Energy in rotational motion
Consider an object rotating about the z-axis. It is moving and thus has kinetic energy. We express the kinetic energy in ...
Consider an object rotating about the z-axis. It is moving and thus has kinetic energy. We express the kinetic energy in terms of the objects moment of inertia
y
x
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1
Energy in rotational motion Imagine that the object is made up of lots of little particles and let mi = ri = vi =
mass of the ith particle perpendicular distance from the axis of rotation to the ith particle speed of the ith particle vi = riω
y ri
mi x
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Energy in rotational motion The kinetic energy of the ith particle rotating about a given axis is 1 1 Ki = mivi2 = miri2ω 2 2 2 And the total rotational kinetic energy is Krot =
= = – Typeset by FoilTEX –
X1 1 1 2 2 2 2 m1 r 1 ω + m2 r 2 ω + · · · = miri2ω 2 2 2 2 i ! 1 X miri2 ω 2 2 i
1 2 Iω 2
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Energy in rotational motion
Recall previously that we considered kinetic energy of a system of particles in which the velocity of the ith particle could be expressed ~CM + ~vi ~vi = V
rel
and the total kinetic energy became Ktotal =
1 2 M VCM + |2 {z }
X1 i
2
mivi2 rel
{z } | motion of CM motion relative to CM – Typeset by FoilTEX –
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Energy in rotational motion
If the relative motion represents particles all moving at the same angular velocity (i.e., vi rel = riω), then the kinetic energy becomes 1 1 2 M VCM + Iω 2 2 2 = Ktranslational + Krotational
Ktotal =
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Moment of Inertia
I=
X
miri2
i
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Clicker Question Consider a single mass with the axes of rotation shown. configurations have the same moment of intertia?
Do these
A. yes B. no
m
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d
m
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Moment of Inertia
I=
X
miri2
i
The moment of inertia depends on the distribution of mass, and therefore it is important to specify axis of rotation. Always specify the moment of inertia with respect to some axis of rotation.
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Clicker Question
Which configuration has greater moment of inertia?
A
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B
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Clicker Question
Which configuration has greater kinetic energy? configurations are rotating with angular speed ω.
A – Typeset by FoilTEX –
Assume both
B 10
Clicker Question
Which configuration would be harder to start rotating from rest?
A
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B
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Clicker Question A puck of mass m moving at a speed v hits an identical puck which is fastened to a pole using a string of length r. After the collision, the puck attached to the string revolves around the pole. Suppose we now lengthen the string by a factor of 2, as shown on the right, and repeat the experiment. Compared to the angular speed in the first situation, the new angular speed is A. twice the length
r
m
m
2r
B. the same
v
v
C. half as much
m
m
D. none of the above – Typeset by FoilTEX –
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Example A, B, and C are heavy connectors linked by molded struts. What is the moment of inertia of the body about an axis A. through point A, perpendicular to the plane of the diagram? m B =0.10 kg B
B. coinciding with rod BC? 0.50 m
0.30 m
A m A =0.30 kg – Typeset by FoilTEX –
0.40 m
C m C =0.20 kg 13
Moment of inertia of continuous bodies The moment of inertia for a system of point particles is I=
X
miri2
i
For a continuous body, this generalizes to Z I = r2dm where dm represents small mass elements. To do the integral, express r and dm in terms of the same integration variable. – Typeset by FoilTEX –
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Example
Calculate the moment of inertia of a uniform thin rod of mass M and length L about the axes shown.
y
y
M
M x L
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x L
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Conservation of Energy with rotational motion
If all forces are conservative, mechanical energy is still conserved: E =K +U with K = Ktranslational + Krotational
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Clicker Question
A block is released from rest at a height h and slides frictionlessly down a ramp. A ball is released from rest from the same height and rolls without slipping down another ramp. Which has a greater speed when it reaches the bottom? A. block B. ball C. neither, they have the same speed at the bottom
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Example
In a lab experiment to test conservation of energy in rotational motion, we wrap a light, flexible cable around a solid cylinder with mass M and radius R. The cylinder rotates with negligible friction about a stationary horizontal axis. We tie the free end of the cable to an object of mass m and release the object with no initial velocity at a distance h above the floor. As the object falls, the cable unwinds without stretching or slipping, turning the cylinder. Find the speed of the falling object and the angular speed of the cylinder just as the object strikes the floor.
