Transcript Document

Direct analogies between (linear) translational and rotational motion:
Quantity or Principle
Linear
Rotation
Displacement
x

Velocity
v

Acceleration
a

Inertia (resistance to
acceleration)
mass (m)
moment of
inertia (I)
Momentum
P = mv
L = I
Momentum rate of change
dP/dt = Fnet
Stated as Newton’s 2nd Law:
F = ma
1/21/15
Oregon State University PH 212, Class 8
dL/dt = net
 = I
1
Rotational Kinetic Energy
Suppose you have a solid sphere sitting at rest. It has no kinetic
energy at all—no motion. Now push on it with a linear force. It
accelerates (in the direction of the force) to a non-zero linear velocity.
Now it’s moving—it has kinetic energy: KT = (1/2)mv2
Where did that energy come from? It was the work done by the
force: W = Fcos·s That’s the displacement multiplied by the force
that is acting in the direction of that displacement.
Now suppose the sphere is again sitting at rest. This time, instead of
pushing on it with a force, you twist on it—exert a torque. It
accelerates (in the angular direction of the torque) to some non-zero
angular velocity. Again, it’s moving, so it must have kinetic energy.
How do we measure and describe this rotational kinetic energy, KR
—and where did it come from?
1/21/15
Oregon State University PH 212, Class 8
2
First, realize that KR is not a “new kind of energy.” It’s just
(1/2)mivi2, for each tiny portion of mass, mi, in the sphere.
But at every point in a rotating object, vi = ri. So the sum
becomes KR = (1/2)miri22 = (1/2)(2)miri2 = (1/2)I2
And where did this energy come from? It was the work done by
the torque: W = ·D (notice the units here). It’s the angular
displacement multiplied by the torque acting in the (rotational)
direction of that displacement.
So we need to expand our definition of the total mechanical
energy, Emech, of an object: Emech = KT + KR + UG + US
And the Work-Energy relationship: Wext = DEmech
Wext = DKT + DKR + DUG + DUS
where Wext now includes all Fextcos ·s and all ext·D.
1/21/15
Oregon State University PH 212, Class 8
3
Direct analogies between (linear) translational and rotational motion:
Quantity or Principle
Linear
Rotation
Displacement
x

Velocity
v

Acceleration
a

Inertia (resistance to
acceleration)
mass (m)
moment of
inertia (I)
Momentum
P = mv
L = I
dP/dt = Fnet
dL/dt = net
F = ma
 = I
Work
F•Ds
•D
Kinetic energy
(1/2)mv2
(1/2)I2
Momentum rate of change
Stated as Newton’s 2nd Law:
1/21/15
Oregon State University PH 212, Class 8
4
A solid sphere initially sitting at rest has a mass of 50 kg and a radius
of 2 m. If a torque of 20 N·m is applied until the sphere has rotated
(about its center) for exactly 100 radians, what angular velocity will
it then have?
[For a solid sphere rotating about its center, I = (2/5)MR2.]
1/21/15
1.
 = 7.07 rad/s
2.
 = 5.00 rad/s
3.
 = 4.29 rad/s
4.
 = 1.68 rad/s
5.
None of the above
Oregon State University PH 212, Class 8
5
HW3, item 5d: One end of a thin, uniform rod, 1.40 m in
length, is attached to a pivot. The rod is free to rotate about
the pivot without friction or air resistance. Initially, it is
hanging straight down (in the “6 o’clock” position). What
initial horizontal speed vT.i, must the lower tip of the rod be
given so that the rod rotates upward, about its pivot and just
reaches the “12 o’clock” position, momentarily halting there?
1/21/15
Oregon State University PH 212, Class 8
6