Name: Centripetal and Centrifugal Force
Uploaded: 2019-07-19
Duration: 2 min 42 s
Description: The meaning of centrifugal force is the apparent force that is felt by an object moving in a curved path that acts outwardly away from the centre of rotation

Question

Express Newton's second law in component form. Give its significance.

Accepted Answer

Newtons's second law of motion states that the rate of change of momentum of a body is proportional to the applied force and the change takes place in the direction of applied force. Mathematically :

$\vec{F} \propto \frac{d \vec{p}}{dt}$ or, $\vec{F} = k \frac{d \vec{p}}{dt}$ . Taking $k =1$ we get, $\vec{F} = \frac{d \vec{p}}{dt}$ .

Considering mass $m$ of the body is constant, we get,

$\vec{F} = \frac{d \vec{p}}{dt} = m \frac{d \vec{v}}{dt}$ or, $\vec{F} = m . \vec{a}$

Component form of newton's second law is given below:

$\vec{F} = F_{x} \hat{i} + F_{y} \hat{j} + F_{z} \hat{k}$

$\vec{p} = p_{x} \hat{i} + p_{y} \hat{j} + p_{z} \hat{k}$

$\vec{a} = a_{x} \hat{i} + a_{y} \hat{j} + a_{z} \hat{k}$

$∴ F_{x} \hat{i} + F_{y} \hat{j} + F_{z} \hat{k} = \frac{d (p_{x} \hat{i} + p_{y} \hat{j} + p_{z} \hat{k})}{dt}$ $\Rightarrow \frac{d (p_{x} \hat{i})}{dt} + \frac{d (p_{y} \hat{j})}{dt} + \frac{d (p_{z} \hat{k})}{dt}$

$∴ F_{x} \hat{i} + F_{y} \hat{j} + F_{z} \hat{k} = {ma}_{x} \hat{i} + {ma}_{y} \hat{j} + {ma}_{z} \hat{k} = m (a_{x} \hat{i} + a_{y} \hat{j} + a_{z} \hat{k})$

By equating the components, we get

$F_{x} = \frac{{dp}_{x}}{dt} = {ma}_{x}$ , $F_{y} = \frac{{dp}_{y}}{dt} = {ma}_{y}$ , $F_{z} = \frac{{dp}_{z}}{dt} = {ma}_{z}$

The component form of Newton's second law signifies that any force can be decomposed in components along the three axis and if any of the three components of force is zero, then the acceleration along the direction of that component is also zero.

Express Newton's second law in component form. Give its significance.

Important Questions on Laws of Motion

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