Experiment 11. Moment of Inertia

Size: px

Start display at page:

Download "Experiment 11. Moment of Inertia"

Stuart Dennis
5 years ago
Views:

1 Experiment Moment of nertia A rigid body composed of concentric disks is constrained to rotate about its axis of symmetry. The moment of inertia is found by two methods and results are compared. n first method moment of inertia is determined oretically by applying formula for moment of inertia of a single disk to each of four disks and adding results. n second method moment of inertia is determined erimentally by measuring acceleration produced by a constant torque on body. The erimental determination of moment of inertia is only valid if friction is negligible. n Part of eriment an estimate of angular acceleration due to friction is obtained. The validity of above approximation is examined. Theory Part Consider a pulley made of four disks. The moment of inertia of a homogeneous disk about axis of symmetry is disk = M R Figure. The body consists of four concentric disks. where M is mass of disk and R is radius. The moment of inertia of a rigid system of concentric disks is n = M R + M R+ = M i R i () where sum extends over all disks each of which has mass M i and radius R i. f mass density ρ is uniform (i.e. constant throughout body) mass of each disk is given by M i = ρ V i = ρ wi π Ri where V i is volume of each disk and w i is width. Substituting this into () moment of inertia of each disk is n πρ i = wi Ri () and total oretical moment of inertia of rotating body is

2 πρ = wi Ri. (3) f density toger with width and radius of each disk are known n moment of inertia of body can be computed. The moment of inertia determined in this manner will be referred to as oretical value ( o ). Theory Part The erimental value of moment of inertia can also be found by exerting a constant torque on body. A mass m is attached to a string which is wrapped around body at some radius R. (Refer to Figure.) R will be one of disk radii R i. f m is released from rest and falls a distance d during a time t acceleration of m is given by d = t a. () Once value of acceleration is known moment of inertia is determined by = m R Theory Part g a - (5) Figure. Experimental determination of moment of inertia. assuming that friction in supports is negligible. A rough measure of friction in supports can easily be found. Suppose body (without mass) is initially spun and N revolutions occur during time T N required for body to come to rest. Assuming angular acceleration due to resistance is constant its magnitude is n given by πn α f =. (6) T This should be approximately true for apparatus. The ression (6) represents in reality average value of angular acceleration due to friction. The magnitude of average resistive torque due to friction in supports is τ f =α f. ncluding this torque in derivation of erimental moment of inertia (5) corrected result is N

3 = m R g - a α f + a /R (7) This is assuming resistive torque is constant. n erimental determination of moment of inertia when m is attached to body resistive torque will have an additional contribution that is proportional to tension of string which connects m to body. This contribution is negligible however if m<<m total.) According to (7) validity of (5) rests upon degree to which following is true: Apparatus o mounted rotational body o vernier caliper o string o masking tape α f a /R <<. o 00-gram slotted weight o two meter stick o stopwatch (8) The pulley is made of aluminum and consists of four concentric disks. (Refer to Figure.) The density of material is given by total mass divided by total volume. The result is 3-3 ρ =.70 x 0 kg m. (0) The string attached to mass m supplies a force that can be applied at any of different radii of pulley. The string can be attached to cylinder with some masking tape. Procedure Parts & ) Using vernier caliper measure and record width w and diameter d of each disk. Obtain 3 or significant figures. Write down uncertainties for each measurement. Number disks according to Figure. ) Cut a piece of string long enough to allow weight hanger to reach floor. Attach a 00-gram weight to string. This is mass m. 3) Attach or end of string to rim of one of disks. Ensure that string will not slip. Rotate body allowing string to wrap around disk without overlapping. Using two-meter stick position bottom of weight so that its vertical distance d from floor is approximately meter ) Steady weight. Release body and measure time t required for weight hanger to strike floor. The timing must be performed very carefully; start timer 3

4 exactly when body is released and stop it just when body strikes floor. Do not permit string to become tangled in supports. Perform two more trials using different students as timers and obtain an average and uncertainty for this measurement. 5) Repeat procedure on anor radius. Your data will consist of three time-to-fall measurements for each different radius. Note that values of m and d are to remain constant. Procedure Part (optional) ) Remove string from body and place a small piece of tape on rim of largest disk. The tape will serve as a reference mark to be used in counting revolutions. ) Spin body as fast as possible while still being able to count revolutions. The angular velocity will n be comparable to a typical final angular velocity in Part. 3) Start time and count number N of revolutions during time T N required for body to come to rest. Record values of N and T N. The data need only be approximate and only one trial will suffice. Analysis Part Construct a data table that contains width diameter radius and moment of inertia of each disk using () and (0). Use S (MKS) units. nclude value of ρ in title of table. Number disks according to Figure. Determine and clearly display value of total oretical moment of inertia o to three significant figures. Analysis Part For each radius R at which a force was applied compute average value of time-offall t. Using () compute acceleration for each value of R. Then using (5) compute for each set of values of R and a. Use S units. Construct a data table that contains values of R t A and. (This table should be separate from table for oretical moment of inertia.) nclude both measured and averaged times-of-fall. Since values of m and d are constant y should be included in title of table. Determine and display values obtained for. Also determine uncertainties for oretical value and for erimental values. For erimental results use fractional uncertainty formula. δt δ = t δr + R δd +. d We assume here that uncertainty in mass m is negligible.

5 Results Report in a table of results erimental and oretical values of moment of inertia ir uncertainties and percentage difference between oretical and each erimental value. Find percentage difference using equation: - x 00%. nclude a one-dimensional error bar comparing single oretical value with erimentally determined values. Analysis Part (optional) Using (6) compute and display approximate value of angular acceleration α f due to friction. Express result in units of rad/sec. For each value of R used compute angular acceleration a/r and corrected values. nclude se values in results table and error bar graph in Part. Questions and Conclusion ) n ory section of your lab report draw free body diagrams for m and for pulley and derive () and (5). Explain your steps. ) n ory section of your lab report derive (6) and (7). Explain your steps. 3) n erimental method lain why acceleration is greater when you attach string to a larger radius. ) Your two erimental results should each give same value for entire of object and not just i of disk to which string was attached. Explain. 5) Was eriment a success? (Did oretical and erimental values agree to degree of uncertainty). Were re systematic errors that could lain differences? 6) Did two different erimental results (derived from wrapping string around different radii) agree with each or? Can you think of any systematic errors that could lain differences? 7) Does friction correction improve agreement between oretical and erimental values? Explain. 5

Rotational Dynamics Smart Pulley

Rotational Dynamics Smart Pulley The motion of the flywheel of a steam engine, an airplane propeller, and any rotating wheel are examples of a very important type of motion called rotational motion. If