Download Rotational Motion Objectives: After reviewing this section you should

Rotational Motion
Objectives: After reviewing this section you should be able to
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define rotational inertia
define center of gravity and center of rotation
define and calculate torque
define angular momentum
apply Newton’s laws and momentum to rotational motion
explain conservation of angular momentum
Rotational Motion
You will recall that circular motion is the motion of an object in a curved path
about an external axis. This week we will explore a similar situation. Rotational
motion is motion of an object in a curved path about an internal axis. The word
'angular' is used to describe motion in a circular path. If you are moving in a
curved path, you are passing through some angle if you take the center of the
path as the center of circle. Circular motion and rotational motion might seem like
the same thing - and they are very similar. A familiar example will help distinguish
them in your mind.
The Earth moves in (roughly) uniform circular motion about the sun. In this
example the sun is the axis of rotation, and it is external (not part of the Earth).
Gravity provides the centripetal force that keeps the Earth in its orbit. The Earth
also spins or rotates about its axis. This rotation is around an internal axis, and
the entire mass of the Earth is spinning as the Earth rotates.
An object which spins around an internal axis has some unique properties. Have
you ever noticed that a bicycle is much easier to balance when it is moving than
when it is at rest? This is because the spinning wheels have rotational inertia and
angular momentum. These provide stability in the same way that a gyroscope
maintains its axis of rotation.
Rotational Inertia
Remember from our discussion of Newton's Laws that objects with mass have
inertia. They tend to maintain their same state of motion. Rotating objects also
have inertia. The property of an object to resist changes in its rotational state of
motion is called rotational inertia. In simpler terms, a rotating object tends to
keep rotating about its axis unless it is interfered with by an external influence.
Rotating objects tend to keep rotating, and non-rotating objects tend to remain
non-rotating.
Like linear inertia, rotational inertia depends upon the mass of the object.
Rotational inertia also depends on how the mass is distributed about the axis of
rotation. If you have ever tried to lift a weight attached to the end of a long
handle, you will notice that it seems much heavier than if you lift the same mass
with a short handle. A tight rope walker uses a long pole while balancing on the
rope. This helps him maintain his balance because most of the mass of the pole
is far away from the center of rotation. The pole has a large rotational inertia,
helping the tight rope walker to resist changes in his motion (prevent him from
falling). Simply stated, if the mass is farther from the axis, the rotational inertia is
larger.
The rotational inertia (I) of objects of different shapes can be found using the
objects mass and radius. Some examples are shown below:
Example:
Which of the following cylinders would you expect to win the race down the
ramp? You can assume the mass of each is the same.
The ring has greater rotational inertia, so it will take longer to get rolling down the
ramp. It will therefore lose. Its greater inertia, however, will keep it rolling longer .
Torque
Torque (rhymes with fork) is the rotational counterpart of force. A torque is a
force which causes an object to turn or rotate about its axis. When you spin a
top, you are applying a torque to get the top spinning. Torque differs from force
just as rotational inertia differed from inertia. The distance between the applied
force and the axis of rotation is important. This distance is referred to as the lever
arm.
Torque is the product of the lever arm and the force applied.
Torque = lever arm x force
The torque is greatest when the applied force is perpendicular to the lever arm. If
you have ever changed a tire on your car, you are familiar with the benefits of
increasing the length of the lever arm to increase the amount of torque. Imagine
trying to remove the lug nuts on your car tire with just your bare hands! The tire
iron adds a long lever arm and increases the torque to make the job easier.
A see-saw is a good example of torque. You know that to balance the see-saw
you have to get just the right combination of mass and distance from the center.
In the example shown below, the see-saw is balanced when the 250 N child is 3
meters from the center and the 500 N child is 1.5 m from the center. Both have
equal torque (750 Nm).
Example:
What torque is generated on a bolt when a force of 500 Newtons is applied at a
right angle to a 25 cm wrench?
Torque = Force x lever arm = (500N)(0.24 m) = 125 Nm
PhET Simulation - Torque
Angular Momentum
Remember that a mass moving in a straight line has momentum. A spinning
object also has momentum. A mass rotating about some axis has angular
momentum. Angular momentum is a measure of the rotational property of
motion. It depends upon the object's rotational inertia and its angular (rotational )
velocity. Recall from last week that angular velocity is the number of rotations per
second.
Angular momentum = rotational inertia x rotational velocity
For the case of an object that is small compared to the radial distance to its axis
of rotation, the angular momentum can be simply expressed as the linear
momentum (mv) times its radius (r).
Angular momentum = mvr
Just as an external force is required to change the linear momentum of an object,
an external net torque is required to change the angular momentum of an object.
We can restate Newton's first law of inertia for rotating objects in terms of angular
momentum.
An object or system of objects will maintain its angular momentum unless acted
upon by an unbalanced external torque.
Rotational Physics:
Applying what we've already learned, angular velocity (ω) is the time rate of
angular displacement. It is angular displacement (Ө) (measured in revolutions,
degrees or radians) per unit of time or the time rate of angular displacement.
1 Revolution = 360o = 2π Radians = 1 Wavelength
Angular displacement, theta (Ө), describes the angle or degrees an object may
be rotated about some fixed axis. This is comparable to displacement in linear
motion.
Angular velocity, Omega (ω) is measured in rev/s, o/s, or rad/s. In linear of
straight line motion we had we had velocity. Remember that velocity is how fast
an object goes in a particular direction. Now we are concerned about how fast
the object rotates or spin. This direction of spin will be clockwise or counter
clockwise.
Angular acceleration, alpha (ά) , is the rate of change of angular velocity. It is
measured in rev/s2, o/s2, rad/s2 or some other combination of appropriate units.
Animation - Rotational Velocity Vector
Conservation of Angular Momentum
Angular momentum is also conserved. If you have ever watched a figure skate
spinning, you will have noticed that when she pulls her arms in tight she spins
more quickly, and when she stretches her arms out wide she spins more slowly.
This demonstrates the conservation of angular momentum. Since angular
momentum is a product of rotational inertia and rotational velocity, changing one
will change the other. Consider the example below. When the mass is farther
from the center, the rotational inertia increases and the rotational velocity
decreases. This also explains why divers in a tight tuck can spin more quickly
when diving.
Content and graphics from Hewitt Conceptual Physics Seventh Edition.
Tutorial - Rotational Motion
PhET Simulation - Torque