### 2. Measurement of mass

Depending upon the manner in which mass of a body is measured, there are two different concepts about the mass of a body. These are: a) Gravitational mass and b) Inertial mass. The methods of measurements of these masses are given as under:

### 1. Measurement of gravitational mass:

a) Gravitational mass is the mass of the body which determines the gravitational pull exerted on it due to the earth. In order to measure the gravitational mass of a body, a common balance is used. The body is placed in the left pan, while the standard weights in the right pan. The gravitational force on the body and standard weights are made equal and the mass of the body is determined.

b) We can measure the Gravitational mass of a body by using a spring balance also. Actually, the spring balance measures the weight of a body. When a body is suspended from a spring balance, it stretches the spring. The elongation produced in the spring is directly proportional to the weight of the body. Because value of g is constant at a place, the elongation is proportional to the mass of the body. Hence, just by calibrating the elongation; gravitational mass of the body can be determined.

### 2. Measurement of inertial mass:

a) Consider a heavy body placed on a smooth horizontal surface. When a small force is applied on the body, it remains in rest position due to inertia. A greater force is needed to overcome the inertia of the body and put it in motion. If an applied force F produces an acceleration a in the body, then F = ma or m =F/a.

c) The inertial mass of a given body can be measured with the help of an inertial balance. It is based on the principle that the inertia of matter is proportional to the inertial mass of the object.

d) The inertial balance consists of a pan and a block of wood joined to each other with the help of two flat metallic strips. One end of each metallic strip is clamped to a table in such a way that flat faces of strips are vertical in the following figure. The body whose inertial mass is to be determined is placed in the pan. When the pan is displaced horizontally in any direction and released, it starts vibrating horizontally. **It has been found that frequency of vibration depends on the following factors:**

i. The length of the projecting part of the inertial balance.

II. The modulus of elasticity of the metallic strip and

III. The total mass of the pan and the object placed on it.

e) When the metallic strip is put into vibration with different objects and periods of vibration noted in each case, it is found that mass if the object and the square of the time period are directly proportional to each other. If m_{1} be the inertial mass of one body plus the mass of pan and T_{1} the time period, then T_{1}^{2} ∝ m_{2}

M_{1}/m_{1}=t_{1}^{2}/t_{2}^{2} or m_{2} = m_{1}×T_{2}^{2}/T_{1}^{2}

If m_{1} is the inertial mass of a known object then inertial mass m_{2} of unknown object can be determined by measuring the time periods in two cases.