at which work is done or energy is transferred. When the car is moving at a constant speed, the power required by the winch to counteract the gravitational force is given by:

• Power (P) = Force (F) × Velocity (v)

At constant speed, the force exerted by the winch is equal to the gravitational force component we just calculated. The velocity (v) of the car is given as 2.20 m/s. Thus, we can calculate the power:

P = F_parallel × v

Substituting the values:

P = 3446.6 N × 2.20 m/s

Calculating this gives:

P ≈ 7582.52 W

Converting Power to Kilowatts

Since power is often expressed in kilowatts (kW), we can convert watts to kilowatts by dividing by 1000:

P ≈ 7.58 kW

Final Thoughts

Therefore, the winch motor must provide approximately 7.58 kW of power when the ore car is moving at a constant speed of 2.20 m/s up the inclined shaft. This calculation highlights the importance of understanding the forces involved and how they relate to power requirements in mechanical systems.