Determining the Mass of a Given Body by the Principle of Moments

Mass of a given body by the principle of moments – Objective

To determine the mass of a given body using a meter scale by applying the principle of moments.

Mass of a given body by the principle of moments – Materials Required

Meter scale

Given body (whose mass is to be determined)

Known masses (slotted weights 5x 50 g)

Stand or support with a knife edge or fulcrum

Thread or string

Measuring tape or ruler

Mass of a given body by the principle of moments – Formula

The weight of the given body  W1 and the known weight W2 are given by 

 W1 = mg

 W2 = Mg

According to the principle of moments:    

Anticlockwise moment = Clockwise moment

W1 x r1  =  W2 x r2

m x g x r1  =  M x g x r2

m x r1  = M x r2

Solving for m

m = M x r2/ r1

m= mass of the given body (unknown)

M= mass of the known weight

r1 = distance of the given body from the fulcrum

r2 = distance of the known weight from the fulcrum

g = acceleration due to gravity (constant)

This equation allows us to determine the mass of the given body  m using the known mass M and the distances r1  and  r2

Mass of a given body by the principle of moments – Theory

A key idea in physics, especially when studying mechanics and rotational equilibrium, is the principle of moments. It is predicted  that the sum of the clockwise and anticlockwise moments about any point must equal one another for a body to be in rotational equilibrium. The law of the lever is another name for this idea.

The principle of moments states that for a body in equilibrium, the sum of the anticlockwise moments about any point is equal to the sum of the clockwise moments about the same point. Mathematically, it is expressed as:

Anticlockwise moment = clockwise moment

W1 x r1  =  W2 x r2

Where:

W1 and  W2  are the weights (or masses) acting on either side of the fulcrum. r1  and r2 are the distances of these weights from the fulcrum.

Key Concepts:

Moment of a Force:

The moment of a force (or torque) is a measure of its tendency to cause a body to rotate about a specific point or axis.

Mathematically, the moment 𝛕 is given by:

𝛕 = Force x perpendicular distance from the pivot

𝛕 = F x r

Here, F is the force applied, and r  is the perpendicular distance from the pivot (fulcrum) to the line of action of the force.

Clockwise and Anticlockwise Moments:

Clockwise moment:

A body experiences a clockwise moment when a force causes it to rotate in clockwise direction.

Anticlockwise moment:

A body’s moment is referred to as an anticlockwise moment if a force causes it to rotate counterclockwise.

Rotational Equilibrium:

When there is no net rotational force exerted on a body, it is said to be in rotational equilibrium.This means the sum of all clockwise moments is equal to the sum of all anticlockwise moments.

Sum of the anticlockwise moment  =  sum of the clockwise moment

Application to the Experiment:

The fulcrum, or knife edge, serves as the pivot point in this experiment, while the metre scale serves as a lever.Moments are created by the forces applied to either side of the fulcrum by the given body and the known weights.

The moment due to the given body (anticlockwise) equals the moment due to the known weights (clockwise) when the metre scale is balanced (in equilibrium).This equation allows us to determine the mass of the given body  m using the known mass M and the distances r1  and  r2

Importance of principle of moments:

In physics and engineering, the principle of moment is frequently applied to examine systems in rotational equilibrium, including balancing, beams, levers and balances.

It helps in determining unknown forces or masses by balancing moments.

Using a metre scale and known weights, this experiment offers a quick and precise way to determine an object’s mass.

Assumptions

When balanced, the metre scale is homogeneous and exactly horizontal.

There are no extra forces introduced by the frictionless fulcrum.

There is very little mass in the thread or string that is used to hang the weights.

Throughout the experiment, the acceleration caused by gravity, or ‘g’ remains constant.

Mass of a given body by the principle of moments – Procedure

To set up the metre scale:

Position the scale horizontally on a fulcrum or knife edge. Make sure the scale is horizontal and balanced.To ensure that the scale is in balance without any extra weights.

Connect the given body:

Use a thread to secure the given body, whose mass needs to be ascertained, to one end of the metre scale.Take note of where the body is on the scale.

Balance the scale:

Position established standard weights on the opposite side of the fulcrum until the metre scale is once more horizontal and balanced.Note the location of the specified body and the known weights from the fulcrum.

Calculate the mass:

Use the principle of moments to calculate the mass of the given body.

Let  m be the mass of the given body, M be the mass of the known weight,  r1  be the distance of the given body from the fulcrum, and  r2  be the distance of the known weight from the fulcrum.

According to the principle of moments:

m x r1  = M x r2

m= M x r2/ r1

Repeat for Accuracy:

Repeat the experiment by changing the position of the fulcrum or using different known weights to ensure accuracy.Calculate the average mass from the repeated trials.

Observations:

Record the distances r1 and  r2 for each trial.

Note the known mass  M used in each trial.

Mass of a given body by the principle of moments – Tabular Column

Trial  No.	Known Mass	distance of the  unknown body from the fulcrum	distance of the known weight from the fulcrum	mass of the given body                             (unknown)
Symbol       →	M	r1	r2	m = M x r2 /r1
Units      →	g	cm	cm	g
1.	50	20.9	39.6	94.7
2.	100	30.3	29.4	97.0
3.	150	29.8	19.4	97.7
4.	200	42.6	20.9	98.1
5.	250	46.8	18.2	97.2

Mass of a Body by the Principle of Moments-Calculations

The mass of the given body can be accurately determined using the principle of moments by balancing the moments of the known and unknown masses on a meter scale.

Use the formula  m=M x r2/r1 to calculate the mass of the given body for each trial.

Determine the average mass from the calculated values.

Mass of a Body by the Principle of Moments-Result

The mass of the given body is determined to be 96.94 grams

Mass of a Body by the Principle of Moments-Precautions:

Ensure the meter scale is perfectly horizontal before taking readings.

The fulcrum should be sharp and frictionless to avoid errors.

Use a thread or string that is light and does not add significant weight to the system.

Repeat the experiment multiple times to minimize errors.

Link to this experiement

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Understanding Torque: An Essential Concept and numerical

WHAT IS TORQUE? HOW DOES TORQUE WORK IN A VEHICLE?