Calculating Work Done by a Force Acting at an Angle

When a force acts on an object, and the object is displaced, the work done by the force can be calculated using the formula ( W Fd cos theta ), where ( F ) is the force, ( d ) is the displacement, and ( theta ) is the angle between the force and the direction of displacement.

Example: Force at 60 Degrees to Displacement

Consider a scenario in which a force of 50 Newtons (N) acts on a body, displacing the body through a distance of 10 meters (m) in a direction making an angle of 60 degrees with the force.

To find the work done, let's apply the formula:

Given data:

Force, ( F 50 , text{N} ) Displacement, ( d 10 , text{m} ) Angle, ( theta 60^circ )

Substituting the values into the formula:

Work done, ( W 50 times 10 times cos 60^circ )

( W 500 times cos 60^circ )

( cos 60^circ 0.5 )

( W 500 times 0.5 250 ) Joules (J)

Note: The given values in the problem might have slightly different figures, but the methodology remains the same.

Breaking Down the Force into Components

If the force is not aligned directly with the direction of displacement, it is necessary to break the force into its components. The work done is only the component of the force that is directly in line with the displacement.

The component of the force in the direction of the displacement can be calculated using trigonometry:

Component of force in the direction of displacement ( F cos theta )

In the given problem:

Component of force 50 N (cos 60^circ) 50 N (times 0.5) 25 N

Work done 25 N (times) 10 m 250 J

Understanding Work Done

Work is defined as a force that acts through a distance. It is only performed along the axis or path along which the body actually moves. Hence, we only consider the component of the force that is parallel to the direction of motion. The perpendicular component of the force does not contribute to the work done and often results in heat or friction.

Example:

Work is ( Fd cos theta ) If the force is perpendicular to the direction of motion, ( theta 90^circ ) and ( cos 90^circ 0 ), so no work is done. If the force is parallel to the direction of motion, ( theta 0^circ ) and ( cos 0^circ 1 ), the work is equal to the product of force and displacement.

In the context of the problem, the angle given is 60 degrees, and trigonometric methods are used to determine the component of the force that is useful in calculating the work done.

Conclusion

Understanding and applying the formula ( W Fd cos theta ) is crucial for solving work-related problems in physics. By breaking the force into components, we can accurately calculate the work done by a force acting at an angle to the direction of displacement.