Subject: Physics
Mass of a body is the quantity of matter contained in it. The effect of a force applied on a body depends on its mass. It has appeared in two entirely different contexts: Newton’s second law of motion and Newton’s law of gravitation. So meaning of mass is clear if we define mass as (i) inertial mass and (ii) gravitational mass.
The inertial mass of a body is the measure of the ability of the body to oppose the change in the velocity i.e. acceleration produced by an external force. It is a measure of the body’s resistance to being accelerated.
As we know,
$$F=ma \: \text {and is}\: a= 1 \text {then}\: F = m $$
Gravitational mass is defined by the Newton’s law of gravitation. The gravitational pull of the earth of mass M and radius R on a body of mass m is given as
\begin{align*} F &= \frac {GMm}{R^2} \\ \text {or,} m &= \frac {F\times R^2}{GM} \\ \end{align*}
The mass m of the body, in this case, is the gravitational mass of the body. This relation shows that the gravitational force increases if the mass of the object increases i.e. the measure of gravitational force on the body is called gravitational mass.
Difference between Inertial Mass and Gravitational Mass
S.N | Inertial Mass | Gravitational Mass |
1. | It is the mass of the material body which measures its inertia. | It is the mass of the material body which determines the gravitational pull upon it. |
2. | It can be measured when the body is in motion. | It can be measured at the rest position of the body. |
3. | The inertial mass is measured according to Newton’s second law of motion. | By Newton’s law of gravitation, gravitational mass is measured. |
The weight of an object is defined as the gravitational force with which the object is attracted by the earth towards its centre. If m be the mass of the object and g be the acceleration due to gravity, then the weight of the object at a certain place is given by
$$ W = mg $$
It is a vector quantity being a force.
Difference between Mass and Weight
S.N | Mass | Weight |
1. | It is the measure of inertia. | It is the measure of gravity. |
2. | It is a constant quantity. | It may vary from place to place. |
3. | It is a scalar quantity. | It is a vector quantity. |
4. | It can never be zero. | It is zero at the places where g=0. |
5. | Its SI-unit is kg and CGS-system are gram. | Its SI-unit is Newton and CGS-system is dyne. |
Acceleration due to gravity varies from place to place. These are discussed below.
\begin{align*} g &= \frac {GM}{R^2}\\ \end{align*} The acceleration due to gravity g’ at P is \begin{align*}g’ &= \frac {GM}{(R+h)^2} \dots (iii) \\ \end{align*} Combining equations} (I) {and} (ii)we get\begin{align*}\frac {g’}{g} &= \frac {GM}{(R+h)^2} \times \frac {R^2}{GM} = \frac {R^2}{(R+h)^2} \\ g’ &= g\times \frac {R^2}{(R+h)^2} \dots (iii)\\ \end{align*}
Since (R+h)>R, we have g’
\begin{align*} g’ &= \frac {g}{\left (\frac {R+h}{R} \right )} = g \left ( 1+ \frac hR \right )^{-1} \\\end{align*}
Expanding ( 1+ \(\frac {h}{R })^{-1} \) by binomial theorem and neglecting the higher power of h/R which has small value, we get \begin{align*} \\ g’ &= g\left ( 1-\frac {2h}{r} \right ) \dots (iv) \\ \end{align*}
Let us consider an object of mass m at a point P on the earth surface at latitude of Ð¤. When the earth is not rotating its weight is , W= mg acts along the radius of the earth towards the centre.
When the earth rotates with angular velocity () on its axis, the object at P will also rotate about the centre C of the radius of the earth. So, the object will experience the centrifugal force, and the object is under the action of two forces centrifugal force and its weight.
The resultant of two forces is shown by PB in the figure which is apparent weight of mg of the object. By using parallelogram law of vector addition,
\begin{align*} PB^2 &= PO^2 + PA^2 + 2PO \times PA \cos (180 - \phi ) \\ \text {or,} \: m^2g’^{2} &= (mg)^2 + (mR\omega ^2 \cos \phi )^2 + 2\times mg \times mR\omega ^2 \cos \phi \times (-\cos \phi ) \\ \text {or,} \: g’ &= g \left [ 1 + \frac {R^2 \omega ^4 \cos ^2 \phi }{g^2} -\frac {2R\omega ^2 \cos ^2 \phi }{g} \right ]^{1/2}\\ \text {Since} \frac {R\omega ^2}{g} \\ \end{align*} is a small quantity, the terms containing the factor\begin{align*} \frac {R^2\omega ^4}{g^2} \text {can be neglected.} \\ \text {Hence,} \: g’ &= g\left [ 1 -\frac {2R\omega ^2 \cos ^2 \phi }{g} \right ]^{1/2}\\ \end{align*}
Expanding the above equation by using binomial theorem\begin{align*} g’ &= g \left [ 1 -\frac 12 \frac {2R\omega ^2 \cos ^2 \phi }{g} + \dots \right ]\\ \end{align*} Higher powers of \begin{align*}\: \frac {2R\omega ^2 \cos ^2 \phi }{g}\\ \end{align*} can be neglected and\begin{align*} \\ g’ &= g \left [ 1 -\frac {R\omega ^2 \cos ^2 \phi }{g} \right ]\\ \text {or,} \: g’ &= g- R\omega ^2 \cos ^2 \phi \\ \end{align*} This equation shows that due to the rotation of the earth, the acceleration due to gravity decreases. It also shows that the acceleration due to gravity increases as the latitude increases \begin{align*} {At the equator,} \phi = 0, so \cos 0 = 1 \\ \therefore g_e &= g – R\omega ^2 \\ \text {At the poles,} \phi = 90^o , \text {and} \cos 90^o = 0 \\ \therefore g_p &= g \\ \end{align*} There is no effect on the acceleration due to gravity at the poles due to the rotation of the earth
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