1. Rotation (III) Torque and angular acceleration “Moment of inertia”
Text Sections : 10.7, and part of 10.4
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2. Angular velocity vector: parallel to the axis of rotation, following a similar right-hand rule:
rotation direction w
Angular acceleration vector: parallel to the angular velocity, if |w| is increasing.
Force causes linear acceleration: Fnet = ma
Torque causes angular acceleration: tnet = Ia ?
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3. Force causes linear acceleration: Fnet = ma
Q: How much torque does it take to rotate a particular object?
Better: How much torque does it take to change the rate of
rotation?
What property of an object determines the response (angular acceleration) to an unbalanced external torque?
Force causes linear acceleration: Fnet = ma
Torque causes angular acceleration: tnet = Ia ?
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4. Ft = mat = m(ra ), since at = ra
The angular acceleration of a particle is proportional to the net torque applied to it.
Example: A particle accelerates in a circle. Break the net force on it into radial and tangential components. Only Ft causes tangential acceleration:
Ft = mat = m(ra ), since at = ra
Multiply by r : rFt = mr 2a
or torque = (mr 2) a Fr r Ft
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5. For a rigid body, a is the same for all particles
For a rigid body, a is the same for all particles. The net force on each particle is composed of internal forces f, and external forces Fext.
Fext f -f a ri mi
The total torque is the sum of the torques on the individual particles:
Torques due to the internal forces f and –f cancel when we sum over all particles. The quantity in brackets is the “moment of inertia”:
and Newton’s 2nd law for rotation is
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6. The Moment of Inertia: Units: kg m2
(r is the distance from a particle to the axis of rotation.)
“I” measures the rotational inertia of an object. It depends on:
the total mass. I is proportional to mass.
2) how the mass is distributed (distance from the rotation axis). I is proportional to (linear size)2, and is larger if the mass is concentrated farther from the rotaion axis.
3) which axis the object rotates about.
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7. I=MR2 I = ½ MR2 I = 1/12 ML2 I = 1/3 ML2 R R
Moments of inertia for uniform objects can be calculated using integral calculus. A few results (see table 10.2 in the text):
I=MR2
I = ½ MR2 L
I = 1/12 ML2 L
I = 1/3 ML2 R R
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8. Quiz
Three point particles, each of mass m, are arranged in the x-y plane in an equilateral triangle of side 2b as shown. What is the moment of inertia:
1) about the x axis? - clicker
2) about the y axis?
3) about the z-axis? x y b 2b
mb2
2mb2
3mb2
4mb2
5mb2
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9. Quiz
Two spheres are made of the same material. However, all the dimensions of sphere B are twice as large as those of sphere A. If the spheres are launched with the same torques, applied for equal times, sphere A accelerates faster than B by a factor of : 2 4 8 16 32
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10. Quiz
Two spinning tops have equal radius and mass, but whereas top B is a thin disc, top A has light spokes connecting the hub to an outer ring. The tops are spun by applying equal torques to the spindle. Which spins the fastest?
Top View A B
spin at same rate
not enough info. A B
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11. Quiz
Two spinning tops are made from circular disks with a pointed spindle inserted through the center. The tops have the same radii and total mass, but since they are made of different material, top A is thicker. The tops are launched by equal torques to the spindles. Which spin is the fastest? A B A B
spin at the same rate
not enough info
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12. Quiz
A uniform board is attached to the edge of a table at one end by a hinge. A coin is placed on the other end, and the free end of the board is held so that the board is horizontal.
coin
When the board is released:
the board falls faster than the coin
the board and coin fall freely together
the coin would fall faster, but presses on the board
it depends on the length of the board
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13. Example
Two (thin, uniform) sticks of different lengths are held nearly upright (at a slight angle to the vertical) on a table and released simultaneously. Which hits the ground first? (Try this at home.)
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14. Summary
The moment of inertia measures the rotational inertia of a body.
Particle: I = mr2
Extended Body:
Newton’s 2nd law for rotation about a fixed axis:
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