A 1000 kg elevator accelerates upward at 1.0 m/s² for 10 m, starting from rest. How much work does the tension in the elevator cable do on the elevator?

Verified step by step guidance

1. Step 1: Identify the forces acting on the elevator. The tension in the cable (T) is responsible for lifting the elevator upward, while the gravitational force (mg) acts downward. The net force (F_net) is responsible for the upward acceleration.
2. Step 2: Use Newton's second law to calculate the tension in the cable. The net force is given by \( F_{\text{net}} = ma \), where \( m \) is the mass of the elevator and \( a \) is the acceleration. The tension in the cable can be expressed as \( T = F_{\text{net}} + mg \). Substitute \( m = 1000 \ \text{kg} \), \( a = 1.0 \ \text{m/s}^2 \), and \( g = 9.8 \ \text{m/s}^2 \) into the equation.
3. Step 3: Calculate the work done by the tension force. Work is defined as \( W = F \cdot d \cdot \cos \theta \), where \( F \) is the force, \( d \) is the displacement, and \( \theta \) is the angle between the force and displacement vectors. Since the tension force and displacement are both upward, \( \cos \theta = 1 \). Substitute \( T \) from Step 2 and \( d = 10 \ \text{m} \) into the equation.
4. Step 4: Simplify the work equation. Combine the values for \( T \) and \( d \) to express the work done by the tension force. Ensure the units are consistent throughout the calculation.
5. Step 5: Interpret the result. The work done by the tension force represents the energy transferred to the elevator to overcome gravity and provide upward acceleration over the 10 m displacement.

Key Concepts

Newton's Second Law of Motion

Newton's Second Law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration (F = ma). In this scenario, the elevator's upward acceleration requires a net force that must overcome both the gravitational force acting downward and provide the necessary upward force for acceleration.

Work-Energy Principle

The Work-Energy Principle states that the work done on an object is equal to the change in its kinetic energy. In the case of the elevator, the work done by the tension in the cable must account for both the work against gravity and the work that contributes to the elevator's upward acceleration, affecting its kinetic energy.

Gravitational Force

Gravitational force is the force exerted by the Earth on an object, calculated as the product of the object's mass and the acceleration due to gravity (approximately 9.81 m/s²). For the elevator, this force acts downward and must be considered when calculating the total tension in the cable, as it influences the net force required for the upward acceleration.