0.19 Phy1170: energy -- work  (Page 5/7)

To do any work, a force must cause a displacement.

More on the stubborn dog

Let's go back and think a little more about the stubborn dog. Assume that you exert a force of 3.46 newtons on the leash and the leash makes an angle of 30degrees with the horizontal floor. This causes the dog to slide 2 meters across the floor, but doesn't lift the dog off the floor. What is the work that you doon the dog?

The horizontal component Fx of the force F is given by

Fx = F*cos(30 degrees), or

Fx = 3.46*cos(30 degrees)

Enter the following into the Google calculator search box:

3.46*cos(30 degrees)

and with rounding, you will learn that the horizontal component of the force that you exerted on the leash is

3 newtons.

How much work was done?

The work done on the dog is the product of the horizontal component of the force that you exert on the leash and thehorizontal displacement distance of 2 meters.

Assuming that you were pulling the leash with a constantforce of 3.46 newtons, at an angle of 30 degrees relative to the horizontal floor, you would do 2*3=6 joulesof work on the dog.

Be careful how you handle the angle theta

The angle theta is the angle between the line of action of the force and the direction of the displacement. It is only because I set the scenario up that waythat the 30-degree angle in the previous stubborn-dog scenario is 30 degrees relative to the horizontal.

In that scenario, the line of action of the 3.46-newton force was at 30 degrees relative to the horizontal, but thedisplacement was horizontal. Therefore, it was necessary to take the angle into account when computing the work done on the dog. In other words, the work doneon the dog was:

W = 3.46N * 2m * cos(30 degrees) = 6 joules

One more discussion about the stubborn dog

Assume that instead of pulling the dog across a horizontal floor, you are pulling the dog up a ramp that has an angle of 30 degrees relative to thehorizontal.

In this case, you pull on the leash at an angle of 30 degrees relative to the horizontal, which is parallel to the ramp. Once again, you pull witha force of 3.46 newtons. This time, you cause the dog to slide 2 meters up the ramp.

In this case, the angle between the direction of the displacement and the line of action of the force is 0 degrees. The cosine of 0 degrees is +1.Therefore, the work done on the dog is

(3.46 N) * 2 m * cos(0 degrees) = 6.92 joules

(You may recognize the format of some of the mathematical expressions in the above paragraphs as being the format of the output of the Google calculator.)

Sample calculations

I will present and explain several scenarios in this section involving force and work.

In some cases, I will use the Google calculator to perform the arithmetic, and the format of the mathematical expressions will be the format of the outputfrom the Google calculator (except that I will usually discard some of the digits to the right of the decimal point).

Single force on a body

A horizontal force

A horizontal force of 10 N towards the right is applied to a 15-kg body causing the body to move 5 m to the right at constant speed. How much work is done on the body?