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UCD Physics Laboratory: Investigation of Rotational Motion Student Name: Student Number: Demonstrator Name Lab Date/Time:

Goal You will investigate, independently for yourself, some key concepts of rotational motion,

in particular: torque (), angular acceleration (), angular displacement () and moment of inertia (I). They are the rotational analogues of force (F), acceleration (a), displacement (s) and mass (m), respectively. You don’t need prior knowledge of this topic to participate in this lab, but you are encouraged to read and think about this script ahead of the lab and to do the warmup exercises in advance. Introduction and warmup exercises A full revolution of 360o is equivalent to 2π radians. What is the angular displacement (in radians, rad) of a spinning disk that completes 2 full revolutions? __________ What is the angular velocity (in radians per second, rad/s) of a spinning disk that steadily completes 2 full revolutions in 10 seconds? __________ What is the angular acceleration (in rad per sec per sec, rad/s2) of a spinning disk that changes its angular velocity from 2 rad/s to 7 rad/s in 10 seconds? __________ For motion in any chosen dimension, a net force, F (S.I. units of N) results in a proportional acceleration, a (S.I. units of ms-2). This we know as Newton’s second law, F = ma, where m is mass (S.I. units of kg), a measure of inertia. For rotational motion about any chose axis, we can write a similar Newton-equation for

which a net torque, (S.I. units of Nm) results in proportional angular acceleration, (S.I. units of rad/s2) where ‘I’ is called the moment of inertia (S.I. units of kgm2):

I eq.1

Applying equation 1, what is the moment of inertia (also give units) where torque,

= 2.0 Nm and angular acceleration, α = 3.0 rad/s2? ________________ This similarity between the above Newton’s equations leads to very similar equations of motion governing both linear and rotation motions. We introduce any such equations governing rotational motion as they are required through the script.

Torque is given by = F r, where F is the applied force, and where r is the shortest distance to the line of action of the force as measured from the axis of rotation. The moment of inertia, I, is a measure of an object’s resistance to a change in its rotational motion about an axis. It is the rotational-analogy to mass, which is a measure of a body’s resistance to change in linear motion.

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You will use two pieces of apparatus to investigate these equations (as parts 1 and 2). The first apparatus lets you apply and calculate a constant torque acting on an object consisting of a pair of cylindrical weights located at the ends of a bar. You’ll measure angular displacement over time from which angular acceleration can be worked out. From this you’ll determine a moment of inertia, I (using equation 1). The second apparatus lets you investigate how I depends on the distribution of mass about the axis of rotation. You’ll change the position of the weights and you’ll use a spring to move the bar back and forth in an oscillation that you time. Part 1: Measure moment of inertia, I, from torque and angular acceleration.

Place cylindrical masses on the bar at r = 0.25 m, their furthest position from the axis of rotation. The bar is attached to an axle which is free to rotate. The mass attached to the line wound around this axle are allowed to fall, resulting in a torque about the axle.

Determining the value of torque used in your experiment

The value of the torque caused by the falling masses is Fr where F is the weight of the mass attached to the string and r is the radius from the centre of the axle to the string where it meets the axle (the shortest distance to the line of action of the force). This value

of remains constant over time given that the weight of the falling mass, F, also remains

constant. Calculate the value of by filling in the following boxes in order:

Mass, m attached to string (kg)

Force, F = m g (N)

Radius, r of axle (m)

Torque, = F r (Nm)

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Measuring the angular acceleration Wind the string attached to the mass around the axle until the mass is close to the pulley. Release it and measure the time, T, for the bar to perform the first complete rotation, the second complete rotation, the third and so on. Remember, a full revolution of 360o is equivalent to 2π radians. Write these values into the third column of the following table. Leave the final column bank for now. You will return to filling in values for the final column shortly.

Number of

Rotations

Angular displacement,

(rad)

Measured time, t (s)

𝑡2 (s2)

0 0 0 0

1 2π

2 4π

3 6π

4 8π

5 10π

6 12π

The equations of motion under constant acceleration look similar for linear and rotational motion. You may be familiar with the equation motion in a straight line (linear motion),

𝑠 = 𝑢𝑡 +1

2𝑎𝑡2, where 𝑠 is the linear displacement (distance covered), 𝑢 is the initial linear

velocity, 𝑎 is the linear acceleration (a constant value for the motion) and 𝑡 is the time taken. The equation for circular motion is analogous:

𝜃 = 𝜔1𝑡 +1

2∝ 𝑡2

where 𝜃 is the angular displacement (angle covered), 𝜔1 is the initial angular velocity, ∝ is the angular acceleration (a constant value for the motion) and 𝑡 is the time taken. This equation simplifies because the bar starts from rest and so the initial angular velocity in

this equation, 𝜔1 = 0, giving:

𝜃 = (1

2∝) 𝑡2 eq.2

In the above table, you have recorded the angular displacement, 𝜃 for a time, t. Now

complete in the final column with the corresponding values you calculate for 𝑡2. It makes

sense to do this because you will now use equation 2 and these values for 𝑡2 to find the

angular acceleration, . The best way to do this is to plot a graph and determine the slope

of the graph. Plot this graph of 𝜃 on the y-axis against values of 𝑡2 on the x-axis, on the following page.

Using a graph to find is a good idea for two reasons. First, it allows you to find an

accurate value of using all of your data. Second, this is a visual method, something one third of your brain is well optimised for processing. Should some data turn out to be erroneous, you will literally see this with a graph.

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Create your graph of 𝜃 on the y-axis against values of 𝑡2 on the x-axis. Do the following plot by hand on this page, or, use JagFit (see back of manual) and attach your printed graph to this page. Important: take care to label axes correctly and include units.

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Finding ∝ from your plot In everyday language we use the word slope to describe a property of a hill. For example, we say that a hill has a steep or gentle slope. The slope tells you how much a value on the y-axis changes for a change on the x-axis. A straight line can be described by the general equation ‘y = mx + c’, where m is the slope (steepness) of the line i.e., the change in y divided by the change in x for the same part of the line, and c is the y-value where the line intercepts (crosses) the y-axis i.e., where x = 0.

Given you’ve plotted 𝜃 on the y-axis against values of 𝑡2 on the x-axis and given we

expect the equation, 𝜃 = (1

2∝) 𝑡2 to apply, then you can expect your data to approximate

to a straight line where m = 1

2∝ and your line goes through the origin (c = 0):

‘y = m x’

𝜃 = (1

2∝) 𝑡2 eq.2

Fit a straight line as best as you can to your data using a ruler. A straight line is the simplest curve to fit to data. Do this by eye, trying to get your ruler a close as possible on average to all the plotted points. Your eye/brain is well adept at this task, and your tutor will account for your fitting by hand. Determine the slope of the line that you have drawn

(fitted by eye) and from this, determine the angular acceleration, ∝. Also write in the correct units for your value:

Average angular acceleration, ∝𝑎𝑣 (as a value with units) = ________ Note that you are dividing a value in radians (rad) by another value in s2 to determine the slope (and so the angular acceleration). Note the similarity of these units to the units for linear acceleration.

But what of the uncertainty in your measurement of ∝? To find out about this, fit two more lines on your plot, the first being the steepest slope that you estimate can still reasonably represent the distribution of points that you plotted, and the second being the least slope. This is a matter for your judgement. Your tutor will account for reasonable fits and also for your fitting by hand. Determine the slopes of these two lines

and write the corresponding maximum and minimum values for ∝ (as values with units):

∝𝑚𝑎𝑥= ________ ∝𝑚𝑖𝑛= ________

Give a value for the uncertainty in angular acceleration, Δ𝛼 in the following form (as a value with units):

Δ𝛼 =1

2(∝𝑚𝑎𝑥−∝𝑚𝑖𝑛) = ________

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And so put your measured value of ∝ into the following standard (compact) form, using this estimated uncertainty, Δ𝛼 (along with units):

∝ = ( ∝𝑎𝑣 +/- Δ𝛼) units

∝ = ( _______ +/- _____ ) _____ Finding the Moment of Inertia

So far, you have determined the torque, and measured the angular acceleration, 𝛼. Use these two values and equation 1 rearranged, 𝐼 = 𝜏 𝛼⁄ (see intro), to determine the

(average) moment of inertia, 𝐼𝑎𝑣, of the net rotating object and write this here (as a value with units):

𝐼𝑎𝑣 = ___________ What is the uncertainty in this value of 𝐼 that you’ve determined? You will learn more on how to handle and combine uncertainties in second semester, so we won’t cover

uncertainties in depth here. Suffice it to say that for the present equation though (𝐼 =

𝜏 𝛼⁄ ), and for the case of an uncertainty in 𝛼 being many times greater than that in 𝜏, we

can make the following approximation for the uncertainty in the moment of inertia, Δ𝐼:

Δ𝐼 =𝐼𝑎𝑣

𝛼𝑎𝑣Δ𝛼 eq.3

where Δ𝛼 is the uncertainty you determined in angular acceleration.

Use equation 3 to determine Δ𝐼 and write this here (as a value with units): ___________ Finally put the moment of inertia, I into the following standard form, using this estimated

uncertainty, Δ𝐼 (along with units):

𝐼 = ( 𝐼𝑎𝑣 +/- Δ𝐼) units

𝐼 = ( _______ +/- _____ ) _____ This completes part 1, well done. Now on to part 2 of this experiment, in which you’ll

investigate how the moment of inertia, 𝐼 depends on where the cylinders are placed.

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Part 2: Investigate how moment of inertia, I, depends on the distribution of mass Take the metal bar on the bench and roll it between your hands. Now hold it in the middle and rotate it about its centre so that the ends are moving most. Which way to rotate the bar is easier? About which axis of rotation does the bar have a higher value of I?

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Attach the weights and the bar to the rotational apparatus known as a torsion axle. This consists of a vertical axle connected to a spring which opposes any departure from the angle of rotational equilibrium.

Note: The apparatus is delicate. So as not to damage the spring, please keep the rod to within half a rotation from equilibrium.

When you rotate the bar, the spring causes a torque about the axis of rotation which acts to restore the bar to the equilibrium angle. Usually the bar overshoots, resulting in oscillation. This is simple harmonic rotational motion. The period of the oscillation, T, is

determined by the torsion constant, 𝜅, and the moment of inertia, I, of the object rotating, in this case the bar and cylindrical weights. These parameters are related by:

𝑇 = 2𝜋√𝐼

𝜅 eq.4

The torsion constant, 𝜅 is written on each torsion axle (you may find this written as ‘D’).

Note it here: 𝜅 = ________

Because Newton’s law is very similar for both linear and rotational motion (see introduction), this simple harmonic rotational motion is very similar to the way a mass on a linear spring undergoes simple harmonic linear motion. For comparison, for a mass on

a linear spring, 𝑇 = 2𝜋√𝑚

𝑘 where m is the mass and k is the spring constant (stiffness).

cylinders

bar

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Now investigate the influence of mass distribution. To do this, vary the position of the cylinders along the bar and measure the period of oscillation, T. To improve the precision with which you measure T, take an average over 10 oscillations for each table entry.

r

Position of each cylinder along rod, m

T

Measured period of oscillation, s

I

Moment of inertia for net rotating object (cylinders + bar),

kgm2

Icyl

Moment of inertia for the cylinders, kgm2

0.05

0.10

0.15

0.20

0.25

For filling in column 3, use equation 4 to calculate the moment of inertia, 𝐼 for the net rotating object i.e., the two cylinders plus the rod. You’ll need to rearrange this equation

to have 𝐼 on the left hand side… For filling in column 4, it turns out that straightforwardly, 𝐼𝑐𝑦𝑙 = 𝐼 − 𝐼𝑏𝑎𝑟,

where 𝐼𝑏𝑎𝑟 = 0.00414 kgm-2 is the moment of inertia of the bar only. On the following page, plot your tabulated values of Icyl versus the position, r. Then on the subsequent page, plot your tabulated values of Icyl versus the position squared, r2.

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Create a graph of Icyl versus the position, r. Do the following plot by hand on this page, or, use JagFit (see back of manual) and attach your printed graph to this page. Important: take care to label axes correctly and include units. Note, your tutor will account for fitting where done by hand.

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Create a graph of Icyl versus the position squared, r2. Do the following plot by hand on this page, or, use JagFit (see back of manual) and attach your printed graph to this page. Important: take care to label axes correctly and include units. Note, your tutor will account for fitting where done by hand.

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What do you conclude about a relationship between Icyl and the position of the masses? Explain by fitting and drawing straight lines using a ruler to each of your graphs and comparing between these two cases. ______________________________________________________________________

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Conclusions and implications Once started in a spin, an ice skater then spins faster by pulling in her arms. This comes from her changing her moment of inertia. Referring to your last answer above, explain whether her spinning faster comes from the moment of inertia increasing, or decreasing. ______________________________________________________________________

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You have measured moment of inertia for an object and you have investigated how it varies as you change the configuration of the object. In doing so, you have also measured moment of inertia for the same object twice, using two different methods. This object is for the case where the weights are positioned at r = 0.25 m. Write here your two values for the moment of inertia of the net object, I (r = 0.25 m), derived from each of parts 1 and 2. Compare between these two values and discuss whether or not they agree. ______________________________________________________________________

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Gyroscopes are used widely to measure acceleration accurately. A standard design for a precision gyroscope uses a spinning disc. It’s an advantage to have a high moment of inertia for this disc, however it’s generally an advantage for a gyroscope to be light too. Explain a design for a gyroscope-disc which maximises this moment of inertia for any given weight. For this, refer to what you have observed independently in this lab i.e., your measurements and graphs. ______________________________________________________________________

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