Transcript Rotational Motion and Equilibrium

Work,
Power and
Energy in
Rotational
Motion
AP Physics C
Mrs. Coyle
Relating Torque to Work and Power
• Consider a rigid body rotating dθ during
an infinitesimal time interval dt.
Work and Torque
• During the time dt, the object moves a distance
ds= r d θ
• The work dW is:
dW= F · ds = F · r d θ = (Fcos ɸ) r d θ
• The torque τ = F sin (90- ɸ) r = (Fcos ɸ) r
dW = τ dθ
(Analogous to dW= Fds for translational motion)
Power and Torque
dW = τ dθ
dW = τ dθ
dt
dt
So power is
P= τ ω
(Analogous to P= Fv for translational
motion)
Work –Kinetic Energy Theorem for
Rotational Motion
ΣW= ½ I ω2f - ½ I ω2i
The net work done by external forces in
rotating a symmetric rigid object
about a fixed axis equals the change in
the object’s rotational kinetic energy.
Ex: #42
Ex: #42
• A top has a moment of inertia of 4.00 x10 -4 kg
m2 and is initially at rest . It is free to rotate
about the stationary axis AA’. A string,
wrapped around a peg along the axis of the
top, is pulled in such a manner as to maintain
a constant tension of 5.57N. If the string does
not slip while it is unwound from the peg,
what is the angular speed of the top after
80.0cm of string has been pulled off the peg?
Ans: 149rad/s
Ex: # 44
A cylindrical rod 24.0cm long with mass 1.20kg
and radius 1.50cm has a ball of diameter
8.00cm and mass 2.00kg attached to one end.
The arrangement is originally vertical and
stationary with the ball on top. The system is
free to pivot about the bottom end of the rod
after being given a nudge.
a) After the rod rotates through 90o , what is its
rotational kinetic energy?
#44 cont’d
b) What is the angular speed of the rod and the
ball?
c) What is the linear speed of the ball?
d) How does this compare to the speed if the
ball had fallen freely through the same
distance of 28cm?
Ans: a)6.90J , b)8.73rad/s, (I tot=0.181kgm2 ) ,
c)2.44 m/s, d)1.0432 times greater
Ex:#45
Ex: #45
• An object with a weight of 50.0N is attached
to the free end of a light string wrapped
around a reel off radius 0.250m and mass
3.00kg. The reel is a solid disk, free to rotate in
a vertical plane about the horizontal axis
passing through its center. The suspended
object is released 6.00m above the floor.
a) Determine the tension in the string, the
acceleration of the object and the speed with
which the object hits the floor.
#45 Cont’d
b) Verify your last answer by using the principle
of conservation of energy to find the speed
with which the object hits the floor.
Ans: a)11.4N, 7.57m/s2 , 9.53 m/s, b) 9.53m/s