A rope of mass M and length L is being rotated about one end in a gravity free space. A pulse is being created at one of the ends. The angle through which rope will be rotated in the time when pulse reaches the opposite end of rope for first time and for the angular velocity of rope 'ω', is
a) π/√2 b) π/2 c) π/(2√2) d) π2
ans is option a
pls explain me how?
ANSWER
To solve this problem, let's break it down step by step:
1. Consider a rope of mass M and length L being rotated about one end in a gravity-free space. This means that there are no external forces acting on the rope, so we can assume that the rope maintains a constant angular velocity, ω.
2. A pulse is being created at one end of the rope. This pulse will travel through the rope to the opposite end. Since the rope is rotating, the pulse will have a certain angular velocity as it moves through the rope.
3. Now, we need to determine the time it takes for the pulse to reach the opposite end of the rope for the first time. To do this, we need to calculate the time it takes for the pulse to travel the entire length of the rope.
4. The time it takes for a pulse to travel a distance can be determined using the formula: t = d/v, where t is the time, d is the distance, and v is the velocity.
5. In this case, the distance the pulse needs to travel is the length of the rope, L. The velocity of the pulse can be approximated as the speed at which the opposite end of the rope is rotating at any given time, which is ωL. So, we have: t = L/(ωL) = 1/ω.
6. Now, we need to find the total angle through which the rope rotates in this time. The angular displacement can be calculated using the formula: θ = ωt, where θ is the angle and t is the time.
7. Substituting the value of t from step 5 into the equation, we get: θ = ω * (1/ω) = 1 radian.
8. Therefore, the angle through which the rope will be rotated when the pulse reaches the opposite end of the rope for the first time is 1 radian, which is equivalent to π/√2 radians.
Hence, option (a) π/√2 is the correct answer.