To determine the rigidity modulus of the material of the given wire by dynamical method using a Torsional pendulum.
:Torsional Pendulum, Stop Watch, Vertical Pointer, Screw Gauge and Vernier Calipers.
Torsional Pendulum consists of a uniform metal disc (or cylinder) suspended by a wire whose rigidity modulus is to be determined. The lower end of the wire is gripped in a chuck fixed at the center of the disc and the upper end is gripped in another chuck fixed to a wall bracket as shown in the fig.
The disc is turned through a small angle in the horizontal plane to oscillations about the axis of the wire. The period of oscillations given by
T = 2π √I/C-------------------------------------------(1)
Where I is the moment of inertia of the disc about the axis of rotation and C is the couple per unit twist of the
But C = πna ^{4}/2L-------------------------------------------(2)
Where a is the radius of the wire L is its length and n is the rigidity modulus. From (I) and (II) we have
n = 8πI/a ^{4} L/T ^{2}-------------------------------------------(3)
In the case of a circular disc (or cylinder) whose geometric axis coincides with axis of rotation of the moment of inertia I is given by
I=MR ^{2}/2
Where M is the mass of the disc and R is the radius .On substituting the value of I in the Eqn. (III), we get
n = 8π/2 MR ^{2}/a ^{4} L/T ^{2}-------------------------------------------(4)
A meter wire whose ‘n’ is to be determined is taken without any kinks. The disc is suspended from one end of the wire .The other end of the wire is passed through the chuck fixed to the wall bracket and is rigidly fixed .The length ‘L’ of the wire between the chucks is adjusted to a convenient value (say 50 cms). A pin is fixed vertically on the edge of the disc and a vertical pointer is placed in front of the disc against the pin to serve as a reference to count the oscillations.
The disc is turned in the horizontal plane through a small angle, so as to twist the wire and released. There should not be any up and down and lateral movements of the disc. When it is executing Torsional oscillations, time for 20 oscillations is noted twice and the mean is taken. The period (T) is then calculated 1/T ^{2}
The experiment is repeated for different values of ‘L’ and in each case the period is determined. The value of L/T^{2} is calculated for each length. The observations are tabulated. From the observations mean the value of L/T^{2} is calculated.
The mass ‘M’ of the disc is measured with a physical balance and its radius ‘R’ is calculated with Vernier calipers. The radius of the wire ‘a’ is determined very accurately with screw gauge at three of four different places and means value is taken since it occurs in fourth power.
Substituting these values in eqn (IV) ‘n’ is calculated. A graph is drawn taking the value of ‘L’ on the ‘x’ axis and the corresponding values of T^{2} on the Y- axis. It is a straight line graph passing through origin. Slope can be calculated from the graph by inverting the slope we will get L/T2 Substituting this value ‘n’ is calculated.
The wire should be free from kinks.
The disc should not wobble.
Least count of vernier callipers
LC = 1 MSD / n; where, n= Total number of divisions in vernier scale
Least count of screw gauge
LC = 1 PSD / n; where, n= Total number of divisions on head scale
Tabular form:
Determination of Radius of disc
S.No. | MSR(cm) | VSR(cm) | (D) TOTAL = MSR + VSR(LC) |
1 | |||
2 | |||
3 | |||
4 | |||
5 | |||
6 |
Diameter of disc, D
Radius of disc R = D / 2
Determination of radius of wire (a)
S.No. | PSR (mm) | Corrected HSR | (A) TOTAL = PSR + HSR(LC) |
1 | |||
2 | |||
3 | |||
4 | |||
5 | |||
6 |
Diameter of Wire A =
Radius of Wire a = A/2 =
Least count of Vernier callipers(L.C) =----------------cms
Least count of Screw gauge (L.C) =----------------cms
Average radius of the wire (a) =----------------cms
Mass of the disc (M) =------------------------gms
Mean radius of the disc R =----------------cms
Table to find time period
Mean value of L/T ^{2}=
Calculations
m= (8π/2) (MR^{2}/a ^{4}) (L/T ^{2})
m= (8π/2) (MR^{2}/a ^{4}) (1/slope)
Rigidity modulus (n) of the wire dynes/cm ^{2}(By table)
Rigidity modulus (n) of the wire dynes/cm ^{2}(By Graph)
S.No. | Length L | Time for 20 oscillations/Trail I/Trail II/Mean time | Time Period T=Meantime/20 | T ^{2} | LT ^{2} |
1 | |||||
2 | |||||
3 | |||||
4 | |||||
5 | |||||
6 |
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