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Lec 08: Rotation
• Rotation: Angles, Speed
• Centripetal Force and Acceleration
• Seesaws: Torque and Inertia
• Angular Momentum
Angles
1 Revolution = 360o = 2π Radians
Q1. How many radians are 8
revolutions?
(8 Revolutions ) x (2π Radians) = 16π = 50.26 Rad
Q2. How many degrees is 1 radian?
(360 degrees) = 2π = 6.28 Rad
(
???
)=
1 Rad
(???) (6.28 ) = (360) (1)
(????) = 57.3 degrees
speed
=
distance traveled
time it took
Angular Velocity
Rotational
=
speed
angle swept
time it took
Revolutions per Minute
RPM
Q3. How many rad/s is 20 rpm?
Angular
Speed:
(20 rpm) x (2π Radians) = 2.09 Rad/s
Q4. What is 10 rad/s rpm?
(60 seconds)
(10 rpm) x (60 seconds) = 95.5 rpm
(2π Radians)
Circular Motion
Q5. When we move in a circle, do we have an
acceleration?
The change is towards the
center
Centripetal
=
acceleration
(speed)2
(radius)
Q6. What force is the cause for
the centripetal acceleration?
Centripetal vs. Centrifugal Force
Centripetal Force:
In order for an object to rotate along a circle, there
is a force responsible for pulling your that object
towards the center of a circular arc called
Centripetal force
Centrifugal Force:
When we experience a centripetal acceleration,
our feeling is actually being pushed away from
the center.
Puzzles
P1. When you turn your car, there is a force responsible for
pulling your car towards the center of a circular arc called
Centripetal force. It is not a new force, but rather one or
more forces combining to cause that acceleration. Which is
it?
(a) Gravity
(b) The force from your hands turning the wheel
(c) The friction between your car tires and the road?
P2. What is the angular speed of the Earth as it rotates
about its axis?
P3. When you put water in a kitchen blender, it begins to
travel in a 5.0-cm radius at a speed of 1 m/s. How quickly is
the water accelerating?
Puzzles
P4. Why do we not fall during a loop-the-loop ride on a
roller coaster?
P5. If Earth pulls the Moon towards itself, why does not the
Moon fall toward the Earth?
P6. Draw vectors for (a) gravitational force; (b) the force
exerted by the seat, and (c) the acceleration felt by the
passengers.
Center of Mass (Gravity)
• Pivot/Fulcrum
• Center of Mass: A point about which an
object naturally spins
• Note: The Center of Mass may be outside
the object
• Two ways to find the Center of Mass:
• Balance the object
• Find about which point it naturally rotates
Torque = (Force ) x (Lever Arm)
Q6. Which wrench requires you to apply less force in
order to loosen a bolt. A long wrench or a short wrench?
(a) The long wrench
(b) The short wrench
(c) It doesn’t matter
Simple machines: LEVER
(FORCE) x (distance) = (force) x (DISTANCE)
SEESAWS
(FORCE) x (lever arm) = (force) x (LEVER ARM)
Example:
Two children with different masses must sit at
different distance from the fulcrum
Puzzles
P7. Balance the seesaw. If the boy is 10 kg and sits 3 m
away from the pivot, wher shall the 15-kg girl sit in order
for the seesaw to be balanced?
(FORCE)(lever arm) = (force) (LEVER ARM)
P8. (A see-saw) A 10-lb objects is placed 1 ft to the left of
fulcrum. How far from the fulcrum should a 5-lb force be
applied to balance it.
1 ft
10 lb
??? ft
5.0 lb
(FORCE)(lever arm) = (force) (LEVER ARM)
P9. (A wheel barrow) A 10-lb objects is placed 1 ft to the
left of fulcrum. How far from the fulcrum should a 5-lb
force be applied to balance it.
??? ft
1 ft
10 lb
5.0 lb
Puzzles
P10. Why is it difficult to unscrew a bottle cap?
P11. You are trying to replace a light at your skating rink. As
you reach up overhead to unscrew the light you twist it, but
you begin to rotate yourself. What caused you to rotate?
P12. Why do helicopters
have two rotors? Which
Law is most involved in
the explanation?
ANGULAR MOMENTUM
ROTATIONAL INERTIA ( I ): How easy it is to
make an object change its rotation
ANGULAR (ROTATIONAL) MOMENTUM :
(ROT. INERTIA ) x (ROTATIONAL SPEED)
TORQUE – MOMENTUM PRINCIPLE
(NEWTON’S 1ST LAW) : In order to change the
rotational momentum, a torque is necessary
Spinning Ice skaters
Example: An ice skater is
spinning gracefully about herself
at 1 rev/seconds while her hands
are extended and her rotational
inertia is about 2.25 kg.m2. How
fast would she be spinning if she
pulls her hands in and in the
process decreases her rotational
inertia to 1.8 kg.m2?
(2.25 kg.m2 ) ( 1 rev/s ) = (1.8 kg.m2 ) ( ??? Rev/s)
Collapsing Neutron Stars
A neutron star is formed when an object such as
our Sun collapses. Suppose our Sun were to
collapse so that its rotational inertia becomes 10
billion times less of its original value. If the star
originally completed one revolution in 25 days
(approx. 0.00000046 rps), how fast does it
revolve after the collapse?