Transcript Rotational Dynamics - Mr. Shaffer at JHS

Rotational Dynamics
The action of forces and torques on rigid object:
Which object would be best to put a screw into a very
dense, hard wood?
A
B= either
C
Torque = Amount of Force x Lever Arm
T=FL
Nm = N x m
If a child pulls the end of the end of the wrench with 10 N of
force and an adult pulls at the ½ point of the wrench with
15 N, who pulls with the most torque and by how much
more?
( ) = child
= adult
The child provides (.25) Nm more Torque
Child T = 10N (.1) = 1 Nm
Adult T = 15N (.05) = .75 Nm
.1 m
Torque
If a person uses a wrench with 40 Nm of
torque to put in a screw, how much force
would be needed using a screwdriver to
put n the screw?
.02 m
T=FL
40 Nm = F .01 (the lever arm of the screwdriver – ½ diameter)
F = 4,000 N (4000N x .22 N/lb = 880 lbs of force)
(Compare this with 88 lbs of force with the wrench – 40Nm/.1=F)
.1 m
Rigid Objects in Equilibrium
For a rigid object to be in equilibrium, the objects linear motion and
angular motion must not be accelerating.
To identify if a rigid object is in equilibrium:
Find all the external forces acting on an object, if all the forces along
the x and y axis = 0 Fx= 0 Fy= 0, the net torque on the object
should also = 0 T = 0
Ignoring the mass of the board, how long far
away from the fulcrum must Johan stand
to balance the car?
To balance the system, the torque on each end
must be equal.
FL =FL
490N (L) = 4,900 (3)
500 kg
L = 30m
50 kg
?m
3 meters
If the board has a mass of 500 kg (evenly
distributed, where is the center of mass of
the system?
Center of Mass = m1r1 + m2r2 + m3r3
m1+m2+m3
50 kg
500 kg
1m
2m
30m
9.4 Moment of Inertia is the measure of mass distribution about
an object rotating about an axis.
I = mr2
(this can change due to the shape of the object)
see page244 in your book
What is I for a ring that has a mass of .01 kg and a diameter of
.02m?
What is I for a solid disc that has a mass of .01 kg and a diameter
of .02m?
Consider a hollowed out ball (A) (like a tennis ball) and a solid
ball (B)(like a baseball) that have the same mass and radius,
rolling across a floor, which comes to a stop first? (C) = come
to a stop at the same time
Rotational Motion about a fixed axis
F = ma (linear)
T = Iα (rotational)
torque = Moment of Inertia x angular acceleration
(needs to measured in radians)
360o = 2π radians
How many radians are in 180o?
Angular acceleration = ωf2 – ωo2
ωf - ωo
α
2Ɵ (in radians)
t
ω=angular velocity ω = v/r
A circular saw accelerates from rest to 80 rev/sec in 240
revolutions, the blade has a mass of 1.5 kg and a radius of .2
meters. What torque must be applied to the motor to
accelerate the blade?
T = Iα
I=
α=
3 bars that pivot at the red point have weights on
them with 3 units of weight distributed at
different points along the bar.
A
B
C
A = the most
B = 2nd most
C = the least
A constant force is applied upward to the
end of each bar. Rank which one
accelerates the most to least.
Work = Fd (linear)
Wrotational = TƟ (in radians)
How much work is done by a pulley
that produces 10 Nm of torque for
one revolution?
KE = ½ mv2 (linear)
(rotational) KErotational = ½ Iω2
What is the KE of a hollow ring with mass 2kg and
.25m radius rotating at 20 rad/sec.?
Total Energy = Translational KE + Rotational KE +mgh
E = ½ mv2 + ½ Iω2 + mgh
The ring mentioned above rolls down a ramp meters
high, how fast is the ring going at the bottom?
½ mv2 + ½ Iω2 = mgh
½ m v2 + ½ mr2 (v/r) = mgh
5m
V=
2mgh
m + I/r2
Momentum Linear = mv
Angular Momentum
L = Iω
For an object moving with a constant
angular velocity around an axis, the
momentum is conserved should the
velocity or moment of inertia change.
Why is it important for a diver who wants to do many flips
before hitting the water to get in a tuck position?
A disc (bottom) spins at a constant
rate. A smaller disc is dropped on the
larger disc. How does the new object
(small and large together) behave?
What if you dropped the hollow ring
instead of the smaller disc?
Torque = F l (lever Arm)
For an object to be balanced, the torques on each side of the
center of mass must be equal.
Rigid objects in equilibrium – All forces are 0, not translational
accel, no angular accel, no torque
Center of Gravity – The of mass of a rigid object is the point
where its entire weight can be considered.
Newton’s 2nd Law (F=ma) for rotating objects:
T = I α where t= torque, I=moment of inertia, α angular accel
Moment of inertial = distribution of mass of a rotating object.
I = mr2 (this can change based on shape of object)
Rotational Work = TƟ, like W=Fd, T is torque, Ɵ is distance
around the axis (in radians)
Rotational Kinetic Energy, KEr = ½ Iω2, angular velocity
Total Energy of an object = ½ mv2 + ½ Iω2 + mgh
Angular Momentum, L = Iω