Let us sketch the diagram for the motion of the particle of...

Let us sketch the diagram for the motion of the particle of mass $m$ along the circle of radius $R$ and indicate $x$ and $y$ axis, as shown in the figure. (a) For the particle, change in momentum $\Delta \vec{p}=m v(-\vec{i})-m v(\vec{j})$ so, $|\Delta \vec{p}|=\sqrt{2} m v$ and time taken in describing quarter of the circle, Hence, $\langle\vec{F}\rangle=\frac{|\Delta \vec{p}|}{\Delta t}=\frac{\sqrt{2} m v}{\pi R / 2 v}=\frac{2 \sqrt{2} m v^{2}}{\pi R}$ (b) In this case $\overrightarrow{p_{i}}=0$ and $\overrightarrow{p_{f}}=m w_{t} t(-\vec{i})$ so $|\Delta \vec{p}|=m w_{t} t$ Hence, $|<\vec{F}\rangle \mid=\frac{|\Delta \vec{p}|}{t}=m w_{t}$

Let us sketch the diagram for the motion of the particle of mass $m$ along the circle of radius $R$ and indicate $x$ and $y$ axis, as shown in the figure.
(a) For the particle, change in momentum $\Delta \vec{p}=m v(-\vec{i})-m v(\vec{j})$
so, $|\Delta \vec{p}|=\sqrt{2} m v$
and time taken in describing quarter of the circle,
Hence, $\langle\vec{F}\rangle=\frac{|\Delta \vec{p}|}{\Delta t}=\frac{\sqrt{2} m v}{\pi R / 2 v}=\frac{2 \sqrt{2} m v^{2}}{\pi R}$
(b) In this case $\overrightarrow{p_{i}}=0$ and $\overrightarrow{p_{f}}=m w_{t} t(-\vec{i})$
so $|\Delta \vec{p}|=m w_{t} t$
Hence, $|<\vec{F}\rangle \mid=\frac{|\Delta \vec{p}|}{t}=m w_{t}$

Key Concepts

Circular Motion

Circular motion refers to the movement of an object along a curved path with a constant radius. In this type of motion, parameters such as the radius, tangential speed, centripetal acceleration, and angular displacement are used to analyze the object's dynamics. It sets the stage for understanding how force and momentum change as the object travels along the circular path.

Momentum Change in Motion

Momentum, being a vector quantity defined as the product of mass and velocity, is crucial in analyzing physical systems. The change in momentum involves differences in both magnitude and direction, making it essential to use vector subtraction methods. This concept is central when studying how forces alter the state of motion of an object as it navigates through different segments of its path.

Impulse-Momentum Theorem

The impulse-momentum theorem states that the impulse experienced by an object, which is the product of a force applied and the time duration over which it is applied, is equal to the change in momentum of the object. This theorem is fundamental in understanding how forces act over specific intervals to produce changes in an object's motion, whether in linear or circular trajectories.

Vector Analysis in Mechanics

Vector analysis in mechanics involves decomposing physical quantities like displacement, velocity, and momentum into their components along defined coordinate axes. This method is crucial for accurately determining resultant changes in momentum or forces in multi-dimensional motion, as seen in scenarios where objects follow curved paths such as in circular motion.

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