Work done in lifting an object
y=h A force F lifts the mass at constant speed
through a height h.
F
The displacement is h
a=0
The applied force in the direction of the
y=0 displacement is:
F = mg (no acceleration)
mg
The work done by the force F is:
W = Fh = mgh
But the kinetic energy has not changed the gravity force mg
has done an equal amount of negative work so that the net work
done on the mass is zero
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Work done in lifting an object
y = h Alternative view: define a different form of energy
Gravitational potential energy, PE = mgy
F
a=0 Define:
Mechanical energy = kinetic energy + potential energy
y=0
Mechanical energy, E = mv2/2 + mgy
mg
Then:
Work done by applied force, F, is (change in KE) + (change in PE)
So W = Fh = KE + PE
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Check, using forces and acceleration
y=h v
Net upward force on the mass is F mg
F
Apply Newtons second law to find the acceleration:
a
F mg = ma,
y=0 vo
so, a = (F mg)/m
mg One of famous four equations
v2 = v20 + 2ah
So, v2 = v20 + 2(F mg)h/m
(m/2)
mv2/2 mv20/2 = Fh mgh That is, DKE = Fh DPE
Or, W = Fh = DKE + DPE
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Or, W = Fh = DKE + DPE
If there is no external force, W = 0 and
0 = DKE + DPE
so that DKE = DPE
As the mass falls and loses potential energy, it gains an equal
amount of kinetic energy.
Potential energy is converted into kinetic.
Energy is conserved overall.
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6.26 m = 0.6 kg, yo = 6.1 m
Ball is caught at y = 1.5 m
mg
a) Work done on ball by its weight?
Weight force is in same direction as the displacement so,
Work = mg displacement = 0.6g (6.1 - 1.5 m) = 27 J
b) PE of ball relative to ground when released?
PE = mgyo = 0.6g (6.1 m) = 35.9 J
c) PE of ball when caught?
PE = mgy = 0.6g (1.5 m) = 8.8 J
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d) How is the change in the balls PE related to the work done by
its weight?
Change in PE = mg(y - yo) (final minus initial)
Work done by weight = mg (displacement) = mg(yo - y) = PE
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Conservation of Mechanical Energy
In the absence of applied forces and friction:
Work done by applied force = 0
So, 0 = (change in KE) + (change in PE)
And KE + PE = E = constant
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Example:
No applied (i.e. external) forces
E = KE + PE = constant
KE = mv2/2
PE = mgy
So E = mv2/2 + mgy = constant, until the ball hits the ground
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Check:
vx = v0 cos q = constant, in absence of air resistance
v2y = v20y 2g(y y0) = (v0 sin q)2 2g(y y0) when object is at height y
v2 = v2x + v2y = (v0 cos q)2 + (v0 sin q)2 2g(y y0)
v2 = v20 2g(y y0), as sin2 q + cos2 q = 1
(m/2)
So, mv20/2 + mgy0 = mv2/2 + mgy and KE + PE = constant
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6.34
Find the maximum
height, H.
Ignore air resistance.
Conservation of mechanical energy: KE + PE = constant
At take-off, set y = 0: E = mv20/2 + 0
At highest point, y = H: E = mv2/2 + mgH
So, E = mv20/2 = mv2/2 + mgH
(v20 v2)/2 (142 132)/2
H= = = 1.38 m
g 9.8
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6.38 v = 0 at highest point
y=
= yo
Find the speed of the particle at A (vo). There is no friction.
Conservation of mechanical energy: E = KE + PE = constant
At A: E = mv20/2 + mgy0 = mv20/2 + 3mg
At highest point: E = KE + mgy = 0 + 4mg
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At A: E = mv20/2 + mgy0 = mv20/2 + 3mg
At highest point: E = KE + mgy = 0 + 4mg
So, E = mv20/2 + 3mg = 4mg
mv20/2 = mg
!
v0 = 2g = 4.43 m/s
What happens at B doesnt matter, provided there is no loss of energy
due to friction!
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Conservative Forces
Gravitational potential energy depends only on height
The difference in PE, mg(ho - hf) is independent of path taken
Gravity is a conservative force
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Conservative Forces
Alternative definitions of conservative forces:
The work done by a conservative force in moving an object is
independent of the path taken.
A force is conservative when it does no net work in moving an
object around a closed path, ending up where it started.
In either case, the potential energy due to a conservative force
depends only on position.
Examples
Gravity
Elastic spring force
Electric force
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Non-conservative Forces
The work done by a non-conservative force depends on the path.
Friction the longer the path taken, the more the (negative)
work done by the friction force.
Defining a potential energy relies on a conservative force so that
the work done in moving an object from A to B depends only on
the positions of A and B.
Examples of non-conservative forces
Static and kinetic friction forces
Air resistance
Tension, or any applied force
Normal force
Propulsion force in a rocket
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