Physics 2A

Rotational Motion and Moment of Inertia

©

2003 Las Positas College, Physics Department Staff

Rotational Motion and Moment of Inertia

Purpose: To determine the rotational inertia of a disc and of a ring and to compare these with the theoretical values.

Equipment:            

Rotating Table, Disc, Ring Hooked Mass Set Long Rod Right Angle Clamp Cylindrical Rod Clamp Table Clamp Smart Pulley/Photogate Stop Watch 2-meter Stick Vernier Calipers Carpenter’s Level String

Theory: By now you’ve probably figured out that every variable that was defined when talking about linear motion has an analogue in rotational motion. Instead of distance traveled, d, we have angle turned, θ. Rather than speaking simply of velocity, v, we talk of angular velocity, ω. Every aspect of linear motion has its partner in rotational motion. And not only are they analogous, but they’re related, too! …usually by some power of the radius of the circle. In fact, here’s how it goes…

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Physics 2A

Rotational Motion and Moment of Inertia

©

2003 Las Positas College, Physics Department Staff

Linear Variable

Symbol

Angular Variable

Symbol

Relationship

distance

d

angle

θ

d=θr

velocity

v

ω

v=ωr

acceleration

a

α

a=αr

force

F

τ

τ = r×F

mass

m

I

I = ∫ r 2 dm

momentum

p

L

L = r×p

angular velocity angular acceleration torque rotational inertia angular momentum

Figure 1 Relationships between Linear and Angular (Rotational) Variables

The same basic equations of linear motion can be used for rotational motion with a simple change of variables: x f = xi + vo t + 12 at 2 v f = vo + at

→

θ f = θ i + ω o t + 12 αt 2 ω f = ω i + αt

Eq. 1

The difference between angular and linear motion, and the relation between them, is something that most students are able to understand fairly easily. Almost everyone has played on a playground merry-go-round, and has noticed the difference in velocity between standing at the center and standing at the edge. The confusion begins to set in with discussions of torque and rotational inertia, the questions being specifically why do we need to differentiate between the simple Force and Mass? When discussing linear motion, it is assumed that all objects are “point objects” whose mass is centered symmetrically about a single point, and they have no size to speak of – that is, all objects are shaped essentially like very small balls that don’t spin. Most objects, however, don’t actually fit this category. Most objects (even most balls) are extended objects – they have a measurable size and they may not be totally symmetric. When a force is exerted on an extended object, it matters not only how big the force is, but also where on the object it is applied. Imagine a bar on a table: v

F

F F

Figure 2 The application of a force, and the resulting motion of a bar

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Physics 2A

Rotational Motion and Moment of Inertia

©

2003 Las Positas College, Physics Department Staff

If the force is applied directly to the center of the object, it will translate linearly across the table. However, a push on either side of the center will cause a rotation of the object. When a force is applied away from the axis of rotation (in this case, the center of mass) it causes the object to rotate. This is what we call a torque. The torque is defined as

τ = r×F

Eq. 2

where F is the force applied and r is the distance from the axis of rotation to where the force is applied. Remember that the definition of a cross product is: A × B =| A | | B | sin θ where θ is the angle between vectors A and B. Thus,

τ = r F sin θ

Eq. 3

This means that the largest torque occurs when r and F are perpendicular to each other (sinθ = 1) and the smallest torque is when r and F are parallel (sinθ = 0). As an example, think of using a wrench to turn a bolt:

r

r

F

r

F

Smaller torque τ = |r| |F| sinθ where sinθ Set Up Sensors > LabPro 1. Click on the photogate icons, and verify that “Motion Timing” is selected under “Current Calibrations”. In the same menu, choose “Set Distance or Length…”, make sure that “Smart Pulley (10 Spoke) in Groove” is selected. The program calculates the acceleration and velocity of the falling mass by treating the pulley as a picket fence with the “proper” spacing. 9. Now press Collect and release the mass. Check the results of the computer measurements against your own measurements of accelerations (from height and time). The "by hand" values can differ by as much as 20% from the Logger Pro values. 10. When you are satisfied that you are measuring acceleration correctly, use Logger Pro to measure the acceleration of the mass as it falls and turns the empty table (Configuration I). Repeat this measurement two times and record the three different acceleration values (three different slopes of v vs. t). You can later use the range in these values to determine your uncertainty range for Iexp. 11. It will later be necessary to determine the frictional torque acting on the system. To do this, replace the 200 gram mass with a tiny mass. Start with 5 or 10 grams and increase the mass until the table starts to turn. The minimum mass which will just start the table rotating is mo. Record this value. 12. Add the disk to the table (this is configuration II.) and repeat Steps 5 and 6. To overcome the greater inertia of the table, use m = 500 grams. Don't forget to measure the minimum mass mo that just starts the table + disk rotating. 13. Remove the disk and add the ring to the table (This is configuration III). Repeat Steps 5 and 6 (500 grams may work well for this configuration). Be sure to record the new value of m and mo. 14. Measure r = (diameter of the shaft of the rotating table / 2).

Analysis: 1. For each configuration, calculate the average acceleration, a, of the mass, m, for that configuration. Also calculate uncertainty (deviation) in the experimental acceleration. 2. Using Newton’s 2nd law: ΣF = ma (applied to the hanging mass) and it’s rotational analog ∑ τ = Iα = rT

Eq. 7

(where T is the tension in the string), solve for I by eliminating T (show your work) to get 7 of 13

Physics 2A

Rotational Motion and Moment of Inertia

©

2003 Las Positas College, Physics Department Staff

I exp

r 2 m( g − a ) = a

Eq. 8

3. Calculate the rotational inertia I, using Eq. 8 for the three configurations: I1 = Itable I2 = Itable + Idisc I3 = Itable + Iring 4. Calculate: Idisc = I2 - I1 Iring = I3 - I1 5. Calculate the theoretical values of Idisc and Iring using the formulas in the text. Note that the disk has a weighted "plug" at the center to replace the missing mass in the hole (so it can be considered a solid disk). Also note that the "ring" is actually a hollow cylinder with inner and outer radii (Do you think if makes any difference in the answer in this case if you use the formula for a ring rather than that of a hollow cylinder?). 6. Now, if Iexp does not equal Itheory then part of the problem might be frictional torques acting on the system. Assuming that this frictional force prevents the shaft from rotating, we can measure it directly by hanging a small weight mo. This weight will be the maximum value that can be hung without the table rotating. T = mo g

r Shaft Ff

Figure 7 Finding the Force of Rotating Friction

Since the system is now in equilibrium, we find that τ net = 0 and m0 gr − F f r = 0 . Also, if we call F f r = τ f , then: m0 gr = τ f .

Eq. 9

7. Determine the value τf for each configuration and record. Now go back to Step 2 of the analysis and include τf in your determination for Iexp. Show that the Corrected value is: r 2 m( g − a ) τ f r Iexp = a a Where τf = mogr. 8. Calculate the corrected values for Iexp for each configuration.

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Physics 2A

Rotational Motion and Moment of Inertia

©

2003 Las Positas College, Physics Department Staff

9. Using the corrected values repeat Step 4. 10. Determine the total uncertainty in the experiment: i.e., theory + exp. 11. Compare the theoretical values to the experimental values of Idisc and Iring (are the % differences within the uncertainty?) If not, why not?

Results: Write at least one paragraph describing the following: • what you expected to learn about the lab (i.e. what was the reason for conducting the experiment?) • your results, and what you learned from them • Think of at least one other experiment might you perform to verify these results • Think of at least one new question or problem that could be answered with the physics you have learned in this laboratory, or be extrapolated from the ideas in this laboratory.

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Physics 2A

Rotational Motion and Moment of Inertia

©

2003 Las Positas College, Physics Department Staff

Clean-Up: Before you can leave the classroom, you must clean up your equipment, and have your instructor sign below. If you do not turn in this page with your instructor’s signature with your lab report, you will receive a 5% point reduction on your lab grade. How you divide clean-up duties between lab members is up to you. Clean-up involves: • Completely dismantling the experimental setup • Removing tape from anything you put tape on • Drying-off any wet equipment • Putting away equipment in proper boxes (if applicable) • Returning equipment to proper cabinets, or to the cart at the front of the room • Throwing away pieces of string, paper, and other detritus (i.e. your water bottles) • Shutting down the computer • Anything else that needs to be done to return the room to its pristine, pre lab form.

I certify that the equipment used by ________________________ has been cleaned up. (student’s name)

______________________________ , _______________. (instructor’s name)

(date)

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Physics 2A

Rotational Motion and Moment of Inertia

©

2003 Las Positas College, Physics Department Staff

Data Tables hand check:

distance, h: _________

time, t: _________

determination of acceleration:

computer measurement of acceleration: _________ percent difference between hand and computer measurements: _________ Derivation of Eq. 8:

Derivation of corrected form of Eq. 8:

Configuration I: Empty Table hanging mass, m: _________ acceleration, a1: _________

average acceleration, aavg: _________

acceleration, a2: _________

uncertainty in acceleration, δa: _________

acceleration, a3: _________ minimum mass which will turn table, mo: _________ rotational inertia from Eq. 8, I1:

torque due to friction, τf: _________ Corrected value of I1: total uncertainty: 11 of 13

Physics 2A

Rotational Motion and Moment of Inertia

©

2003 Las Positas College, Physics Department Staff

Configuration II: Table + Disk hanging mass, m: _________

mass of disk: _________

acceleration, a1: _________

average acceleration, aavg: _________

acceleration, a2: _________

uncertainty in acceleration, δa: _________

acceleration, a3: _________ minimum mass which will turn table, mo: _________ rotational inertia from Eq. 8, I2:

experimental rotational inertia of disk, Idisk, exp:

theoretical rotational inertia of disk, Idisk, theory:

torque due to friction, τf: _________ corrected value of I2: corrected value of Idisk, exp: total uncertainty:

percent difference, theory and experiment:

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Physics 2A

Rotational Motion and Moment of Inertia

©

2003 Las Positas College, Physics Department Staff

Configuration III: Table + Ring hanging mass, m: _________

mass of ring: _________

acceleration, a1: _________

average acceleration, aavg: _________

acceleration, a2: _________

uncertainty in acceleration, δa: _________

acceleration, a3: _________ minimum mass which will turn table, mo: _________ rotational inertia from Eq. 8, I3:

experimental rotational inertia of ring, Iring, exp:

theoretical rotational inertia of ring, Iring, theory:

torque due to friction, τf: _________ corrected value of I3:

corrected value of Iring, exp:

total uncertainty:

percent difference, theory and experiment:

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