If a boy(500N) wishes to slide down a rope that has a breaking stand of 350N, then:
(a) with what acceleration must it move downward in order to not break the rope.
(b) if he slide 30m, how long will it take?
(c) how fast will he be going before he hits the ground?
Force up on boy = rope tension T
Force down on boy = m g
acceleration of boy downward = a
then
ma = mg - T
here m = 500/9.81 = 50.97 kg
so
50.97 a = 500 - 350 = 150
a = 150/50.97 = 2.94 m/s^2
d = (1/2) a t^2
30 = 1.47 t^2
t = 4.52 seconds
assuming 30 meters is the ground then
v = a t = 2.94 * 4.52
Now your elevator problem and your rope problem are very much the same so look them over carefully.
To answer these questions, we need to consider the forces acting on the boy while he slides down the rope. The forces involved are the weight of the boy and the tension in the rope.
(a) To prevent the rope from breaking, the tension in the rope must be equal to or greater than the weight of the boy. In this case, the breaking strength of the rope is 350N and the weight of the boy is 500N. Therefore, the tension in the rope must be equal to or greater than 500N for it to not break.
Now, let's consider the equation of motion for the boy sliding down the rope. Since we are assuming the system is vertical, we can use the following equation:
Tension - Weight = Mass * Acceleration
Given that the boy's weight is 500N and the breaking strength of the rope is 350N, we have:
Tension - 500N = Mass * Acceleration
Since the tension must be equal to or greater than 500N, we can rewrite the equation as:
500N - 500N = Mass * Acceleration
0N = Mass * Acceleration
Therefore, for the boy to not break the rope, the acceleration must be 0 m/s^2. This means the boy needs to slide down the rope at a constant speed.
(b) If the boy slides down the rope with a constant speed, it means that his acceleration is 0 m/s^2. In that case, we can use the formula:
Distance = Initial velocity * Time
Since the boy starts from rest, the initial velocity is 0 m/s. The distance he slides down is given as 30m. Therefore, the equation becomes:
30m = 0m/s * Time
This equation shows that the time it takes for the boy to slide down 30m is indeterminate because any time value will satisfy the equation when the initial velocity is zero.
(c) Since we have established that the boy slides down with no acceleration, we can also conclude that his final velocity will be zero when he hits the ground. This is because he must come to a stop at the end of his slide.