A rope AB attached to a small block of neglible dimensions and passes over apulley C so that its free end hangs 1.5 metres above ground. When block rests on the floor. The end A of the rope is moved horizontally in astraight line by a man walking at a uniform velocity V=3metres per second
Find time taken by block to reach the pulley if h= 4.5metres that is from free end to centre of pulley
The man is walking away, so the horizontal distance the end of the rope has moved at time t is 3t meters.
Initially, there are 3 meters of rope from the free end to the pulley. So, at time t, the amount of rope from C to the man is
c^2 = 3^2 + (3t)^2= 9(1+t^2)
Now, the block must rise 4.5 meters, so we need
(4.5+3)^2 = 9(1+t^2)
2.5^2 = (1+t^2)
5.25 = t^2
t = 2.29 seconds
To find the time taken by the block to reach the pulley, we can use the equation for displacement:
s = ut + (1/2)at^2
Given:
Initial velocity (u) = 0 (block is at rest initially)
Acceleration (a) = g (acceleration due to gravity, approximately 9.8 m/s^2)
Displacement (s) = h = 4.5 meters
Since the block is initially at rest, its initial velocity is 0. Using the equation above, we can rearrange it to solve for time (t):
2s = at^2
2h = gt^2
t^2 = (2h)/g
t = √(2h/g)
Substituting the given values:
t = √(2 * 4.5 / 9.8)
t ≈ √0.918
t ≈ 0.958 seconds
Therefore, the block takes approximately 0.958 seconds to reach the pulley.
To find the time taken by the block to reach the pulley, we need to consider the motion of the block and the time it takes for the rope to become taut.
Let's break down the problem into steps:
Step 1: Calculate the time it takes for the rope to become taut.
First, we need to calculate the time it takes for the rope to become taut, given that the end A of the rope is moved horizontally at a constant velocity V = 3 meters per second.
The distance the rope needs to travel to become taut is equal to the height from the ground to the pulley, which is h = 4.5 meters.
Using the equation time = distance / velocity, we can calculate the time taken for the rope to become taut:
time = height / velocity
time = 4.5 meters / 3 meters per second
time = 1.5 seconds
Therefore, it takes 1.5 seconds for the rope to become taut and start lifting the block.
Step 2: Calculate the time taken for the block to reach the pulley.
Once the rope is taut, the block starts moving upward. We need to find the time it takes for the block to travel the remaining distance from the ground level to the pulley, which is (h - 1.5) meters.
We can use the equation of motion for constant acceleration, where the acceleration is due to gravity:
distance = initial velocity * time + (1/2) * acceleration * time^2
In this case, the initial velocity is 0 since the block starts from rest. The acceleration due to gravity is approximately 9.8 m/s^2.
(distance) = (0) * (time) + (1/2) * (9.8 m/s^2) * (time^2)
(h - 1.5) = 4.9 * (time^2)
Now, let's solve for time:
4.9 * (time^2) = (h - 1.5)
4.9 * (time^2) = (4.5 - 1.5)
4.9 * (time^2) = 3
(time^2) = 3 / 4.9
time = sqrt(3 / 4.9)
Using a calculator, the approximate value of time is 0.74 seconds.
Therefore, the time taken for the block to reach the pulley is approximately 0.74 seconds.
Note: This solution assumes that there is no friction or any other forces acting on the block or the rope, and that the rope is massless.