(Mass Moment of Inertia)
To experimentally measure the mass moment of inertial of a solid disc
A solid disc and axle, vertical holding rod, a few meters of inextensible string, a stop watch or two electronic time gates, a weight hanger, assorted weights, and a calculator
The condition for the rotational equilibrium of a rotating disc about its centroidal axis is
ΣT = Iα.
I is the mass moment of inertia of the solid disc about its centroidal axis (the axis that is normal to it and passes through its center of mass). Of course, the center of mass of a uniform solid disc is the same point as its geometric center.
In this experiment a solid disc that can spin about its centroidal axle will be given angular acceleration by applying a constant tangential force to its outer edge for a short period (t1). The disc accelerates during t1. After removing the force at the end of period t1, the disk will decelerate due to friction (mainly at its axle) and comes to stop within a period (t2). By measuring t1 and t2, the angular acceleration of the disc in each phase can be calculated. A schematic diagram of one possible apparatus setup is shown in Fig. 1.
The hanging mass M2 exerts a constant tangential force F = M2g onto to outer edge of the small disc or cylinder (Fig. 2). When M2 moves from A to B during period t1, it accelerates the disc. The linear acceleration a of M2 during the falling time t1can easily be calculated using the distance y = AB.
If α1 is the angular acceleration of the attached discs, it is related to a by a = R1α1 from which α1 can be calculated.
α2, the angular deceleration of the disc during the stopping period, can also be calculated from time t2 or the angular displacement θ2 measured by counting the number of turns during deceleration.
Let ΣT be the net torque acting on the discs causing it to rotate about its centroidal axis. With friction present at the axle, ΣT = Tapplied - Tfriction.
The applied torque is Tapplied = F∙R1 where F = M2g is the tangential force and R1 is the radius of the disc about which the string is wrapped.
1) Gather the appropriate equipment as per the equipment list.
The trick is to measure the falling time (t1) of weight M2g from A to B in order to calculate its falling acceleration, a. Do all necessary preparations to set the apparatus as shown and
2) Let M2 (The weight hanger including the added weights) fall and pull the string that rotates the disc. M2 will have linear acceleration, a1, while the disc gains angular acceleration, α1. The two are related by a1 = Rα1.
3) Calculate a1 from the travel time (t1) and the falling distance y = AB. Time t1 can either be measured by a stop watch or an electronic time gate.
The distance y = AB can be measured by a ruler or a tape measure.
4) Calculate α1.
5) Calculate the applied torque Tapplied = F∙R1 in which F = M2g.
6) Calculate final velocity, vf , of the falling mass and the ωf of the solid disc. These are the final speeds at the end of the acceleration period. As soon as the weight hits the ground or passes point B, it must no longer pull the string and the disc must no longer accelerate. The arrangement must be made for this to happen. Make sure that Point B is the end of the acceleration period after which the rotating disc starts to decelerate and comes to stop.
7) While a group member is measuring the falling time, another member must count the number of turns the disc makes before it comes to stop. This means the number of turns during the slowing down period.
8) Convert the number of turns made to radians and calculate the angular deceleration, α2 , of the disc during the slowing down period.
9) Treat I as an unknown, and use α2 to calculate the frictional torque, Tfriction. The frictional torque, Tfriction, will therefore be in terms of the unknown, I.
Tfriction = - Iα2 (Why?)
10) Apply the formula ΣT = Iα1 (during the acceleration period) to solve for the unknown or the measured value of I.
11) Calculate the accepted value of I directly from I = 0.5MR2 + 0.5M1R12 by using the masses and radii of the discs. If masses are not given, calculate them for each disc using M = ρV where ρ is the mass density of the material of the discs.
12) calculate the %error on I.
13) Repeat the experiment for two more different values of M2, the falling weight.
14) Average the 3 values you found for I in step 10 and calculate a final %error on I.
15) Calculate the maximum angular momentum of the disc during the experiment based on the value of I in Step 14.
g = 9.81 m/s2.
Measured: M, M1, M2, t1, t2 or θ2, R, and R1
|Trial||M (kg)||M1 (kg)||M2 (kg)||t1 (s)||t2 (s)||θ2 (turns)||θ2 (radians)||R (m)||R1 (m)|
Comparison of the results:
Provide the percent error formula used and the calculated percent errors in each case as well as the percent error on the average value of I.
State your conclusions of the experiment.
Provide a discussion if necessary.