# M-5 TORSION PENDULUM AND LATHE

1. PURPOSE:

(1) To determine the rigidity modulus of steel and brass.

(2) To measure the moment of inertia of several bodies.

2. APPARATUS:

Torsion lathe.
Torsion pendulum, with accessories.

3. THEORY:

(A) RIGIDITY MODULUS

Consult a good textbook for amplification of the bare details given here.

The modulus of rigidity n (shear modulus) of a solid is defined by:

 [1]
```
F/A
n = ———
x/ℓ
```

where F is the applied force, A is the area of the face to which the force is applied, x the linear displacement, and ℓ the distance between the upper and lower faces.

When a rod or cylinder it twisted, it undergoes the same sort of shear deformation in each part, and the rigidity modulus may be written:

 [2]
```
2ℓL
n = ————
4
πr φ

```
where ℓ is the length of the rod, r is its radius, L is the applied torque and φ is the displacement angle (in radians).

(B) TORSION PENDULUM

Consider a body suspended by a rod, the rod being coincident with a symmetry axis of the body. See Fig. 3. If the body is given a twist about the rod's axis, it will oscillate with angular harmonic motion with period:

T = 2π√(I/K)
where T is the oscillation period, I the moment of inertia of the body (about its axis of rotation) and K the torsion constant of the wire.

The torsion constant K is defined as the torque required to twist the end of the wire through one radian, K ≡ Δτ/Δθ. K depends on the dimensions of the wire and the rigidity modulus:

 [3]

T = 2π√(I/K)

 [4]
```
2ℓK
n = ———
4
πr

```
where ℓ is the length of the wire and r is its radius.

4. PROCEDURE:

(1) The torsion lathe is a simple apparatus specifically designed to measure the torsion constant. The metal sample to be tested is in the form of a one meter long rod, with special fittings at each end. This rod is clamped at one end, the other end attaches to a graduated wheel with a flexible steel strap around its rim. Weights attached to the steel strap twist the rod, and the vernier scale on the wheel allows the measurement of the twist angle.

Do not exceed the elastic limit of the material being tested. Consult your instructor if you are in doubt how large a load the material can tolerate.

Measure the rigidity modulus of the steel sample and the brass sample.

(2) The torsion pendulum uses the very same sample rods as the torsion lathe, permitting an independent measurement of their rigidity modulus by two methods, static (with the torsion lathe) and dynamic (with the torsion pendulum.

The upper end of the rod is clamped into a special fitting permanently attached to the wall. But before you attach the sample, hang the iron ring over the wall mount, for you'll need to lower it over the rod later, and this will save you the nuisance of disassembling the apparatus then.

The horizontal plate at the bottom has a moment of inertia Io, but its shape is not so simple that the inertia can be determined simply from its dimensions (it has a hub and lock nut which complicate the geometry). However, you have available metal ring and two metal cylinders, of simple shape. These are shown in Fig. 3. The moments of inertia of these objects may be determined quite precisely by weighing them and measuring their dimensions. The period of the pendulum is measured with just the base plate attached. Then the ring is lowered onto the base plate and the period of the pendulum is measured. The ring is lifted back onto the upper support and the two cylinders are placed on the base plate as shown in Fig. 3. The period is measured.

Therefore you can measure the period of the pendulum with two different moments of inertia: (Io + Iring) and (Io + Icyl). With this information Io and K may be determined.

The ring has the simplest geometry, so take its moment of inertia to be

 [5]

Iring = M(R12+R22)/2

from direct measurements of its mass and radii. Use this to determine Io. Use the parallel axis theorem, along with measurements of the mass and dimensions of the cylinders, as an independent check to see whether the dynamic measurement agrees with the geometric calculation.

The parallel axis theorem is:

 [1]

Ia2 = Ic + M ℓ

where M is the mass of the body, Ia is the moment of inertia about any axis, Ic is the moment of inertia about the axis through the center of gravity, and ℓ is the distance of separation of the two axes.

The moment of inertia of a solid cylinder about its own axis is Mr2/2.

(3) Use the experimentally determined value of K to calculate the rigidity modulus n and compare with the static value determined in part (1). Do this for both metal sample rods, steel and brass.

Text© 1992, 2004 by Donald E. Simanek.

Figures from Ingersoll and Martin, Experiments in Physics, McGraw-Hill, 1942.