Expert: Ahmed Salami - 7/11/2011

Question
Hello,

Could you please help me with the following question?

Q. A load of weight 7kN is being raised from rest with constant acceleration by a cable. After the load has been raised 20 metres, the cable suddenly becomes slack. The load continues upwards for a distance of 4 metres before coming to instantaneous rest. Assuming no air resistance, find the tension in the cable before it became slack. (Ans: 8400N)

Thank you

Answer
Hi May,
Consider the forces acting on the body. If the body were at rest the tension in the cable would be equal to the weight of the body. Since it is moving upwards, the tension must be greater than the weight to cause an upward acceleration.
We have,
     T - mg = ma
     T = mg + ma = m(g+a)
The body has been lifted through a height of 20m, its velocity at that height is gotten from;
     v² = 0² + 2a(20)
     v² = 40a
After this point (i.e when the cable becomes slack), this velocity is reduced to zero while the body is now only acted upon by gravity and therefore has a downward acceleration equal to g.
We therefore have,
     0² = v² - 2g(4)
     v² = 8g
Comparing with the equation above
     40a = 8g
       a = g/5
Therefore
     T = m(g + g/5)
     T = m(6g/5)
     T = 6mg/5 = (6)(7)/5
     T = 42/5 = 8.4KN

Alternative method;
By using energy considerations, we use the the statement of the fact that tha total work done by all forces with the exception of the gravitational field is equal to the change in total mechanical energy (sum of kinetic and potential energies)of the body.
Up to the height of 20m just before the slack, the tension is the only force (apart from gravity).
This gives,
     20T = (1/2)mv² + mg(20)  
After the slack and up to the 24m height no force acts (except gravity). There is therefore no additional work done.
Therefore,
     20T + 0 = (1/2)m(0²) + mg(24)
     20T = 24mg
     T = (24)(7)/20
     T = 8.4KN
as before.
I believe you would agree that the latter is simpler.

Regards