Foundation course - PHYSICS
Lecture 6-1: Energy, work and power
Dynamics of solid bodies
• Kinetic energy
• Work
• Power
• Potential energy
• Conservation of mechanical energy
Naďa Špačková spackova@physics.muni.cz
Kinetic energy
Kinetic energy:
• energy associated with the state of motion of an object
• the object is moving faster, its kinetic energy increases
• kinetic energy of an stationary object is zero
• kinetic energy is always positive
17        1 2
Ei = — m v
Units of energy: 1 joule = 1 J = 1 kg.m2.s2
Example:
m = 3 kg
m = 1500 kg
E„ = 6 J
'K
2 mIs
E„ = 300 000 J
20m/s
'K
Work
Work:
• if an object accelerates to a greater speed by applying a force to this object, its kinetic energy increases (and vice versa)
• work W is the energy transferred to or from an object by means of a force acting on the object a
		F
		
X
A constant force exerted on an object results to acceleration of the object:
Fx = max
If acceleration is non-zero, velocity changes:    v = v0 + axt
Distance increases:    d = vnt + — at2
o 2
Work
			F X ^		
					
d
Acceleration, velocity, and distance are related by the equation:
v2 = vo + 2M
Work
This component does no work.
r
Wire
x
Bead
This component does work.
Fg does no work
(9 = 90°)
Only the force component along the object's displacement does the work.
Work is the scalar (dot) product of F and d
W = F d
W = Fdcosd
Work
Work done by many forces:
forces perpendicular to the direction of
motion do no work
(6 = 90°, cos 6 = 0, IV = 0)
FA is in the direction of motion:
e = o°, cos e = i, wA = fa d
FB is in the opposite direction of motion: 6 = 180°, cos 6 = -1, WR = -F d
d = 1 m
FA =3N
WA = 3 J
FB = -1.5 N WB = -1.5 J
The total work done on the box is: W — W + W — F d — F d
Work done when forces change:
Work is equal to the area under the force-displacement graph
CD O
constant force
CD O
Displacement (m)
changing force
Displacement (m)
Kinetic energy
Fd — —mv — — mv0     Work - kinetic energy theorem:
W = AEk
Work is the transfer of energy that occurs when a force is applied through a displacement.
Work on a system is equal to the change in the system's energy.
• the external world does work on a system - work is positive - energy of the system increases
• the system does work on the external world - work is negative - energy of the system decreases
Energy associated with motion = kinetic energy
Energy associated with changing position = translational kinetic energy
Work done by the gravitational force
f T
m9 V
#k,f
U
Work Wg done on an object by the gravitational force:
Wg — mgd cos 0
Rising object: 9 = 180°, cos 0 = -1: Wg = - m g d Falling object: 0 = O°,cos0 = +1:   W =+mgd
Work done in lifting and lowering an object:
AE=E,f. .-E...t.=W r+W .
k        k,final        k,initial applied gravitational
The object is stationary before and after the lift: AE, = 0
1/1/ + 1/1/ =0
w =-w
a g
upward displacement
does 4-positive work
does negative work
does negative ^ work
does positive m9 work
downward displacement
mg
Example:
A box of mass m = 10 kg is pulled up an incline from rest and is stopped after traveling a distance L = 6 m to a height h = 2.5 m.
(a) What is the work done by the gravitational force Fg?
(b) What is the work done by the tension force T?
Checkpoint question:
A pig has a choice of three frictionless slides along which to slide to the ground. Rank the slides according to how much work the gravitational force does on the pig during the descent, greatest first.
Work done
by a spring force
Spring force: variable force
x=0 F = 0
Block attached to spring
x
0
(a)
	x positive Fx negative p		
			
\-x—*■ 0			
(b)
x negative 7^. positive
U—x— 0
x
relaxed state: F = 0
s
Hooke's law for the spring force:
F=—kd
If the length of the spring is parallel with x axis and x = 0 represents the relaxed position, then:
Fx = -kx
The work done by the spring force:
W=--kx2 2
Power
Power is equal to the work done by a force divided by the time required for the change:
Average power P ;
Instantaneous power P: P =
dW dt
Units of power: 1 watt = 1 W = 1 J.s1 = 1 kg.m2.s3
Instantaneous power P;
If is a particle moving along a straight line and acted on by a constant force:
r • u
P =-= F-v = FvcosO
t
P= 3.6 kW P= 1.8 kW
Negative power. Positive power.
(This force is (This force is
removing energy.)      y supplying energy.)
- /\60° ^
Frictionless —v --j—   y >
Example:
A force of 5.0 N acts on a 15 kg body initially at rest. Compute the work done by the force in
(a) the first second
(b) the second second
(c) the third second
(d) the instantaneous power due to the force at the end of the third second.
Machines and their benefits
Machine:
device that makes tasks easier by changing either the magnitude or the direction of the applied force
Mechanical advantage MA:
• input force F.\ the force exerted by a user on a machine _
• output force F : the force exerted by the machine Ft
A machine can increase force, but it cannot increase energy.
Efficiency
In case of a real machine:   W ^W-      The machine is less efficient.
o I
The efficiency of a machine (in %) is equal to the output work, divided by the input work, multiplied by 100.
An ideal machine has equal output and input work, and its efficiency is 100 percent.
Efficiency:
W
e = —^ x 100
e =
w0	!f0	
	iF	k
MA
IMA
MA
e = —— X 100 IMA
Simple machines
Potential energy
Potential energy is associated with the configuration (arrangement) of a system of objects that exert forces on one another
Negative work done by the
gravitational force
Positive work done by the
gravitational force
Conservative forces:
• gravitational and spring forces are conservative
• friction force is not conservative
The force is conservative. Any choice of path between the points gives the same amount of work.
And a round trip gives a total work of zero.
Potential energy is defined as:
Potential energy
Gravitational potential energy (system body-Earth):
yf = h
^Ep = -Wg
h
y = 0
/±Ep = -W = Epf - Epi = mg{yf-y
At a reference point y. = 0 we set E = 0
-E t = mg[y-y
Potential energy
X
SWT-
Elastic potential energy
E nf E „; - /C Xf /C X;
pt      Pi    2    ' 2
At a reference point x(. = 0 we set E = 0:
Example:
A 2.0 kg block of cheese slides along a frictionless track from point a to point b. The cheese travels through a total distance of 2.0 m along the track, and a net vertical distance of 0.80 m. How much work is done on the cheese by the gravitational force during the slide?
The gravitational force is conservative. Any choice of path between the points gives the same amount of work.
Energy: kinetic and potential
1 ?
Translational kinetic energy: EK — — mv Gravitational potential energy:   Ep — mgh
GPE... gravitational potential energy, KE... kinetic energy
Reference levels
Reference Level at Juggler's Hand
Reference Level at Highest Point
Beginning
Middle
Reference level
End
Beginning
0 +
-h
Reference level
Middle
End
KE GPE
KE
The total mechanical energy of the system in each situation is constant.
Conservation of mechanical energy
Mechanical energy of the system is equal to the sum of the kinetic energy and potential energy of the system's objects.
E = Ek + Ep
2.00 m
1.00 m
0.00 m
Mechanical Energy
v2
Ball starts falling. Ball has fallen 1.00 m. Ball hits the ground.
20.0 J 20.0 J 20.0 J 20.0 J   20.0 J
10.0 J   10.0 J
+ 0.0 J = M        M + M = 0.0 J +
GPE     KE        ME       GPE      KE      ME GPE      KE ME
1
AEk = W
AEp = -W
AEk = -AEp
Er2   Ekl —   (Ep2 Epl
Conservation of mechanical energy:
Ek2 + Ed2 — Ekl + E -
In an isolated system, where only conservative forces cause energy changes, the kinetic energy and potential energy can change, but their sum, the mechanical energy remains constant.
Conservation of mechanical energy
A B C D E
Position
Checkpoint question:
The figure shows four situations - one in which an initially stationary block is dropped and three in which the block is allowed to slide down frictionless ramps.
(a) Rank the situations according to the kinetic energy of the block at point B, greatest first.
(b) Rank them according to the speed of the block at point B, greatest first.
Example:
A child of mass m is released from rest at the top of a water slide, at height h = 8.5 m above the bottom of the slide. Assuming that the slide is frictionless because of the water on it, find the child's speed at the bottom of the slide.
The total mechanical energy at the top — is equal to the total at the bottom. -
Summary
Work:
Kinetic energy:
Potential energy:
W = F d
W = AE,
AE =-W
Gravitational potential energy:	*Ep = -Wg		Epiy) = mgh
			
Elastic potential energy:	AEp = -Ws		EP{x) = ^kx2
Conservation of mechanical energy:
Ek2 + Ev2 — Ekl + E x