WORK, POWER, ENERGYGrade 12
Work done by a constant force
When the point at which a force acts moves, the force is said to have done work.
When the force is constant, the work done is defined as the product of the force and distance moved, in the direction of the force.
Work is a scalar quantity and is measured in Joules (J).
Work done = Force x displacement in the direction of the force
Example 1:How much work is done when a force of 5 kN moves its point of application 600mm in the direction of the force.
W = F.Δxcosθ
W = 5000 x 0,6 x cos0
= 3000 J
Example 2:A 20 N force pulls a 2 kg block across a smooth floor as shown below. How much work is done by this force to move it 3 m right?
Note:
1.With no friction this work done will result in an equivalent gain in kinetic energy.
2.If there was friction (and it still moved) then this work done who equal the sum of the gain in kinetic energy and the heat generated at the surface.
Net Work: Sometimes more than one force acts at the same time. We call the work done after taking all the forces into account thenet work done.
Example:A 5 N force acts to the right on an object on a horizontal rough surface. The frictional force is 2 N. The object moved for 2 metres. Calculate the new work done on the object.
There are two equivalent approaches we can adopt to finding the net work done on the object. We can:
- Approach 1:calculate the work done byeach force individuallyand then sum themtaking the signs into account. (Taking right as positive)
W1=F1.∆x cos = (5).(2)(cos0) = 10 J
W2=F2.∆x cos = (2).(2)(cos180) = -4 J
Wnet = W1 + W2
= 10 + (-4) = 6 J
- Approach 2:calculate the resultant force from all the forces acting and calculate the net work done using the resultant force.
Fnet = F1 + F2 = 5 + (-2) = 3 N right
Wnet = Fnet.∆x cos = (3).(2)(cos0) = 6 J
Note: The 5N force does positive work on the object and the 2N force does negative work on the object.
Using the equation for WORK in the data sheet.
W = FΔxcosθ
1. If on the horizontal then θ = 0 or 180 and so cosθ = +1 or -1 for all horizontal calculations involving the equation. YOU MUST PUT IN θ = 0 or 180
2. If on a slope then the work done by or against the weight of the object can be calculated using the component of the weight acting down the slope. This is always Fg.sinθ
Work done by weight = F.∆x.cosθ
= (Fg,sinθ).∆x.cos0
Work done to lift it up slope against weight = (Fg,sinθ).∆x.cos180
3. If pulled (or pushed) at an angle
Then work done horizontally:
W = FΔxcosθ
(The examiner who chose this equation chose it for this situation.)
WORK-ENERGY THEOREM
The work done by a net force on an object is equal to the change in its kinetic energy.(most common example)
ΔEk = Wnet = Fnet.Δx.cosθ
NOTE:- the work done by any force results in a change in energy.
-work done against friction results in heat being formed.
-work done by a gravitational pull results in a loss in potential energy.
-work done against a gravitational pull results in a gain in potential energy.
Example 3:What will be the speed of a 2 kg block if it is lifted by a rope to a height of 5 m by a force of 25,4 N? (Assume to air resistance.)
25,4N
Up positive (Fg = mg = 2 x -9,8 = -19,6 N)
ΔEk = Wnet = Wg + WT
= Fg∆xcos180 + FT∆xcos0
= (19,6 x 5 x -1) + (25,4 x 5 x 1)
= 29 J
ΔEk = ½ mvf2 – ½ mvi2
19,6 N29 = ½ x 2 x vf2 - 0
v = 5,39 m.s-1
Example 4:What is the gain in kinetic energy when a 4000 N force lifts a crate of 160 kg through a distance of 12m if the average frictional force (including air resistance) is 600 N?
Conservative Forces:
A conservative force is one for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken.
(Work is independent of the route.)
Example: Gravitational force.
If a 5 kg object falls down a vertical height of 2 m by two different means – falling straight down and moving at an angleof 30oto the horizontal down a frictionless slope.
2 m∆x = 2/sin30 = 4 m
30
Non-conservative forces: A non-conservative force is one for which work done on the object depends on the path taken by the object.
Example: Friction
Note: When non-conservative forces are present mechanical energy is not conserved, but total energy (of the system) is still conserved. Energy is simply converted to a form from which it cannot be retrieved (eg. heat).
The transferred energy will be equal to: Wnc = ∆Ek + ∆Ep
Example: A sphere of mass 2 kg is shot vertically upwards from ground level and reaches a maximum height of 4 m above the ground. A constant frictional force of 5,4 N acts on the sphere throughout its motion. Calculate the initial velocity of the sphere.
ENERGY
Energy is the capacity to do work.
For example:- water in a reservoir is said to possesses energy as it could be used to drive a turbine lower down the valley.
There are many forms of energy e.g. electrical, chemical heat, nuclear, kinetic, gravitational potential etc.
The SI units are the same as those for work, Joules J.
Mechanical energy
The mechanical energy of a body in the sum of its kinetic and potential energy.
Mechanical energy = EK + Ep
Kinetic energy
Kinetic energy may be described as energy due to motion.
The kinetic energy of a body may be defined as the amount of work it can do before being brought to rest.
Potential Energy
There are different forms of potential energy. Two examples are: i) a pile driver raised ready to fall on to its target possesses gravitational potential energy while (ii) a coiled spring which is compressed possesses an internal potential energy.
Only gravitational potential energy will be considered here. It may be described as energy due to position relative to a standard position (normally chosen to be he earth's surface.)
The potential energy of a body may be defined as:the amount of work it would do if it were to move from its current position to the standard position.
Potential energy cont………..
Note: gravitational field strength is the force per unit mass which a body experiences at its position. On the surface of the earth it is about 9,8 N.kg-1.
FOR A BODY FALLING OR RISING UNDER IDEAL CONDITIONS (conservative forces only), MECHANICAL ENERGY IS CONSERVED.
Example 1:A lead shot of mass 2 kg was dropped from a height of 18 m above the earth. What was its kinetic energy when it was 4 m from the ground?
Mechanical E (4m) = Mechanical E (top) (ideal conditions)
mgh + K = mgh + ½ mv2
2 x 9,8 x 4 + K = 2 x 9,8 x 18 + 0
K = 352,8 – 78,4
= 274,4 J
Example 2:A 500 g ball rolls 5 m down to the bottom of a slope which is 2m high vertically. If the average frictional force on the ball is 0,8 N, what is its speed at the bottom of the slope?
Mechanical E (top) + Wf = Mechanical E (bottom)
mgh + ½ mv2+ Ff∆xcos180 = mgh + ½ mv2
(0,5 x 9,8 x 2) + 0 + (0,8 x 5 x -1) = 0 + ( ½ x 0,5 x v2)
v2 = (9,8 – 4)/0,25
v = 4,47 m.s-1
Example 3:A 0,45 kg ball is projected up at 30 m.s-1. On the way up it passes through some branches and loses 50 J of mechanical energy. How high will it go? (Ignore air resistance.)
The Simple Pendulum
When a pendulum swings potential energy is converted to kinetic energy and back to potential energy. The energy change can be regarded as ideal. The potential energy at the top of the swing is equal to the kinetic energy at the bottom.
U + K (top) = U + K (bottom)
m.g.h + 0 = 0 + ½ m.v2 the mass cancels so….
g.h = ½ v2
v(bottom) = √(2gh)
Example:A bob is dropped from a height of 20 cm. What was its speed at the bottom of the swing?
Kinetic energy and momentum
These may seem the same but there are distinct differences:
1.Momentum is ALWAYS conserved in collisions. Kinetic energy is only conserved in elastic collisions.
2.Momentum is gained as a force acts on a body for a time. Kinetic energy is gained as a force acts on a body over a distance.
Conservation of energy:The principle of conservation of energy state that the total energy of a system remains constant. Energy cannot be created or destroyed but may be converted from one form to another.
POWER
Power is the rate at which work is done.
Another way of saying this is that power is the rate at which energy is changed. This means that when we say a car is a powerful we mean that it can convert chemical energy (from petrol) to other energy (usually kinetic) very fast.
Power rating:-if a bulb is rated 100W.240V it means that if you place a voltage of 240 V across it then it will convert 100 J of electrical energy to light every second.
Example 1:What is the power of a winch if it can drag a 500 kg tree over 30 m at constant speed in 20 s against a frictional force of 2400 N?
If work is being done by a machine moving at velocity v against a constant force, or resistance, F, then since work done is force multiplied by the displacement, work done per second is F.v, which is the same as power.
P = W/Δt = (F.Δx)/Δt = F.(Δx/Δt) = F.v
Example 2:What is the maximum speed a 54 kW car can go against a frictional force of 2000N?
Example 3:What will the final speed of a 20 kg block be if it is lifted from rest to a height of 50 m by a winch with a power rating of 500 W in half a minute )operating at maximum power the whole time)? Assume 5090 J of energy are lost due to various frictional forces and air resistance and only use power, work and energy equations.