Question

A stone of mass $1kg$ is whirled in a horizontal circle attached at the end of a $1m$ long string. If the string makes an angle of 30° with vertical, calculate the centripetal force acting on the stone. ($g=9.8m/s^2$).

Answer 1

Hint:Since, the stone is attached to a string which is whirling so centripetal force will act on it. We should always remember that centripetal force is directed at right angle to the motion and towards the fixed point of the instantaneous center of curvature. So, using the given formula of the centripetal force and the given values we can calculate the centripetal force.

Complete step by step answer:
Given.
Mass of the stone (m) = 1kg
Length of the string (r) = 1m
Angle the string making (θ) = 30°
To find,centripetal force $({F_c})$ acting on the stone.
At first, we need to understand the definition and meaning of Centripetal force
A centripetal force is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Always remember that the centripetal force is directed at right angles to the motion and also along the radius towards the center of the circular path.One common example involving centripetal force is that a body which moves with uniform speed along a circular path.
The magnitude of the centripetal force on an object of mass m moving at tangential speed v along a path with radius of curvature r is
${F_c} = \dfrac{{m{v^2}}}{r}$
Now, for the given question
${F_c} = \dfrac{{1 \times {v^2}}}{1}$ ; since Mass of the stone (m) = 1kg & Length of the string (r) = 1m
Now, $v = \sqrt {rg\tan \theta } $ (from Euler’s Laws of Motion); where g is the acceleration due to gravity.
So, $v = \sqrt {1 \times 9.8 \times \tan {{30}^\circ }} = 2.37m/s$
Hence, Centripetal Force,
${F_c} = \dfrac{{1 \times {{(2.37)}^2}}}{1} \\
\therefore{F_c} = 5.67N$

Hence, Centripetal Force for the given stone of mass 1kg is whirled in a horizontal circle attached at the end of a 1m long string making an angle of 30° with vertical is 5.67N.

Note: Centripetal force acts towards the fixed point of the instantaneous center of curvature of the path and centrifugal force acts away from the fixed point of the
instantaneous center of curvature of the path so don’t confuse in between these two and calculate accordingly. We need to remember the formulas and always use similar units for the given values to get the correct answer.