Work, Power and Energy

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Worked Examples in this section

Example 3.9
Example 3.10
Example 3.11
Moment, couple and torque

The moment of a force F about a point is its turning effect about the point.

It is quantified as the product of the force and the perpendicular distance from the point to the line of action of the force.

Figure 3.4: Moment of a force

In Figure 3.5 the moment of F about point O is

Equation 3.8
A couple is a pair of equal and parallel but opposite forces as shown in Figure 3.6:

Figure 3.6: A couple

The moment of a couple about any point in its plane is the product of one force and the perpendicular distance between them:

Equation 3.9
Example of a couple include turning on/off a tap, or winding a clock.

The SI units for a moment or a couple are Newton metres, Nm.

In engineering the moment of a force or couple is know as torque. A spanner tightening a nut is said to exert a torque on the nut, similarly a belt turning a pulley exerts a torque on the pulley.

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Work done by a constant torque

Let a force F turn a light rod OA with length r through an angle of q to position OB, as shown in Figure 3.7.

Figure 3.7: Work done by a constant torque

The torque TQ exerted about O is force times perpendicular distance from O.

Equation 3.10

 

Now work done by F is

s is the arc of the circle, when qis measure in radians

Equation 3.11

 

The work done by a constant torque TQ is thus the product of the torque and the angle through which it turns (where the angle is measured in radians.)

As the SI units for work is Joules, TQ must be in Nm
 
 

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Power transmitted by a constant torque

Power is rate of doing work. It the rod in Figure 3.7 rotates at n revolutions per second, then in one second the angle turned through is

radians, and the work done per second will be, by Equation 3.11

as angular speed is

then

Equation 3.12
The units of power are Watts, W, with n in rev/s, w in rad/s and TQ in Nm.
 
 

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Worked Example 3.9

A spanner that is used to tighten a nut is 300mm long. The force exerted on the end of a spanner is 100 N.

  1. What is the torque exerted on the nut?
  2. What is the work done when the nut turns through 30° ?
Solution

a)

Calculate the torque by Equation 3.10

b)

Calculate the work done by Equation 3.11

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Worked Example 3.10

An electric motor is rated at 400 W. If its efficiency is 80%, find the maximum torque which it can exert when running at 2850 rev/min.

Solution

Calculate the speed in rev/s using Equation 3.12

Calculate the power as the motor is 80% efficient


 
 

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Work done by a variable torque

In practice the torque is often variable. In this case the work done cannot be calculated by Equation 3.11, but must be found in a similar way to that used for a variable force (see earlier.)

Figure 3.8: Work done by a variable torque

The work done when angular displacement is dqis TQdq. This is the area of the shaded strip in Figure 3.8. the total work done for the angular displacement q is thus the area under the torque/displacement graph.

For variable torque

Equation 3.13
As with variable forces, in general you must uses some special integration technique to obtain the area under a curve. Three common techniques are the trapezoidal, mid-ordinate and Simpson's rule. They are not detailed here but may be found in many mathematical text book.
 
 

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Worked Example 3.11

A machine requires a variable torque as shown in Figure 3.9, Find:

  1. The work done per revolution
  2. The average torque over one revolution
  3. The power required if the machine operates at 30 rev/min

Figure 3.9: Torque requirement for Worked Example 3.12

Solution

a)

From Equation 3.13

for one revolution

b)

Average torque is the average height of figure OABCDE = area /2p

c)

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