EXPERIMENT 103MOMENT OF INERTIAANALYSISExperiment 103 entitled Moment of Inertia which it calculates the disk and ring mass moment of inertia and compare the output that rotates to different axes: diameter and center. The estimation of the rigid body rotational inertia is the moment of inertia. It is the resistance of a rigid body to any change in its rotational motion about a specified axis. For this activity, two different set ups were utilized, horizontal and vertical. We began the activity by setting up the equipment needed.
The mounting rod has been attached to the smart pulley and connected to the photogate. Mass hanger binds with a thread that loop around through the cylinder along the vertical shaft. We plugged the photogate head in a 220 Volts source. On the center of vertical shaft, the disk placed on it and smart timer connected to the photogate head.On verifying the disk’s moment of inertia that rotates on the center which it was mounted on the rotating platform.
The diameter of the shaft measured by the Vernier caliper. We recorded the radius of the shaft as r.In order for us to overcome the existing kinetic friction, we added small amount of mass to the weight hanger and observed a constant speed on its way down, the small mass we added is defined as the friction mass.According to the manual, the timer was set to Accel, Linear Pully to calculate the disk’s moment of inertia that rotates on its diameter. The D-shaped hole at the side of the disk has been inserted and the disk from the vertical shaft has been removed.After gathering the data needed, there were terms and theories to be defined in order to analyze the results correctly. The rotational analogue of mass for linear motion is called moment of inertia. The mathematically equation for moment of inertia is the mass times the square of distance to the rotation axis_I=mr2Equation 1Where m is the mass of the particle and r is the shortest or perpendicular distance relative to the axis of rotation.The basis of all moment of inertia is the point mass relationship since any object can accelerate from a group of point masses. Also, it is a measurement of any object’s resistance that changes in rotation direction.The tensor and the scalar arethe two types of the moment of inertia. The geometric objects which it is related to the linear relations between the scalars, vectors, and others is called Tensors. Just as a scalar is described by a single number, and a vector with respect to a given basis is described by an array, any tensor with respect to a basis is described by a multidimensional array. This experiment is the moment of inertia scalar form will be use.For a group of particles having different mass and shortest or perpendicular distance relative to the axis of rotation, it is given by_I= i=1Nmiri2= m1r12+ m2r22+ m3r32+ (Equation 2) For rigid body consisting of continuous distribution of mass, the moment of inertia can be computed by taking the integral of the masses relative to the axis of rotation.I= r2dm(Equation 3)The mass element , dm, expressed in geometry of the object terms so that the integration can be achieve over the object.Thus, for the moment of inertia of hollow cylinder or ring, recall,=dmdvThus, Equation 3 can be rewritten as_I= r2 dV(Equation 4)Substitute eqn. 4 to eqn. 3 will give the formula for inertia with respect to volume and with density, , as constant and is uniform. I= r2dv (Equation 5)If a disk of mass M, thickness L, and radius R rotated about an axis through its center and perpendicular to its plane as shown in the figure Belowcenter23939500Figure 1. Hollow cylinder or ring.From equation 5, the infinitesimal volume is given by_dv=(2rL dr)Substituting it to the equation will yieldI= 12( R2 L)(R2)(Equation 6)Equation 6 can be simplified in terms of the total mass M which is equal to the product of density and the total volume V. The Total volume of disk and the total mass of the disk can be expressed in the equations below respectively_V= R2LM= V= R2 L The ring or solid cylinder moment of inertia will be, I= 12MR2On the other hand, if a disk of mass M, thickness L, and radius R rotated about an axis through its diameter, its moment of inertia can be defined as, whereas, Iy is the equation 8.The Rotating Rigid Body that relates in Newton’s Second Law of Motion has the force to rotate its body on an axis which is a quantity called Torque.Torque is the calculations on how much force exert on an object that causes to rotate. The pivot point, object rotates on an axis, will be label as O’ and force is F’. The distance is called moment arm which it is denoted as r’ and it is also a vector.Torque can be expressed as, torque=force x lever arm”=F d(Equation 8)By multiplying Newton’s second law of motion Fnet = ma and multiplying it by R yields Fnet R = m R a, where a can be equated to R a and Fnet= “net, therefore formulating,”net=m R2 am R2 is the moment of inertia and as we recall and can be simplified as :”net=I a(Equation 9)Thus, the experimental value of moment of inertia isI= m g-a r2a(Equation 10)Where: m = mass added on pan (without the friction mass)a = acceleration of the ring or disk which taken from the smart timerr = radius of shaft to which the thread is woundOn this experiment, based from the observation the disk moment of inertia on a center is greater than the disk that rotates to its diameter. As we can see and based from the results of the computations, the moment of inertia depends on distribution of weight within an object. From the computed moment of Inertia from equation 10 and equation 8, 93353.9315 gcm2 and 46628.5282 gcm2 respectively, we can see that there is a greater Moment of Inertia on the disk which lied horizontally. CONCLUSIONThe external forces of the axis’ distance or rotation causes the moment of inertia that the output won’t be constant even if the rigid body mass is constant. The net torque is also the cause on the rotational motion of a rigid body changes.The moment of inertia is associated with the angular acceleration from the torque equation where it is the result of inertia moment and angular acceleration. The moment of inertia is greater than the ring for the disk because while a disk and a ring have equal mass and radius, the ring mass is distributed equally at a distance equal to the ring radius. The moment of inertia for the disk is lesser because most of the mass lies closer to the axis of rotation.The experiment of the disk’s moment of inertia that results a greater than the ring due to the larger radius of the disk than the ring; thus, using the formula given, the values would be different in which the bigger moment of inertia was acquired with the radius of the disk than the ring. Likewise, the position of the axis of rotation played a big part in the difference of the moment of inertia of the disk and ring because of the mass distribution in both rigid bodies.