A cable lifts a 1050-kg elevator at a constant velocity for a distance of 34.1 m. What is the work done by (a) the tension in the cable and (b) the elevator's weight?

To find the work done by the tension in the cable and the elevator's weight, we can use the formula:

Work = Force × Distance × Cos(θ)

Where:
Work is the amount of work done,
Force is the force applied,
Distance is the distance over which the force is applied,
θ is the angle between the force and the displacement.

(a) Work done by the tension in the cable:

Since the elevator is moving at a constant velocity, the net force acting on it is zero. This means the upward force due to tension in the cable must be equal and opposite to the downward force due to the elevator's weight. Therefore, the angle θ between the force and the displacement is 180°.

The weight of the elevator can be calculated using the formula:

Weight = Mass × Gravitational Acceleration

Given:
Mass (m) = 1050 kg
Gravitational Acceleration (g) = 9.8 m/s^2

Weight = 1050 kg × 9.8 m/s^2
Weight = 10290 N

Now we can calculate the work done by the tension in the cable:

Work = Force × Distance × Cos(θ)
Work = (10290 N) × (34.1 m) × Cos(180°)
Work = -351609 J (negative sign indicates work done against the force of gravity).

The work done by the tension in the cable is -351609 J.

(b) Work done by the elevator's weight:

Since the elevator is moving at a constant velocity, there is no displacement in the direction of the weight force. Therefore, the angle θ between the force and the displacement is 0°.

Now we can calculate the work done by the elevator's weight:

Work = Force × Distance × Cos(θ)
Work = (10290 N) × (34.1 m) × Cos(0°)
Work = 352179 J

The work done by the elevator's weight is 352179 J.

To find the work done by the tension in the cable, we can use the formula:

Work = Force * Distance

In this case, the force is equal to the tension in the cable. Since the elevator is moving at a constant velocity, the net force acting on it is zero. Therefore, the tension in the cable must be equal to the weight of the elevator.

The weight of the elevator can be found using the equation:

Weight = mass * acceleration due to gravity

Given that the mass of the elevator is 1050 kg, and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the weight:

Weight = 1050 kg * 9.8 m/s^2 ≈ 10,290 N

Now, we can calculate the work done by the tension in the cable:

Work = Force * Distance = Weight * Distance

Work = 10,290 N * 34.1 m ≈ 351,489 N·m ≈ 351,489 J

Therefore, the work done by the tension in the cable is approximately 351,489 Joules.

Now, let's calculate the work done by the elevator's weight. Since the elevator is moving at a constant velocity, the net force acting on it is zero. Therefore, the work done by the elevator's weight is also zero. This is because work is defined as the product of force and distance, and in this case, the force and displacement are perpendicular to each other (the weight acts vertically downward, while the displacement is horizontal), resulting in zero work.

Therefore, the work done by the elevator's weight is zero.