 When we calculated the work of a force, we did not take into account how much time the work was done, e.g. in what time did we accelerate a car to a certain speed and give it kinetic energy or lift a load to a height and give it potential energy or tension a spring with a force constant and give it elastic energy.

Now we will also consider time. For this purpose, we introduce the physical quantity called power. The greater the power e.g. of a machine, the greater the amount of work it does in a given time.

Power is therefore defined as the work done per unit time: The power is denoted by and its SI unit is Watt ( ). From the equation, the unit of power is: which we read as "Joule per second".

From the equation for power, let's make work the subject: the unit of work, expressed in terms of the unit of power ( ), is: Using the knowledge from the chapter, work of a force, the equation for the work of a force can be written in two forms: and consequently, the unit of work is: Power is work per unit time: the unit of power is: which we read as: Watt, Joule per second, and Newton metre per second.

The work is: and its unit is: which we read as: Joule, Watt second, Newton metre.

## Power in a linear motion

If a constant force acts on a body during a linear motion, it moves with uniform acceleration. The Force therefore performs work and with the distance travelled, the body's velocity increases (or decreases - depending on the direction of the force), and thus its kinetic energy.

When a body moves a distance in a short time interval , the work done is given as: We divide both sides of the equation by : We note that: and the instantaneous speed is given as:  In a straight-line motion, the power at constant force depends on the distance and time . At the moment of observation, when the body reaches velocity , the power is: Example

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## Power in rotational motion

In rotation, we assume that a constant torque acts on the body. The angular velocity of the body increases uniformly - see the theory, Torque, if the torque acts in the direction of rotation, decreases uniformly if it acts in the opposite direction, or remains constant if the torque is zero.

When the body rotates through an angular displacement in a short time interval , the work done is given as: We divide both sides of the equation by : We note that: and from the chapter, circular motion, we know that:  Power in rotational motion is the product of torque and angular velocity : Example

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### Pulley

A pulley (or gear) transfers the rotational power from the first roller to the second roller. In doing so, the angular speed (or frequency) of rotation of the second roller and the torque can be changed.

The belt connects two parallel rollers, with radii and as shown in the figure below:

Let an electric motor with power be connected to the first roller. The power is transmitted via a pulley to the second roller. The first roller rotates with angular velocity , and the second with . Therefore:

• the force along the belt is the same everywhere as it is the internal force of the belt.

If the statement were not true, there would have to be a point on the belt - let it be point T - where the force just before point T would be different from the force at point T. Since there would be two different forces acting on the same sections of the belt, we would see a section of the belt a little before point T, moves faster (or slower) than the section at point T. But when the belt rotates, we do not notice this, so the force along the belt must always be the same.

• The angular velocities of both rollers are inversely proportional to the radius.

Let's prove this statement by first realizing that the speed of the belt is equal to the tangential speed of the two rollers (if the belt does not slip). It follows from this that: We note from circular motion that:  We rearrange: • The power is the same on both rollers.

This statement can be shown using the formula of power which is given as: The force is the same on both rollers, as is the tangential speed . It follows that the powers are the same.

• The torque on both rollers is proportional to the radius of the rollers and inversely proportional to the angular velocities (or frequencies) of both rollers.

Let's prove the statement by first writing out the torque equations that apply to both rollers:  We divide the equation and and note that : We note from above that:  Example

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## Power and energy

In the chapter, elastic energy, the energy equation was given as: We divide both sides of the equation by : We note that:  The sum of all energies is the total energy of the body. In total energy, for now, we only consider kinetic, potential, and elastic energy: and write simply: Power tells how fast the total energy of a body changes and is given as: where is the sum of kinetic, potential, and elastic energy.

Example

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