Math Problem Statement

Astronauts in space move a toolbox from its initial position m to its final position m. The two astronauts each push on the box with a constant force. Astronaut 1 exerts a force N and astronaut 2 exerts a force N. What is the total work performed on the toolbox?

Solution

To calculate the total work performed on the toolbox, we use the formula for work:

W=Fdcos(θ)W = F \cdot d \cdot \cos(\theta)

Where:

  • WW is the work done,
  • FF is the total force exerted on the object,
  • dd is the displacement of the object,
  • θ\theta is the angle between the force and the direction of the displacement.

Here’s what we need to know to calculate the total work:

  1. The forces applied by the two astronauts.
  2. The displacement (initial position and final position of the toolbox).
  3. The angle at which the forces are applied relative to the direction of motion (if the forces are in the same direction as the displacement, θ=0\theta = 0^\circ and cos(0)=1\cos(0^\circ) = 1).

If you provide the missing information:

  • The magnitude of the forces (F1F_1 and F2F_2),
  • The displacement (dd),
  • The angles at which the forces are applied (or if they’re in the direction of the displacement),

I can help you calculate the total work done by the astronauts.

Do you have any further details, or would you like clarification?


Here are 5 related questions that expand on the concept of work and force:

  1. How do forces applied at an angle affect the work done on an object?
  2. What happens if the forces are not constant during displacement?
  3. How do you calculate work if one of the astronauts is pushing in the opposite direction?
  4. What is the difference between positive work and negative work?
  5. How does work-energy theorem relate to the total work done on an object?

Tip: When forces are applied at angles to displacement, only the component of the force in the direction of motion contributes to the work.

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Math Problem Analysis

Mathematical Concepts

Physics
Work and Force
Vector Decomposition

Formulas

W = F · d · cos(θ)

Theorems

Work-Energy Theorem

Suitable Grade Level

Grades 10-12