7.12:
Work Done by Many Forces
Consider a baggage tractor pulling a baggage cart with a force of 10,000 N at an angle of 30 degrees. If the cart offers a friction of 2,000 N while being towed, what is the work done on the cart to displace it by 10 m in a straight line?
A free-body diagram is drawn to get a clear understanding of the forces.
The forces perpendicular to the motion of the cart do not cause displacement of the cart, so no work is done by those forces.
The forces in the x-direction are the component of the pulling force and the frictional force, which are summed vectorially.
The total work done is the resultant force along the displacement times the displacement.
Alternatively, the work done by each force can be first calculated and then summed.
The work done by the perpendicular forces is zero. The pulling force and the frictional force are not perpendicular to the displacement and so contribute to the total work.
7.12:
Work Done by Many Forces
The total work done on an object acted upon by multiple forces can be computed using two methods that give the same result. In one method, the work done by each force is first calculated. Then, those values are summed algebraically to calculate the total work done by all the forces. In the second method, the net force is first calculated by a vector sum of all the forces. Then, the work done by this force is obtained.
Since forces perpendicular to the displacement do no work, they do not contribute to the work done on the object. They can simply be ignored in the calculation in the first method. However, in the second method, they need to be first considered in calculating the net force vector, and only then is their contribution seen to be zero.
If mutually opposite forces act on the object's center of mass, they contribute different positive and negative work in the first method. Through the algebraic summation, the overall work done on the object is found to be positive or negative depending on which of the forces is larger in magnitude. In the second method, the net force is first obtained and is determined to act either along or opposite to the net displacement. Then, the work is obtained and is determined to be either positive or negative depending on its angle with respect to the displacement.
Suggested Reading
- Young, H.D and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson: section 6.1; pages 180–181.