Angular Acceleration And Moment Of Inertia Report Examples

Introduction.

The purpose of this experiment is to elaborate how the angular acceleration of an object is measured. The report also explains how the angular acceleration of an object can be made constant. In addition, the investigation aims to establish if there is an association between the moment of inertia and probability distributions.

Procedure

Angular Acceleration
The radius of the stepped pulley and the plastic disks are measured using the vernier calipers. The hanging mass is lowered and the apparatus is observed to ascertain that the disk is accelerating and that the friction is minimum. Logger Pro is used to measure the linear acceleration for five different falling masses. The average acceleration standard deviations of the accelerations are also computed.

Moment of inertia

The pendulum's period is measured for small oscillations about the pivot hole drilled. Three trials are performed for five oscillations.
The width, length, and mass of the block and the height are measured. The width and length are used as the limits in the integral which defines the moment of inertia.
The derivation is used to compute the numerical value which is compared to the experimental value and the percent difference computed.

Data

Acceleration
Experimental Acceleration m/s2
Average Acceleration = 0.043 m/s2
Standard deviation of acceleration m/s2 = 0.018382383

Moment of Inertia.

Experimental moment of inertia (kg*m2) = 0.0035
Theoretical moment of inertia (kg*m2) = 0.00276
Percentage difference = 21.1%

Data analysis

Newton's second law in rotational motion can be defined to be Torque = Iα.
When the angular acceleration is constant, the angular speed obeys kinematic equation that is identical to the straight line motion  =  0 + αt. The  0 is the initial angular velocity and it is zero if the disks starts its motion at rest.
The assumption behind this experiment is to consider that the tension is only due to the hanging mass. Therefore, the analysis assumes a frictionless system with the weight of the vertical pulley being assumed to be zero. The linear acceleration can therefore be considered as a product of the angular acceleration and the radius of the pulley.
T – mg = -ma
For a massless vertical pulley, a = αrs
Therefore, Torque = rs T = Iα where, T is the tension, rs is the radius of the step pulley and I is the moment of inertia.

Substituting for the tension

Torque = rs (mg – mαrs) = Iα

Rearranging the formula,

a = mgrs2/ (I + mrs2)

The angular acceleration can then be obtained from the formula,

α = a/ rs
The experimental values of linear acceleration are compared to the theoretical values that were computed using the formula a = mgrs2/ (I + mrs2) for the masses of the different objects considered in the investigation.
The percentage error in the experimental value obtained can be computed using the formula (0.00546/0.043)* 100= 12.69767% The average difference between the theoretical value and the experimental value will be divided by the average experimental acceleration so as to obtain the error.
Comparison of the theoretical angular acceleration and the experimental angular acceleration would yield the same value since both quantities obtained by divided the linear acceleration by the same value rs

Moment of inertia.

The rotational inertia of an object is expressed by the second moment of the magnitude of the position vector from the axis with respect to the mass element. That is I = ʃ r2 dm
in the experiment we are considering a block and as such the differential element is two-dimensional since there will be a change in the length and the width of the block during rotation. The differential element of mass will therefore be replaced with the differential area element.
Thus, dm= σ dA where dA=dxdy. σ-represents the differential mass element divided by the differential area element.
The double integral is expanded with the limits of x being 0.179 and 0 and the limits of y being 0.091 and 0. integrating and substituting these values yields a value of 0.002756 kg*m2 which is the theoretical value of the moment of inertia.

The experimental value of the moment of inertia is computed using the formula,

T = 2 pie * √(I/Mgh) the period of the pendulum is observed and since the mass of the object is known and the distance from the center of object to the axis of rotation was measured, the moment of inertia can be obtained by rearranging the formula. The experimental value of the moment of inertia is obtained to be 0.0035 kg*m2.
The percentage error is computed using the formula (0.0035-0.00276)/0.0035 = 0.2114286 ≈ 21.14%

Discussion

Acceleration
The experimental acceleration was measured as the object was dropped. The standard deviation is a measure of how far the individual values of acceleration were dispersed from the average value. The value of standard deviation observed in this experiment is within the expected limits considering that different masses were used and as such the mean value is not the average value of the replication of different measures of the same object. The relationship between angular acceleration and linear acceleration is employed in the study to present a method for measuring angular acceleration.
The moment of inertia of a rotating object can be expressed as a probability distribution. The second part of the experiment uses probability distribution to determine the moment of inertia of a rectangular block. The rotational inertia of an object is expressed by the second moment of the magnitude of the position vector from the axis with respect to the mass element. The experiment utilizes the relationship between the periodic time of the oscillation of a pendulum and the moment of inertia to compute the experimental value of moment of inertia.
The errors incurred in the experiment were mainly experimental errors. The acceleration of the object was only recorded once for every mass. Therefore were experimental errors which could be reduced by replicating the experiment preferably three times for every mass and computing the average. In the moment of inertia experiment, the number of replicates for the period of the pendulum was three and there is a great difference between he first two values and the third value.
The percentage error indicate that there is a high amount of experimental errors in the experiment and replicating the procedure would help reduce the percentage error in the experiment. The first two measures are tend too increase the experimental value of the moment of inertia. This is because the moment of inertia is directly proportional to the period of the pendulum and therefore an increase in the period of oscillation increases the moment of inertia.

Conclusion

The rotational inertia of an object is expressed by the second moment of the magnitude of the position vector from the axis with respect to the mass element. The angular acceleration of a rotating object is computed form the linear acceleration by dividing it with the radius of the step pulley. The angular acceleration is kept constant through the use of a uniform angular velocity throughout the experiment.