NAEST 2020: Kinematics of a Cylinder Rolled between the two

Question: In Case-1, the lower hand is fixed on the table. The upper hand presses the cylinder and is moved ahead ensuring no slipping of the cylinder either at the lower hand nor at the upper hand.
The distance moved by the upper hand is $L_1$, by the centre of the cylinder is $L_2$ and by the lower hand is $L_3$, in the same time. Within experimental errors,

1. $L_1 = L_2$
2. Speed of the topmost point of the cylinder is equal to the speed of the upper hand at any instant.
3. Speed of the topmost point of the cylinder is double of the speed of the centre of the cylinder.
4. $L_1 = 2 L_2 $
5. $L_1 = L_3$
6. $L_2 = L_3$

Question 2: Consider Case-2. The symbols $L_1$, $L_2$, $L_3$ have the same meanings as in Case-1. Within experimental errors,

1. $L_1 = L_2$
2. $L_1 = L_3$
3. Speed of the topmost point of the cylinder is equal to the speed of the upper hand at any instant.
4. Speed of the topmost point of the cylinder is double of the speed of the centre of the cylinder at any instant.
5. $L_2 = L_3$
6. $L_1 = 2 L_2$

Solution 1: Distance moved by the hand means distance moved by any point of the hand (say fingertip). Similarly, the speed of the hand means the speed of any point on the hand.

It can be read from the scale that $L_1 = 2L_2$ i.e., the distance moved by the upper hand is twice the distance moved by the centre of the cylinder. Thus, option (D) is correct.

The hands are not slipping on the cylinder. Thus, the speed of the topmost point of the cylinder is equal to the speed of the upper hand. Thus, option (B) is correct.

The cylinder's point in contact with the lower hand is at rest. Thus, the speed of the topmost point of the cylinder is twice the speed of itc centre. Thus, option (C) is correct.

Solution 2: In part 2, the centre of the cylinder is fixed, the upper hand is moving forward and the lower hand is moving backward. The distance $L_2 = 0$ and $L_1 = L_3$ as seen from the readings on the scale. Thus, option (B) is correct.

The speed of the topmost point of the cylinder is equal to the speed of the upper hand as there is no slipping. Thus, option (C) is correct.

The cylinder of radius $r$ is rotating about a fixed axis with angular velocity $\omega$. The speed of its centre is zero. The velocity of its topmost point is $\omega r$ in the forward direction and the velocity of the lowest point is $ \omega r$ in the backward direction.

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