Work

What happens when the applied force and the motion are not parallel? If there is an angle between the direction of motion and the applied force then to determine the work done we have to calculate the component of the applied force parallel to the direction of motion. Note that this means a force perpendicular to the direction of motion can do no work.

Calculate the work done on a box, if it is pulled 5 m along the ground by applying a force of $F$ =10 N at an angle of ${60}^{\circ }$ to the horizontal.

1. Analyse the question to determine what information is provided
   • The force applied is $F$ =10 N
   • The distance moved is $s$ =5 m along the ground
   • The angle between the applied force and the motion is ${60}^{\circ }$

   These quantities are in the correct units so we do not need to perform any unit conversions.

2. Analyse the question to determine what is being asked
   • We are asked to find the work done on the box.
3. Calculate the component of the applied force in the direction of motion

   Since the force and the motion are not in the same direction, we must first calculate the component of the force in the direction of themotion.

   

   From the force diagram we see that the component of the applied force parallel to the ground is

   $$\begin{array}{ccc}\hfill F_{\left|\right|}& =& F·cos\left({60}^{\circ }\right)\hfill \\ & =& 10\mathrm{N}·cos\left({60}^{\circ }\right)\hfill \\ & =& 5\mathrm{N}\hfill \end{array}$$
4. Substitute and calculate the work done

   Now we can calculate the work done on the box:

   $$\begin{array}{ccc}\hfill W& =& F_{\parallel }s\hfill \\ & =& \left(5\mathrm{N}\right)\left(5\mathrm{m}\right)\hfill \\ & =& 25\mathrm{J}\hfill \end{array}$$

   Note that the answer is positive as the component of the force $F_{\parallel }$ is in the same direction as the motion.

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Work

1. A 10 N force is applied to push a block across a friction free surface for a displacement of 5.0 m to the right. The block has a weight of 20 N. Determine the work done by the following forces: normal force, weight, applied force.

   

2. A 10 N frictional force slows a moving block to a stop after a displacement of 5.0 m to the right. The block has a weight of 20 N. Determine the work done by the following forces: normal force, weight, frictional force.

   

3. A 10 N force is applied to push a block across a frictional surface at constant speed for a displacement of 5.0 m to the right. The block has a weight of 20 N and the frictional force is 10 N. Determine the work done by the following forces: normal force, weight, frictional force.

   

4. A 20 N object is sliding at constant speed across a friction free surface for a displacement of 5 m to the right. Determine if there is any work done.

   

5. A 20 N object is pulled upward at constant speed by a 20 N force for a vertical displacement of 5 m. Determine if there is any work done.

   

6. Before beginning its descent, a roller coaster is always pulled up the first hill to a high initial height. Work is done on the roller coaster to achieve this initial height. A coaster designer is considering three different incline angles of the hill at which to drag the 2 000 kg car train to the top of the 60 m high hill. In each case, the force applied to the car will be applied parallel to the hill. Her critical question is: which angle would require the least work? Analyze the data, determine the work done in each case, and answer this critical question.

   Angle of Incline	Applied Force	Distance	Work
   35 ${}^{\circ }$	$1.1\times {10}^4\mathrm{N}$	100 m
   45 ${}^{\circ }$	$1.3\times {10}^4\mathrm{N}$	90 m
   55 ${}^{\circ }$	$1.5\times {10}^4\mathrm{N}$	80 m

7. Big Bertha carries a 150 N suitcase up four flights of stairs (a total height of 12 m) and then pushes it with a horizontal force of 60 N at a constant speed of 0.25 m $·$ s ${}^{-1}$ for a horizontal distance of 50 m on a frictionless surface. How much work does Big Bertha do on the suitcase during this entire trip?
8. A mother pushes down on a pram with a force of 50 N at an angle of 30 ${}^{\circ }$ . The pram is moving on a frictionless surface. If the mother pushes the pram for a horizontal distance of 30 m, how much does she do on the pram?

   

9. How much work is done by an applied force to raise a 2 000 N lift 5 floors vertically at a constant speed? Each floor is 5 m high.
10. A student with a mass of 60 kg runs up three flights of stairs in 15 s, covering a vertical distance of 10 m. Determine the amount of work done by the student to elevate her body to this height. Assume that her speed is constant.