lecture 15 rotational dynamics. moment of inertia the moment of inertia i: the total kinetic energy...
TRANSCRIPT
Moment of Inertia
The moment of inertia I:
The total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies:
Torque
We know that the same force will be much more effective at rotating an object such as a nut or a door if our hand is not too close to the axis.
This is why we have long-handled wrenches, and why doorknobs are not next to hinges.
A more general definition of torque:
Fsinθ
Fcosθ
You can think of this as either:
- the projection of force on to the tangential directionOR
- the perpendicular distance from the axis of rotation to line of the force
Torque
If the torque causes a counterclockwise angular acceleration, it is positive; if it causes a clockwise angular acceleration, it is negative.
You are using a wrench to
tighten a rusty nut. Which
arrangement will be the
most effective in
tightening the nut?
a
cd
b
e) all are equally effective
Using a WrenchUsing a Wrench
You are using a wrench to
tighten a rusty nut. Which
arrangement will be the
most effective in
tightening the nut?
a
cd
b
Because the forces are all the same, the only difference is the lever arm. The arrangement with the largest largest lever armlever arm (case bcase b) will provide
the largest torquelargest torque.
e) all are equally effective
Using a WrenchUsing a Wrench
The gardening tool shown is used to pull weeds. If a 1.23 N-m torque is required to pull a given weed, what force did the weed exert on the tool?
What force was used on the tool?
Force and Angular Acceleration
Consider a mass m rotating around an axis a distance r away.
Or equivalently,
Newton’s second law:
a = r α
The L-shaped object shown below consists of three masses connected by light rods. What torque must be applied to this object to give it an angular acceleration of 1.2 rad/s2 if it is rotated about (a) the x axis,(b) the y axis(c) the z axis (through the origin and perpendicular to the page) (a)
(b)
(c)
The L-shaped object shown below consists of three masses connected by light rods. What torque must be applied to this object to give it an angular acceleration of 1.2 rad/s2 if it is rotated about an axis parallel to the y axis, and through the 2.5kg mass?
Dumbbell IDumbbell I
a) case (a)a) case (a)
b) case (b)b) case (b)
c) no differencec) no difference
d) it depends on the rotational d) it depends on the rotational inertia of the dumbbellinertia of the dumbbell
A force is applied to a dumbbell for a certain period of time, first as in (a) and then as in (b). In which case does the dumbbell acquire the greatercenter-of-mass speed ?
Dumbbell IDumbbell I
a) case (a)a) case (a)
b) case (b)b) case (b)
c) no differencec) no difference
d) it depends on the rotational d) it depends on the rotational inertia of the dumbbellinertia of the dumbbell
A force is applied to a dumbbell for a certain period of time, first as in (a) and then as in (b). In which case does the dumbbell acquire the greatercenter-of-mass speed ?
Because the same force acts for the same time interval in both cases, the change in momentum must be the same, thus the CM velocity must be the same.
Static Equilibrium
X
Static equilibrium describes an object at rest – neither rotating nor translating.
If the net torque is zero, it doesn’t matter which axis we consider rotation to be around; you choose the axis of rotationThis can greatly simplify a problem
Center of Mass and Gravitational Force on an Extended Object
center of massmjm1
xj
...
xj
mjm1
Fj = mj g
...X
axis of rotation
xcm
F = Mg
X
axis of rotation
So, forget about the weight of all the individual pieces. The net torque will be equivalent to the total weight of the object, pulling from the center of mass of the object
Center of Mass and BalanceThis fact can be used to find the center of mass of an object – suspend it from different axes and trace a vertical line. The center of mass is where the lines meet.
Balancing RodBalancing Rod
1kg
1m
A 1-kg ball is hung at the end of a rod
1-m long. If the system balances at a
point on the rod 0.25 m from the end
holding the mass, what is the mass of
the rod?
a) ¼ kg
b) ½ kg
c) 1 kg
d) 2 kg
e) 4 kg
1 kg
X
CM of rod
same distance mROD = 1 kg
A 1-kg ball is hung at the end of a rod
1-m long. If the system balances at a
point on the rod 0.25 m from the end
holding the mass, what is the mass of
the rod?
The total torque about the The total torque about the
pivot must be zero !!pivot must be zero !! The CM
of the rod is at its center, 0.25 0.25
m to the right of the pivotm to the right of the pivot.
Because this must balance the
ball, which is the same same
distance to the left of the pivotdistance to the left of the pivot,
the masses must be the
same !!
a) ¼ kg
b) ½ kg
c) 1 kg
d) 2 kg
e) 4 kg
Balancing RodBalancing Rod
When you arrive at Duke’s Dude Ranch, you are greeted by the large wooden sign shown below. The left end of the sign is held in place by a bolt, the right end is tied to a rope that makes an angle of 20.0° with the horizontal. If the sign is uniform, 3.20 m long, and has a mass of 16.0 kg, what is (a) the tension in the rope, and (b) the horizontal and vertical components of the force, exerted by the bolt?
When you arrive at Duke’s Dude Ranch, you are greeted by the large wooden sign shown below. The left end of the sign is held in place by a bolt, the right end is tied to a rope that makes an angle of 20.0° with the horizontal. If the sign is uniform, 3.20 m long, and has a mass of 16.0 kg, what is (a) the tension in the rope, and (b) the horizontal and vertical components of the force exerted by the bolt? Torque, vertical force, and horizontal force are all zero
But I don’t know two of the forces!
I can get rid of one of them, by choosing my axis of rotation where the force is applied.
Choose the bolt as the axis of rotation, then:
(b)
Linear momentum was the concept that tied together Newton’s Laws, is there something similar for rotational motion?
F = ma implies Newton’s first law: without a force, there is no acceleration
Now we have
2
2
2 0 00 0
20 0
2 00
13
2 32 cos
2 3 2Wood:
1 2 2 4: 0 2
2 3 3
Ball:1
: 0 and sin2
10 2
2
fall fall fall fall
fall fall fall
fall fall
I ML
MgL MgL gMg L
I ML L
L Ltt tt
g g
t y t y gt y L L
LL gtt
g Wood !fallt
Angular Momentum
Consider a particle moving in a circle of radius r,
I = mr2
L = Iω = mr2ω = rm(rω) = rmvt = rpt
Angular Momentum
For more general motion (not necessarily circular),
The tangential component of the momentum, times the distance
Angular Momentum
For an object of constant moment of inertia, consider the rate of change of angular momentum
analogous to 2nd Law for Linear Motion
Conservation of Angular Momentum
If the net external torque on a system is zero, the angular momentum is conserved.
As the moment of inertia decreases, the angular speed increases, so the
angular momentum does not change.
Figure SkaterFigure Skater
a)a) the samethe same
b)b) larger because she’s rotating larger because she’s rotating fasterfaster
c) smaller because her rotational c) smaller because her rotational inertia is smallerinertia is smaller
A figure skater spins with her arms extended. When she pulls in her arms, she reduces her rotational inertiaand spins faster so that her angular momentum is conserved. Comparedto her initial rotational kinetic energy, her rotational kinetic energy after she pulls in her arms must be:
Figure SkaterFigure Skater
a)a) the samethe same
b)b) larger because she’s rotating larger because she’s rotating fasterfaster
c) smaller because her rotational c) smaller because her rotational inertia is smallerinertia is smaller
A figure skater spins with her arms extended. When she pulls in her arms, she reduces her rotational inertiaand spins faster so that her angular momentum is conserved. Comparedto her initial rotational kinetic energy, her rotational kinetic energy after she pulls in her arms must be:
KErot = I 2 = L (used L = I ).
Because L is conserved, larger
means larger KErot.
Where does the “extra” energy come from?
KErot = I 2 = L (used L = I ).
Because L is conserved, larger
means larger KErot.
Where does the “extra” energy come from?
As her hands come in, the velocity of her arms is not only tangential... but also radial.
So the arms are accelerated inward, and the force required times the Δr does the work to raise the kinetic energy
Conservation of Angular Momentum
Angular momentum is also conserved in rotational collisions
larger I, same total angular momentum, smaller angular
velocity
Rotational WorkA torque acting through an angular displacement does work, just as a force acting through a distance does.
The work-energy theorem applies as usual.
Consider a tangential force on a mass in circular motion: τ = r F
s = r ΔθW = s F
Work is force times the distance on the arc:
W = (r Δθ) F = rF Δθ = τ Δθ
Rotational Work and Power
Power is the rate at which work is done, for rotational motion as well as for translational motion.
Again, note the analogy to the linear form (for constant force along motion):
a) case (a)a) case (a)
b) case (b)b) case (b)
c) no differencec) no difference
d) it depends on the rotational d) it depends on the rotational inertia of the dumbbellinertia of the dumbbell
A force is applied to a dumbbell for a certain period of time, first as in (a) and then as in (b). In which case does the dumbbell acquire the greater energy ?
Dumbbell IIDumbbell II
a) case (a)a) case (a)
b) case (b)b) case (b)
c) no differencec) no difference
d) it depends on the rotational d) it depends on the rotational inertia of the dumbbellinertia of the dumbbell
A force is applied to a dumbbell for a certain period of time, first as in (a) and then as in (b). In which case does the dumbbell acquire the greater energy ?
Dumbbell IIDumbbell II
If the CM velocities are the same, the translational kinetic energies must be the same. Because dumbbell (b) is also rotating, it has rotational kinetic energy in
addition.
A 2.85-kg bucket is attached to a disk-shaped pulley of radius 0.121 m and mass 0.742 kg. If the bucket is allowed to fall, (a) what is its linear acceleration? (b) What is the angular acceleration of the pulley? (c) How far does the bucket drop in 1.50 s?
A 2.85-kg bucket is attached to a disk-shaped pulley of radius 0.121 m and mass 0.742 kg. If the bucket is allowed to fall, (a) What is its linear acceleration? (b) What is the angular acceleration of the pulley? (c) How far does the bucket drop in 1.50 s?
Pulley spins as bucket falls
(c)
(b)
(a)
The Vector Nature of Rotational Motion
The direction of the angular velocity vector is along the axis of rotation. A right-hand rule gives the sign. Right-hand Rule:
your fingers should follow the velocity vector around the circle
Optional materialSection 11.9
The Torque VectorSimilarly, the right-hand rule gives the direction of the torque vector, which also lies along the assumed axis or rotation
Right-hand Rule: point your RtHand fingers along the force, then follow it “around”. Thumb points in direction of torque.
Optional materialSection 11.9
The linear momentum of components related to the vector angular momentum of the
system
Optional materialSection 11.9
Applied torque over time related to change in the vector angular momentum.
Optional materialSection 11.9
a) remain stationarya) remain stationary
b) start to spin in the same b) start to spin in the same direction as before flippingdirection as before flipping
c) start to spin in the same c) start to spin in the same direction as after flippingdirection as after flipping
You are holding a spinning bicycle wheel while standing on a stationary turntable. If you suddenly flip the wheel over so that it is spinning in the opposite direction, the turntable will:
Spinning Bicycle WheelSpinning Bicycle Wheel