Chapter 5 - Dynamics of Uniform Circular Motion
UCM – Motion of an object travelling at a constant
speed in a circular path with fixed radius
Axis – Center point around which circular motion
occurs
Rotation – Circular motion around an internal axis
Revolution – Circular motion around an external axis
Period – T – Time it takes to make one complete
oscillation.
SI Unit – seconds
What is the period of a car’s engine when the
tachometer reads 2000.rpms?
Circumference – Distance around the outside of a
circle
C = 2pr
Or
C = pd
The “speed” we are actually talking about is called;
Tangential velocity – vt – The velocity of an object
tangent to its circular path.
Does it depend on radius, if so, why?
Absolutely: Radius changes the size of the circle that an object
travels in. For an object to cover a certain distance in a specific
time, it must travel at a certain speed.
For an object in UCM the magnitude of tangential
velocity is constant but its direction is constantly
changing.
If velocity (in this case the direction of it) is changing,
what is occurring?
Centripetal acceleration – ac – acceleration towards a
circle’s center (the word centripetal means moving
towards the center, but why the center?)
ac = vt2/r
Centripetal acceleration = velocity2/radius
If there is an acceleration occurring, what also must be
present?
Centripetal force – Fc – force that causes centripetal
acceleration. It isn’t a new force, just a force cause by
something pulling inward, such as friction, weight, or
tension.
Fc = m ac
Chapter 6 – Work and Energy
Work – done when a force, F, creates a
displacement, s.
If force and displacement are in the same direction,
W=Fs
SI
Force
Distance
Work
CGS
Newton
Dyne
(N)
Meter (m) Centimeter
(cm)
N*m
Dyne*cm
Joule (J)
erg
BE
Pound
(lb)
Foot
(ft)
ft*lb
If force and displacement are not in the same
direction, we need to incorporate an angle.
Where q is the angle that exists between the
force and displacement.
W=Fcosqs
Kinetic Energy – KE – Energy of motion
KE = ½(m)(v2)
SI Unit - Joule
Work - Energy Theorem
When work is done on something, its
kinetic energy changes.
F=ma
F(s) = ma(s)
F(s) = m ((v-vo)/(t)) ((½)(v+vo)(t))
F(s) = m (½) ((v-vo)(v+vo))
F(s) = ½ (m) (v2-vo2)
F(s) = ½mv2- ½mvo2
W = ½mv2- ½mvo2
W = KE – KEo
W = DKE
Gravitational Potential Energy – PE – Energy of
position
(energy based on a position above a reference level)
PE = (m)(g)(h)
SI – Joule
PE is related to the work done by gravity.
Wg = -DPE
Wg = -(PE-PEo)
Conservative Force – A force that does no net
work if there is no displacement, i.e. gravity.
Non-Conservative Force – A force that does
work independent of the path, e.g. friction, air
resistance.
Wtotal = WNonconsv + Wconsv
DKE = WNC + Wg
DKE = WNC - DPE
WNC = DKE + DPE
Conservation of Mechanical Energy
If there is no non-conservative work done, Mechanical
Energy is conserved. (used for finding final or initial
heights or speeds)
Eo = E
KEo+ PEo = KE + PE
½m(vo2) + mgho = ½m(v2) + mgh
½(vo2) + gho = ½(v2) + gh
Non-Conservative Forces and Mechanical
Energy
WNC = DKE + DPE
WNC = (KE – KEo)+ (PE – PEo)
WNC = (½m(v2) - ½m(vo2)) + (mgh - mgho)
Power – Work done per unit time (or the
average rate at which work is done)
P = Work/time = W/t
SI Unit – Joule/second = Watt
CGS – erg/second
BE – ft*lb/second
Power can also be measured in horsepower
1 h.p. = 550ft*lbs/second = 745.7 Watts
Chapter 7: Linear Momentum and Impulse
Linear Momentum: p – product of mass
and velocity, like inertia it is a resistance to
change except it incorporates velocity.
p = mv
SI Unit – kgm/s
How do we change momentum?
Impulse – J – product of force and time
J = Ft
SI Unit – Ns
Impulse-Momentum Theorem
J = Dp
Ft = p – po
Ft = mv – mvo
Conservation of Linear Momentum
The linear momentum of an isolated
system remains constant, po = p
*Isolated means that no net external forces
are present.
**A system is two or more things
interacting.
For an isolated system
po = p
po1 + po2 = p1 + p2
m1vo1 + m2vo2 = m1v1 + m2v2
Collisions in 2D
In one dimension, we can say that
po = p
In two dimensions, it holds true that
pox = px
poy = py
In other words, the components of the vector
are conserved!
Chapter 8
Rotational Kinematics
Angular Displacement – q - angle
through which an object rotates.
We denote counter clockwise motion
as positive and clockwise motion as
negative.
SI unit – radian
q (measured in radians) = Arc Length/Radius
or
q = s/r
Converting from degrees to radians:
360o = 2p radians
or
180o = p radians
Angular Velocity – w – rate of angular
displacement.
w = Dq/t
SI Unit – rad/s
Does angular speed depend on radius?
Angular Acceleration – a – rate of angular
velocity changes.
a = Dw/t
SI Unit – rad/s2
Kinematic Equations
Linear
v = vo + at
s=vt
s = vot + ½at2
v2 = vo2 + 2as
Rotational
w = wo + at
q=wt
q = wot + ½at2
w2 = wo2 + 2aq
Tangential Velocity – vt – the speed and
direction of an object moving in a circle. The
direction is always tangent to the circle.
Ch 5
vt = 2pr/T
Ch 8
vt = rw
Centripetal Acceleration – ac – acceleration
towards a circle’s center.
Ch 5
ac = vt2/r
Ch 8
ac = rw2
Tangential Acceleration – at – if tangential
speed changes, an acceleration is occurring.
Ch 5
NONE
Ch 8
at = ra
Total Acceleration – a – vector sum of
centripetal and tangential acceleration.
a2 = ac2 + at2
Rolling Motion
When a wheel in contact with the
ground rotates once, without slipping, it
will travel a distance equal to the
circumference of its outside edge.