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Cement Hopper

maths question, MEI Mechanics 1?

A crane is used to lift a hopper full of cement to a height of 20m on a building site. the hopper has mass 200kg and the cement 500kg. initially the hopper accelerates upwards at 0.05m/s squared, then it travels at constant speed for some time before decelerating at 0.1m/s squared until it is at rest. the hopper is then emptied.

i) find the tension in the crane's cable during each of the three phases of the motion and after emptying.

the cable's maximum safe load is 10,000 N.
ii) what is the greatest mass of cement that can safely be transported in the same manner?

Let T = tension in the cable (N)
Let a = acceleration of the hopper (m/s^2)
Let g = acceleration due to gravity (m/s^2)
Let m = mass (kg)

i) As the hopper accelerates upwards at 0.05m/s^2 :

T - mg = ma
T = ma + mg
T = m(a + g)
T = 700 (0.05 + 9.8)
T = 6895 N

As it travels at a constant speed, a = 0 :
So T = mg
T = (700)(9.8)
T = 6860 N

As it decelerates, a = -0.1m/s^2 :
So, T = ma + mg
T = m(a + g)
T = 700(-0.1 + 9.8)m/s^2
T = 6790 N

When the hopper is empty, and at rest, m = 200 kg, a = 0 :
So, T = mg
T = (200)(9.8)
T = 1960 N

The tension in the crane's cable during each of the three phases of the motion
and after emptying is:

As the hopper accelerates upwards, the crane's cable tension = 6895 N
As hopper travels at a constant speed, the crane's cable tension = 6860 N
As hopper decelerates, the crane's cable tension = 6790 N
When the hopper is empty, and at rest the crane's cable tension = 1960 N

(ii) Let m be the greatest mass (cement + hopper) that can be safely transported.
Since the greatest tension happens in the initial phase of acceleration at 0.05 m/s^2,
use T = ma + mg, solve for m :

T = ma + mg
T = m(a + g)
10000 = m(0.05 + 9.8)
10000 = m(9.85)
m = 10000/9.85
m = 1015 kg

Mass of the cement that can be transported
= 1015 - 200
= 815 kg

Therefore, the greatest mass of cement that can be safety transported is 815 kg.

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