0.19 Phy1170: energy -- work  (Page 3/7)

Suzie and the heavy trunk

Your college dorm is equipped with a rope on a pulley that can be used to lift heavy objects and swing them into a window on the second floor. Suzie hasused the pulley and the rope to lift her heavy trunk up to the level of the window. She asks you to hold the rope and maintain the trunk at that level whileshe goes up the stairs to pull it in the window.

Suzie likes to gossip

Suzie, however, really likes to gossip. On the way up to the window, she meets some friends and starts talking, leaving you holdingonto the rope. After about 30 minutes, you become really tired and frustrated and you lower the trunk back to ground level.

Did you do any work on the trunk?

The answer is that you did not do any work on the trunk. Although you became very hot and tired standing there and holding the rope, you did not cause adisplacement of the trunk. For 30 minutes, you simply held the rope causing the trunk to remain at the same location. At the end of that period, you carefullyallowed the rope to slip through your hands while the force of gravity caused a downward displacement on the trunk.

Was any work done on the trunk?

Was any work done on the trunk from the time Suzie handed you the rope until the trunk was back on the ground?

Yes, the force of gravity caused the height of the trunk to be displaced from the level of the second floor window downward to the ground, so the force ofgravity did work on the trunk.

Was any work done on the trunk from the time that Suzie started raising it until it was back on the ground?

The answer is no. The net displacement of the trunk over that time period was zero. If the displacement was zero, no work could have been done. In this case,the positive work done by Suzie in lifting the trunk to the second floor window was canceled out by the negative work done by gravity when you allowed the trunkto return to the ground.

Measurement of work can depend on the time interval

As you can see, therefore, the amount of work done in some cases can depend on the time interval over which the measurement is made.

Also, as you can see, certain activities (such as holding a rope to keep a trunk from falling) canmake you very tired even though you are not doing any work.

Mathematical representation of work

As stated earlier, we can represent work as

W = F * d * cos(theta)

where

• F represents a force being applied to an object
• d represents the displacement of the object
• theta is the angle between the displacement path d and the line of action of the force.

Work is a scalar quantity

Although both force and displacement are vector quantities, the product of those two vectors with the cosine of an angle produces a scalar quantity. Workhas a magnitude but no direction. The product is often called a scalar product or a dot product.

The scalar or dot product

The scalar product (or dot product ) of two vectors is defined by theequation

vectorA * vectorB * cos(theta)

where theta is the angle between the two vectors when they are drawn tail to tail.