Uniform
Circular Motion
Radians!
Force of Gravity
m1m2
Fg = G 2
R
mp M E
Fg = G
2
RE
MP
g=G 2
RP
M P Gravitational Field
2
m
g=G 2
-11
G = 6.67 ×10 N 2
RP
kg
Find gravitational field on the following:
Find the mass of a 65-kg person on each
Change of Gravitational Field
MP
g=G 2
RP
2
m
-11
G = 6.67 ×10 N 2
kg
Find gravitational field on around Earth at:
a) R = 2 radii of the planet
b) R = 4 radii of the planet
Angular displacement
s
q = rad
r
circle = 2p rad
Angular displacement
s
q = rad
r
s full = 2p r
2p r
q=
rad = 2p rad
r
1 radian in degrees
Which particle has angular position of
5p / 2
Rpm to rad/s
w=
q
Dt
w - [ rad / s ]
If it takes 5 seconds to complete 10 rotations, find:
a) Period of rotation T in seconds
b) Frequency f of rotation
c) Angular speed in rad/s
d) How are T and f related?
e) Angular displacement at 2.0 s
f) Convert e) to degrees
Kinematics of the roulette
wheel
• A small steel ball rolls counterclockwise around the inside of a
30.0 cm diameter roulette wheel. The ball completes exactly 2
rev in 1.20 s.
a) What is the ball’s angular velocity? (Answer: 10.5 rad/s)
b) What is the ball’s angular position at t = 2.00 s? Assume θi = 0.
Express the answer in rad and degrees. (Answer: 2.1 rad, 120)
RPM in a car engine
• When your motor is running at 3000 rmp, what is the
crankshaft’s angular velocity? (Answer: 314 rad/s)
Draw a displacement graph
a)w = 10.5rad / s
b)45rpm
q (t)
Interpret the graph and draw w (t)
Find angular speed of the wheel of the bike at t = 20 s, t = 50 s, t = 100 s, t = 160 s
Interpret the graph and draw
q (t)
Angular and linear
• Derive the formula for v for an object that is moving
around a circle.
v = rw
Two points on a CD
CD diameter is 12.0 cm. The maximum spinning rate
is 540 rpm. What is the speed of a point a) 3.0 cm
from the center, b) 6.0 cm from the center? Find
angular and linear speeds for both.
Which horse would you rather sit on on a merry-go-round? Why?
Acceleration in a UCM
Draw a circle. Choose two points and draw the velocity vector at those points.
Find the vector of the velocity change. What is the direction of the vector? If you
divide that vector by a scalar representing the time it took the particle to move
between those two points, what will you get?
Explain the term ‘centripetal’ that is used for this acceleration.
Deriving ac
Compare
For Emma and Jacob, compare:
a) angular speed,
b) tangential speed,
c) angular acceleration,
d) centripetal acceleration,
e) period of rotation
Quasar Carnival Ride
For safe operation, the maximum sustained acceleration is about 3g. Find the
period of the ride at the maximum acceleration? (Answer 3.1 s)
Critical thinking
What is the source of ac?
Centripetal vs. centrifugal
There is no centrifugal ‘force’ on the rider. What is his body doing?
Washing machines
Where do the socks go???
Explain the physics of the spin cycle in a washing machine
Various forces acting as Fc
Two satellites orbit the Earth on the same orbit. Which would have to move faster: a
heavier one or a lighter one? Prove your point mathematically by deriving the
formula for orbital speed of a satellite.
Keppler’s III law
T µR
2
3
Prove, using Newton’s ULG, assume
orbits to be circular.
“Tarzan” problem
A 65-kg person swings off a bank that is 6.0 m above the water on a tight 5.0 m rope.
Find the force of tension in the rope at the bottom on the ark.
“Rollercoaster” problem
A 45-kg skate boarder executes trick shown in the picture. The radius of the semicircle is 3.0 m and the height to which the boarder rises h = 1.5 m. What is the normal
force that the ramp is exerting on the boarder at the bottom of the ramp.
Force of friction as Fc - unbanked
Calculate the maximum speed a 1500-kg car
can have to safely execute a turn on a 75-m
turn if the coefficient of static friction
between the tires is 0.70. What speed can a
car of similar make have if its mass is 3500
kg? What should the speed be on an icy road
if the coefficient drops to 0.10?
Banked curves
In terms of UCM, explain the purpose of
banked turns of race-car courses.
Draw an FBD and determine which
force(s) act as centripetal in this case.
Derive a formula for Fc in this case.
Derive a formula for the angle on the
banked curve if friction is neglected.
Force of tension
A ‘swing’ ride at a carnival consist of chairs
what a swung in a circle by 15.0-m cables
attached to a vertical pole, as the drawing
shows. Supposed the total mass of chair and
its occupant is 179 kg. a) Determine the force
of tension in the cable. b) Find the speed of
the chair.