In a circular motion of a particle, the tangential...

In a circular motion of a particle, the tangential acceleration of the particle is given by $a_{t}=2 t \mathrm{~m} /$ $\mathrm{sec}^{2}$, where $t$ is the time of motion. The radius of the circle described is $R=2 \mathrm{~m}$. 'lhe particle is initially at rest. $\Lambda$ t time $t$, the total acceleration of the particle makes an angle $45^{\circ}$ with radial acceleration. Choose the correct alternatives. (a) 'lhe value of $t$ is $2^{3 / 3} \mathrm{~s}$. (b) The value of $t$ is $2^{1 / 3} s$. (c) The total acceleration of the particle when $t=1$ s is $\sqrt{17} / 2 \mathrm{~m} / \mathrm{s}^{2}$. (d) The total acceleration of the particle when $t=1$ s is $1 / 4 \mathrm{~m} / \mathrm{s}^{2}$.

In a circular motion of a particle, the tangential acceleration of the particle is given by $a_{t}=2 t \mathrm{~m} /$ $\mathrm{sec}^{2}$, where $t$ is the time of motion. The radius of the circle described is $R=2 \mathrm{~m}$. 'lhe particle is initially at rest. $\Lambda$ t time $t$, the total acceleration of the particle makes an angle $45^{\circ}$ with radial acceleration. Choose the correct alternatives.
(a) 'lhe value of $t$ is $2^{3 / 3} \mathrm{~s}$.
(b) The value of $t$ is $2^{1 / 3} s$.
(c) The total acceleration of the particle when $t=1$ s is $\sqrt{17} / 2 \mathrm{~m} / \mathrm{s}^{2}$.
(d) The total acceleration of the particle when $t=1$ s is $1 / 4 \mathrm{~m} / \mathrm{s}^{2}$.

Key Concepts

Resultant (Total) Acceleration

The resultant or total acceleration in circular motion is the vector sum of the tangential and centripetal accelerations. Since these two acceleration components are perpendicular to each other, the magnitude of the total acceleration can be computed using the Pythagorean theorem. This resultant acceleration shows both the changes in speed and the change in direction of the moving particle.

Circular Motion

Circular motion refers to the movement of an object along the circumference of a circle. In this type of motion, the object’s position continuously changes direction while maintaining a constant distance (the radius) from a fixed center point. The analysis of circular motion involves understanding how velocities and accelerations are directed relative to the circle's geometry.

Tangential Acceleration

Tangential acceleration is the component of acceleration that acts along the tangent to the path of the object in circular motion. It is responsible for changes in the speed (magnitude of velocity) of the object. This acceleration is directed along the direction of the velocity at any point on the circular path and does not affect the direction of the motion directly.

Centripetal (Radial) Acceleration

Centripetal acceleration, also known as radial acceleration, points from the object toward the center of the circular path. It is responsible for changing the direction of the velocity vector, thus keeping the object moving along a curved (circular) path. Its magnitude is determined by the square of the object's speed divided by the radius of the circle.

Angle between Acceleration Components

The angle between the total acceleration and one of its components (usually the radial acceleration) can be determined by taking the inverse tangent of the ratio of the tangential acceleration to the centripetal acceleration. In problems involving circular motion, knowing this angle helps in understanding the relative contributions of speeding up (or slowing down) and the change in direction due to the circular path.

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