How Quickly Can a Rock Be Hoisted with Limited String Strength?

[8parks11]

a) What is the minimum time in which one can hoist a 1.00-kg rock height of 10.0m if the string used to pull the rock up has a breaking strength of 10.8 N? Assume the rock to be initially at rest.


b) If the string is replaced by one that is 50% stronger, by what percentage will the minimum time for the hoist be reduced.


a)

i just used a= 2(x-xo)/t^2 and then i got 2(10)/t^2 = 10.8/1
so i got 10.8=20/t^2 and so t would be root 10.8 but the answer is 4.5seconds... how?



b) i need to get a) to solve this...

 

[physics girl phd]

Hint 1: Draw a free body diagram for the rock.
Hint 2: There is some provided info in the problem you haven't used yet.

 

[kyle8921]

I would approach this problem by first analyzing the given information and setting up equations to solve for the minimum time needed to hoist the 1.00-kg rock.

Firstly, we know that the maximum tension the string can withstand is 10.8 N. We also know that the rock is initially at rest, so its initial velocity is 0 m/s. We can use the equation F=ma to calculate the minimum force needed to hoist the rock, which is equal to its weight, mg. Therefore, we have:

F = mg = (1.00 kg)(9.8 m/s^2) = 9.8 N

Since the maximum tension the string can withstand is 10.8 N, we can conclude that the minimum force needed to hoist the rock is 9.8 N. Now, we can use the equation F=ma again, but this time we will use the minimum force needed (9.8 N) as the force and the acceleration as the unknown variable. This gives us:

9.8 N = (1.00 kg)a

Solving for a, we get a = 9.8 m/s^2.

Next, we can use the kinematic equation x = xo + vot + 1/2at^2 to calculate the minimum time needed to hoist the rock. Since the rock starts from rest (vo = 0 m/s) and reaches a height of 10.0 m (x = 10.0 m), we have:

10.0 m = 0 + 0t + 1/2(9.8 m/s^2)t^2

Solving for t, we get t = 2.02 seconds. This is the minimum time needed to hoist the rock to a height of 10.0 m with a string breaking strength of 10.8 N.

Moving on to part b, if the string is replaced with one that is 50% stronger, its breaking strength will be 10.8 N x 1.5 = 16.2 N. Using the same approach, we can calculate the minimum force needed to hoist the rock with this stronger string, which is 9.8 N. However, since the force is now greater, the acceleration will also be greater. Using the same equation as before, we get:

9.8 N = (1