Question

A bucket of water of mass $ 21kg $ is suspended by a rope wrapped around a solid cylinder $ 0.2{\text{ m}} $ in diameter. The mass of the solid cylinder is $ 21kg $ . The bucket is released from rest. Which of the following statements are correct?
(A) The tension in the rope is $ 70{\text{ N}} $ .
(B) The acceleration of the bucket is $ \dfrac{{20}}{3}m/{s^2} $
(C) The acceleration of the bucket is $ \dfrac{{40}}{3}m/{s^2} $
(D) The tension in the rope is $ 140N $

Answer 1

Hint :As there are two unknowns here, so we will be using two equations, one will be obtained by balancing the forces on the bucket and the other will be obtained by the torque or moment equation for the solid cylinder around which the rope is wrapped.

Complete Step By Step Answer:
As the bucket will be experiencing the force due to gravitation which will be equal to,
 $ f = m \times g $ ,
Taking ‘g’ as $ 10m{s^{ - 2}} $ and m is given as $ 21kg $ , so the force will be,
 $ f = 21 \times 10 = 210N $
This force will be pulling the bucket downward.
Now writing the equation for bucket,
 $ \sum F = ma $
 $ 210 - T = 21a{\text{ }} - - - - - - - - - (1) $
Where T is the tension on the rope and let a is the acceleration of the bucket.
For calculating the tension on the rope let us take the moment equation around the solid cylinder into consideration,
 $ \tau = I\alpha $ ,
Where, $ \tau $ is the torque about the solid cylinder and is given by $ T \times r $
I is the moment of inertia of the solid cylinder and is given by $ \dfrac{{m{r^2}}}{2} $
And $ \alpha $ is the angular acceleration and is given by $ \dfrac{a}{r} $
Now putting the above values in the equation of torque, we will get,
 $ T \times r = \dfrac{1}{2}m{r^2} \times \dfrac{a}{r} $ ,
Simplifying this will give us,
 $ T = \dfrac{{ma}}{2} $
 $ T = \dfrac{{21a}}{2}{\text{ - - - - - - - - - - - - (2)}} $
Putting this in equation $ 1 $ ,
 $ 210 - \dfrac{{ma}}{2} = 21a $ ,
As we know the value of m,
 $ 210 - \dfrac{{21a}}{2} = 21a $
Rearranging the above equation will give us,
 $ 210 = 21a + \dfrac{{21a}}{2} $
On solving we will get, $ a = \dfrac{{20}}{3}m{s^{ - 2}} $
Now putting this value of a in equation $ 2 $ ,
 $ T = \dfrac{{21}}{2} \times \dfrac{{20}}{3} $ ,
 $ T = 70N $
Hence, option A and B are the correct answers.

Note :
Though the exact value for the acceleration due to gravity is $ 9.8m{s^{ - 2}} $ , but for convenience we have taken 10 to solve this question. It is a common practice and one should not get confused in the same. We can have a better understanding of the forces and their direction by making the FBD.