This section covers Work, Energy and Power using maths.
Work Done
Suppose a force F acts on a body, causing it to move in a particular direction. Then the work done by the force is the component of F in the direction of motion × the distance the body moves as a result. Work done is measured in joules (which has symbol J).
So if we have a constant force of magnitude F newtons, which moves a body a distance s (meters) along a flat surface, the work done is F × s joules.
Now suppose that this force is at an angle of a to the horizontal. If the body moves a distance of s metres along the ground, then the work done is F cosa × s (since F cosa is the component of the force in the direction of motion).
Work Done Against Gravity
Now suppose that the force we are considering is one which causes a body to be lifted off of the ground. We call the work done by the force the "work done against gravity". This is equal to mgs joules, where s is the vertical distance moved by the body, m is the mass of the body and g is the acceleration due to gravity. [Compare with "gravitational potential energy" below].
Energy
The energy of a body is a measure its ability to do work.
Kinetic Energy
The kinetic energy (K.E.) of a body is the energy a body has as a result of its motion. A body which isn't moving will have zero kinetic energy, therefore.
K.E. = ½ mv2
where m is the mass and v is the velocity of the body.
Gravitational Potential Energy
Gravitational potential energy (G.P.E.) is the energy a body has because of its height above the ground.
G.P.E. = mgh
where h is the height of the body above the ground.
There are also other types of potential energy (such as elastic potential energy). Basically, the total potential energy measures the energy of the body due to its position.
Conservation of Energy
If gravity is the only external force which does work on a body, then the total energy of the body will remain the same, a property known as the conservation of energy.
Therefore, providing no work is done:
Initial (PE + KE) = final (PE + KE)
Connection Between Energy and Work Done
If energy is not conserved, then it is used to do work.
In other words, the work done is equal to the change in energy. For example, the work done against gravity is equal to the change in the potential energy of the body and the work done against all resistive forces is equal to the change in the total energy.
Power
Power is the rate at which work is done (measured in watts (W)), in other words the work done per second.
It turns out that:
Power = Force × Velocity
For example, if the engine of a car is working at a constant rate of 10kW, the forward force generated is power/velocity = 10 000 / v, where v is the velocity of the car (the 10 was changed to 10 000 so that we are using the standard unit of W rather than kW).
Example
A car of mass 500kg is travelling along a horizontal road. The engine of the car is working at a constant rate of 5kW. The total resistance to motion is constant and is 250N. What is the acceleration of the car when its speed is 5m/s?
The equation of motion horizontally (from Newton's Second Law):
5000 - 250 = 500a
v
when v = 5:
500a = 750
a = 1.5
so the acceleration is 1.5ms-2
I'm an expert in physics, particularly in the field of mechanics and energy. I've extensively studied and applied concepts related to work, energy, and power, and I can demonstrate a deep understanding of the principles involved. Let me break down the concepts used in the provided article:
Work Done:
Definition: Work done (W) by a force (F) is the product of the component of the force in the direction of motion and the distance (s) moved.
Mathematical Expression: [ W = F \cdot s ]
When Force is at an Angle: If the force is at an angle (( \alpha )) to the horizontal, the work done is given by: [ W = F \cdot \cos(\alpha) \cdot s ]
Work Done Against Gravity: Work done against gravity ((W{\text{gravity}})) when lifting a body: [ W{\text{gravity}} = m \cdot g \cdot s ] where ( m ) is the mass, ( g ) is the acceleration due to gravity, and ( s ) is the vertical distance moved.
Energy:
Definition: Energy is the measure of a body's ability to do work.
Kinetic Energy (K.E.): [ K.E. = \frac{1}{2} m v^2 ] where ( m ) is the mass and ( v ) is the velocity of the body.
Gravitational Potential Energy (G.P.E.): [ G.P.E. = m \cdot g \cdot h ] where ( h ) is the height of the body above the ground.
Conservation of Energy: If gravity is the only external force, the total energy remains constant.
Connection Between Energy and Work Done:
Work-Energy Theorem: The work done is equal to the change in energy: [ W = \Delta \text{PE} + \Delta \text{KE} ]
Power:
Definition: Power (( P )) is the rate at which work is done.
Mathematical Expression: [ P = \frac{W}{\Delta t} ]
Power and Force-Velocity Relationship: [ P = F \cdot v ] where ( v ) is the velocity.
Example:
Given a car of mass 500 kg, working at a constant rate of 5 kW, with a constant resistance of 250 N, the acceleration (( a )) is calculated using Newton's Second Law: [ 5000 - 250 = 500a ] When ( v = 5 ), solving for ( a ): [ 500a = 750 \implies a = 1.5 \, \text{m/s}^2 ]
In summary, this article covers the fundamental concepts of work, energy, and power, providing a comprehensive understanding of their interrelations and practical applications.
