Experiment no 4

Experiment No 4
Determine the moment of inertia of flywheel by falling weight method
Flywheel is a solid disc of significant size and weight mounted on the shaft of machines such as steam engines, diesel engines, turbine etc. Its function is to minimize the speed fluctuations that takes place when load on such machines suddenly decreases or increases. The flywheel acquires excess kinetic energy from the machines when load on the machine is less or its running idle and supplies the stored energy to the machine when it is subjected to larger loads. The capacity of storing / shedding of kinetic energy depend on the rotational inertia of the flywheel. This rotational inertia is known as moment of inertia of rotating object namely wheels. The moment of inertia about the axis of rotation can be analytically estimated as
I =∫r2dm
For the known geometry and mass density of the material used, in SI system of units the unit of moment of inertia is Kg.m2
In lab the object of the experiment is to estimate the moment of inertia of the given flywheel experimentally
A string carrying a suitable mass m at its one end having a length less than the height of the axle from the ground is wrapped completely and evenly round the axle. When the mass m is released, the string un winds itself, thus, setting the flywheel in rotation. As the mass m descends the rotation of the flywheel goes on increasing till it becomes maximum when the string leaves the axle and the mass drops off.
Let H be the distance fallen through by the mas before the string leaves the axle and the mass drops off, and let v and ω be the linear velocity of the mass and angular velocity of the flywheel respectively at the instant the mass drops off. Then, as the mass descends a distance h, it losses potential energy, mgh which is used up.
(1) Partly in providing kinetic energy of translation 1/2mv2 to the falling mass itself.
(2) Partly in giving kinetic energy of rotation ½ I ω2 to the flywheel ( I = M.I of the flywheel )
(3) Partly in doing work against friction

If the steady work done against friction is F per turn, and, if the number of rotations made by the flywheel till the mass detaches is equal to n1 , the work done against friction is equal to n1 F. hence by the principle of conservation of energy, we have
mgh =1/2mv2+1/2 I ω2 + n1 F Eq-1
after the mass has detached, the flywheel continues to rotate for considerable time t before it is brought to rest by friction. If it makes n2 F and evidently it is equal to the kinetic energy of the flywheel at the instant the mass drops off. Thus
n2F = ½ I ω2
F=1/2 I ω2 / n2
And substituting this value of F in equation 1 we get mgh= 1/2mv2 + ½ I ω2 + ½ I ω2 n1/n2
When I= 2mgh –mv2/ ω2 (I+n1/n2) Eq-2
I= 2mgh –mr2 ω2 / ω2(I+n1/n2) Eq-3
If r be the radius of the flywheel , ν=rω
After the mass has detached, its angular velocity decreases on account of friction and after some time t, the flywheel finally comes to rest. At the time of detachment of the mass
The angular velocity of the wheel is ω and when it comes to rest its angular velocity is zero. Hence, if the force of friction is steady, the motion of the flywheel is uniformly retarded and the average angular velocity during this interval is equal to ω/2. Thus
ω/2 = 2πn2 / t
ω = 4πn2 / t substituting ω from equation’s we get
I = mn2/ (n1+n2) {ght2/8π2n22 –γ2}
Thus observing the time t and counting the equations n1 and n2 made by the flywheel, its moment of inertia can be calculated from equations. Wheel, its moment of inertia can be calculated from equations.

Apparatus: 1 flywheel mounted on the shaft fitted on the wall
2 known masses (1kg, 2kg, 3kg)
3 a string with a pin on one end and a hook on the other end
4 a stop watch
5 a vernier calliper
Exercise: 1 measure the radius (r) of the axle with the use of calliper
2 Push the pin of the string into the hole of axle, wind the string closely on the axle and count the number of turns (n1)
3 Attach a known mass (m) at the free end of the string
4 Release the mass. Note the time (t) of the fall from instant of release to the instant it gets detached
5 Count the total number of revolution (n) made from the instant the mass is released to the instant it comes to rest. In order to help counting revolutions a mark with red paint has been made on the wheel and a fixed pointer close to the wheel has been provided.
6 The steps (d) and (e) should be repeated thrice and an average value should be taken for ‘t’ and ‘n’
7 Enter the observations in the respective column in the observation table for a mass (m)
8 Change the mass (m) at free end of the string
(A) Repeat the steps from 4 to 8
(B) Avoid large mass (m) otherwise accuracy in measurement would be affected.