Experiment 11
Introduction:
When a body moves in a circle and its angular speed changes, the body experiences an angular acceleration. Torque gives the body an angular acceleration. Rotating a body necessitates to overcome rotational inertia of the body, known as its moment of inertia. In highly symmetrical cases it is feasible to develop formulas for calculating moment of inertia of a body. In this part of the experiment, a constant torque was applied to a circular disk that could rotate on a low friction bearing. The disk span up to a higher angular speed. Object of this experiment was to calculate the disk's angular acceleration and relate it to the applied torque.
Theory:
A hanging mass fell and accelerated the disk by unwinding a string wound around one of three small (stepped) pulleys mounted to the disk. A photogate was used to measure the angular speed of the vertical pulley. From this, logger pro was used to calculate the linear speed of the string and its acceleration. It was assumed that v is the linear speed of the string; the sub-script V stands for the vertically mounted pulley and S for the stepped pulley. Then the angular speed of each pulley is written as follows. Each r is the radius of each pulley.
ωV = v/rV and ωS= v/rS (1)
The tension in the string was the force exerted the torque and it was not simply the weight of the hanging mass. The mass of the vertical pulley and all friction was ignored. Then Newton's second Law gives for the falling weight:
T - mg = -ma
Ignoring the mass of the vertical pulley,
a = α rS (2)
Now, applying the definition of torque to the disk and using the rotational version of Newton's second law of motion gives:
τ = rS. T = I.α
Substituting for the tension gives: τ = rS.(mg-m α rS) = I.α
Substituting from Eqn. (2) and solving for the linear acceleration gives
α = mg rS 2/( I + mrS 2) (3)
For rotational motion τnet = I.α , where I is the rotational mass of the object or the rotational inertia or the moment of inertia. Moment is a statistical term which measures how spread out the mass is distributed from the axis of rotation. Differently shaped objects have different rotational inertias and the same object even has different moments of inertia depending on the choice of the rotation axis. In this experiment, a solid disk of mass M and radius R is used with an axis perpendicular to the plane of the disk and through the disk's center. In this case, the moment of inertia is I = MR2/2
Procedure:
Following figure depicts the apparatus used in this experiment: a heavy disk that can rotate in a horizontal plane on a low friction bearing.
Angular Acceleration:
A vernier caliper was used to measure diameter of stepped pulley used and rS was obtained by dividing the diameter by two. Mass M and radius R of plastic disk were measured. Moment of inertia of plastic disk was then calculated. The apparatus, like above figure, was set up. White, kite string or fishing line was used as the string and it was wrapped around the screw a few times. After this, the software for using the photogate and the 10-spoke pulley was set up by executing several menu commands. The length of the string was so adjusted that the weight hanger does not strike the floor when all of the string has unwound. A foam pad was placed beneath the hanging weight to avoid damage if the string would break. Slotted masses and the Beck mass hanger were used. A mass was lowered before recording data. The apparatus was examined to make sure the disk accelerated and the friction was minimal. Logger pro was used to measure the linear acceleration of 10-spoke pulley for five different falling masses and to calculate an average acceleration for when the hanging weight fell. Data and standard deviation were recorded in a spreadsheet for each trial. For each trial, the linear acceleration was calculated from equation 3 and value was compared with measured one. Theoretical acceleration was entered in the spreadsheet.
Moments:
Rotational inertia is a measure of how hard it is to start an object rotating about a given axis. A certain group of integrals of functions that describe how something is distributed is called moments. In this experiment it is mass, not probability, that is distributed in space.
T = 2π(I/Mgh)^0.5 (4)
Finally, formula of moment of inertia was derived for a plane rectangle of length b and width a.
I = M/3.(a^2 + b^2)
After this, number for M, a and b were substituted into the formula derived and a numerical value was computed. Percentage difference was calculated between this value and the one obtained from equation no. 4.
Data Analysis:
where, I = 1/2 x mass of wheel x (radius of wheel) 2
Percent difference for Trial-1 is 0.018 i.e 0.018/0.036 x100% = 50 %
Percent difference for Trial-2 is 0.009 i.e 0.009/0.048 x100% = 18.75 %
Percent difference for Trial-3 is 0.036 i.e 0.036/0.064 x100% = 56.25 %
Percent difference for Trial-4 is 0.052 i.e 0.052/0.08 x100% = 65 %
Percent difference for Trial-5 is 0.067 i.e 0.067/0.096 x100% = 69.8 %
Aver Period (sec) = (0.741 + 0.661 + 0.777 )/3 = 0.726
Theo Inertia = Mass of Block (kg) x [ (length of block) ^2 + (width of block) ^2 ] /3
= 0.2188/3x (0.18^2 + 0.09^2)
Discussion:
In case of data analysis of Part-1, it has been found that there is massive difference between theoretical values of acceleration from experimental values of acceleration for all trails except trial no.2. There may be various source of error for which this discrepancy has occurred. One of them may be not measuring the stepped pulley diameter accurately. Incorrect measurement of mass and radius of disk may result in the error. During calculation of radius R of the plastic disk, slight difference in the moment of inertia formula was ignored and it might result including the shape of stepped pulley. This might cause the difference of experimental and theoretical value.
For part two, the % difference between theoretical and experimental values was small and thus acceptable.
