PHY 1401 – General Physics I
Rotational Motion
Leader:
___________________ Recorder:
__________________________
Skeptic:
___________________ Encourager:
________________________
Materials
In this
experiment we will investigate several aspects of rotational motion including
angular velocity, angular acceleration, and torque. For parts II – V we will use an apparatus
consisting of a turntable rotating on a low friction bearing.
We are all probably familiar with the
concept that there are 360° in a circle. This forms the basis of how we measure an
angle. If we place an angle with its
vertex at the center of a circle, it will cut out a fraction of the
circle. For instance the 30° angle shown
in figure 1 cuts out 1/12 of the circle.
Thus any angle that cuts out 1/12 of a circle will have a measure of
1/12 * 360° = 30°.
Figure 1
30° = 1/12 * 360°
Is there anything special about saying
there are 360° in a circle? The answer is that
the circle was divided into 360° by the ancient Babylonians 4400 years ago
and it stuck. The Babylonians noticed
that the sun took about 360 days to go around Earth and thus divided a circle
into 360°. Other systems are
possible. For example, the military used
to use a system in which a circle was divided into 400 gradients or 400
grad.
Q1. If an angle cuts out 1/12 of a circle, what
would be its measure in gradients?
Q2. What would be the measure of a right angle in
gradients?
A system of measure that mathematicians
and physicists find very useful is radian measure. A simple way to think about radian measure is
that instead of 360° in a circle, there are 2p radians.
Q3. If an angle cuts out 1/12 of a circle, what
would be its measure in radians? (Leave your answer in terms of p.)
One of the main
reasons that mathematicians and physicists find radian measure useful is that
when measured in radians there is a very simple relationship for determining
angle.
Notice that if
you place an angle at the center of a circle, then it cuts out an arc along the
circle. The length of the arc is denoted
by s. The angle in radians can then be determined
from the formula , and this result is very useful in math and physics.
Q4. For the following four circles, determine the
Indicated Angle in two ways.
i) Multiply the fraction of a circle the angle
cuts out by 2p.
ii) Measure the radius and arc length and use the
information to determine the angle.
iii) Compare your two estimates of the measure of
the angle.
Part II Angular Velocity
If an
object is rotating, then its angle is changing with time. In that case we say that the object has an
angular velocity. Angular velocity is
defined as (1). The apparatus we are using consists of a
horizontal turntable rotating on a low friction bearing. Once you start the apparatus turning it will
remain turning at a fairly constant rate.
Q5. Given a
piece of tape and a stopwatch, describe a procedure for determining the angular
velocity of the turntable as it spins.
Q6. Your
answer for Q5 may have been to time one rotation of the turntable. What sources of error might occur with that
procedure and how could you lessen the source of those errors.
Q7. Based on
your answer to Q6 write a modified procedure for determining the angular
velocity of the turntable which will make the errors you identified smaller.
Q8. Employ
the procedure and measure the angular velocity of the turntable when you spin
it at three different speeds. Record
the angle in radians. Organize your data
into a data table and be sure to include units.
In this
part we will examine the conditions necessary to produce an angular
acceleration. With the turntable
initially at rest, use your index finger to apply a force along the tangent at
the edge of the turntable.
Q9. Describe
the resulting motion of the turntable.
Trying to
keep the force about the same, apply the force along the tangent but at four
successively smaller radii. Each time
start the wheel from rest.
Q10. Describe
how the resulting motion of the turntable changes as you reduce the
radius. In particular describe the
affect on the angular acceleration.
Starting
the turntable at rest each time, compare the effect of increasing the force
applied along the edge of the turntable.
Q11. Describe
how the angular acceleration changes when you increase the tangential force
along the edge of the turntable.
Now apply
a force along a line directed towards the center of the turntable, i.e.
radially.
Q12. Does
directing the force radially produce the same angular acceleration as directing
it tangentially? Explain.
The
quantity that produces an angular acceleration is called torque. In questions Q10 – Q12 we have investigated
three things which affect torque.
Q13. From
your observations, describe how torque depends on the magnitude of the applied
force.
Q14. From
your observations, describe how torque depends on the radius at which the force
is applied.
Q15. From
your observations, describe how torque depends on the direction at which the
force is applied.
We
usually combine these observations into a single formula for torque given by
Torque = Force x Lever Arm, where force is the
tangential force that is applied and the lever arm is the distance from the
axis of rotation at which the force is applied.
A little more rigorously, torque is defined as t = rF sinq where t is the torque, F is the magnitude of the
applied force, r is the radius at which the force is applied and q is the angle between the radius and the force.
Q16. If a
force is applied tangentially to a circle what is q? What
is sin q?
Q17. If a
force is applied radially to a circle what is q? What is sin q?
Q18. Torque
is maximum for a given force when the angle is _____?
Part IV Angular Acceleration
When the angular velocity of an object changes the object has an angular acceleration. In this part of the lab, we will let the apparatus rotate freely, then slow it for a while using a light pressure due to our finger and then let it rotate freely again. Angular acceleration is defined as (2)
Q19. During which part of the described motion does
the turntable turn at a constant angular velocity?
Q20. During
which part of the described motion is there an angular acceleration?
Q21. The
definition of angular acceleration above includes Δω. How can you determine Δω for the
described motion?
Q22. The
definition of angular acceleration includes Δt. What is the appropriate Δt to use?
Q23. Describe
a procedure for determining the angular acceleration for this procedure using
the same materials as in part II.
Q24. Start
the turntable spinning and then slow it with a light pressure from your
finger. Record your data in a table and
your determination of the angular acceleration in the space below.
Part V Moment of Inertia
In this part we will
examine how mass affects rotational motion
Spin the turntable so that
it is rotating around 1/2 turn per second.
Take a large ball of clay and drop it as close to the center of the
turntable as you can.
Q25. Describe the effect, if any, on the rotation
of the turntable.
Repeat but drop the clay
further from the axis this time.
Q26. Describe the effect, if any, on the rotation
of the turntable.
Repeat two more times
dropping the clay further from the axis, with the last drop being about at the
edge of the turntable.
Q27. Describe the effect, if any, on the rotation
of the turntable.
Q28. Did the effect depend on the distance the
clay was dropped from the axis of rotation?
If so, how?
Again spin the turntable
at about 1/2 rotation per second and drop clay balls of increasing mass along
the edge of the turntable.
Q29. Describe the effect, if any, on the rotation
of the turntable.
Q30. Did the effect depend on the amount of clay
that you dropped? If so, how?
Q31. Summarize your observations:
Dropping the clay ball
further from the axis produced a ____________ change in the rotation. Dropping larger clay balls at the edge
produced a __________ change in the rotation.