Question

A particle is in motion along a curved path. Which of the following is true in this case?
A. If its speed is constant, it has no acceleration.
B. If its speed is constant, the magnitude of its acceleration at any point of its path is inversely proportional to the radius of curvature of the path there.
C. If its speed is increasing, the acceleration of the particle is along the direction of its motion
D. If speed is constant, the direction of its acceleration cannot be along the tangent.

Answer 1

Hint: In a curved path the direction of motion changes continuously. Speed is the rate of change of distance whereas velocity is the rate of change of displacement. Therefore, even when the speed is constant the velocity is not a constant, and since acceleration is the rate of change of velocity there will be an acceleration in a curved path even for a constant speed. In a curved path with constant speed the acceleration is the centripetal acceleration given by the equation ${a_{cen}} = \dfrac{{{v^2}}}{r}$.

Complete step by step answer:
In a curved path with constant speed, the acceleration is the centripetal acceleration. Therefore, option D which states, when speed is constant, the direction of its acceleration cannot be along the tangent. So this statement is true.

The value of centripetal acceleration is given by the equation
${a_{cen}} = \dfrac{{{v^2}}}{r}$
Where $v$ is the linear velocity and $r$ is the radius of curvature of the path.
Thus option B which states, If its speed is constant, the magnitude of its acceleration at any point of its path is inversely proportional to the radius of curvature of the path there. So this statement is true.

Option A is incorrect because due to a change in velocity there will be a nonzero acceleration even when speed is a constant. Velocity is the rate of change of displacement. It is a vector quantity. It depends on both the magnitude and direction of motion. In a curved path even when the speed is constant the direction keeps changing continuously. Therefore, the velocity is not constant in this case.
Acceleration is the rate of change of velocity. Since velocity is changing there is a nonzero acceleration in a curved path even when the speed is a constant.
Option C is incorrect because if speed is also increasing then there will be both tangential acceleration and centripetal acceleration Therefore the acceleration will not be along the direction of its motion. It will be in the direction of the total acceleration which is the vector sum of centripetal and tangential acceleration.

$\therefore$ Option (A) and Option (D) are correct.

Note:
The component of acceleration that is tangential to the movement of the object gives the tangential acceleration. It is always a linear acceleration. But linear acceleration is simply the acceleration along a straight line so linear acceleration need not be always tangential. It can be found using the equation ${a_{\tan }} = \alpha \times r$, where $\alpha $ is the angular acceleration and $r$ is the radius of the rotation. If the speed is not constant in the motion along a curved path then there will be both centripetal and tangential acceleration. Therefore, to find the net acceleration we should take the vector sum of centripetal and tangential acceleration.
$\Rightarrow a = {a_{cen}} + {a_{\tan }}$