What Torque is Required for a Hoist Drum to Lift an 8 Tonne Load?

[robbinz]

Homework Statement

Q
A load of mass 8 tonne is to be raised with a uniform acceleration of 1.1 m/s^2 by means of a light cable passing over a hoist drum of 2 m diameter. The drum has a mass of 1 tonne an a radius of gyration of 750 mm. Find the torque required at the drum if friction is neglected.

A
(87.8 KN-M)

Homework Equations

T=Ia = MK^2a ( The a is ment to be alpha )

F=MA

The Attempt at a Solution

Ok so my attempt at the solution was T = Ia(alpha) + F x R ( Taking F=MA )
But it comes out wayyyy wrong.

So obiously I know the solution as I have given it. But I am confused as how to do the working. Can anyone help ?

p.s. First forum post...yayyy :D

Thanks

 

[Dr.D]

You need a couple of free body diagrams to start things off. Then write the equations of motion for each mass (the drum, the block being raised) and write the relevant kinematic relations. Then it should all fall into place for you.

 

[Mene]

for your question and welcome to the forum! Let's break down the problem and see if we can figure out the solution together.

First, let's define some variables:

m = mass of load (8 tonne)
a = acceleration (1.1 m/s^2)
d = diameter of drum (2 m)
M = mass of drum (1 tonne)
r = radius of gyration (750 mm = 0.75 m)
T = torque required at drum

Now, let's think about the forces acting on the drum. We have the torque (T) from the cable pulling on the drum, and we also have the force (F) from the weight of the load. Since the drum is rotating, we also need to consider the moment of inertia (I) of the drum.

Using the equation T=I(alpha), we can see that the torque is directly proportional to the moment of inertia and the angular acceleration (alpha). Since we are given the mass and radius of gyration of the drum, we can calculate the moment of inertia using the equation I=Mr^2. Plugging in the values, we get:

I = (1000 kg)(0.75 m)^2 = 562.5 kg*m^2

Now, let's consider the force (F) from the weight of the load. Since the load is being lifted with a uniform acceleration, we can use the equation F=ma to calculate the force. Plugging in the values, we get:

F = (8000 kg)(1.1 m/s^2) = 8800 N

Since the drum is rotating, we need to consider the radius of the drum in our calculations. The radius (R) of the drum is half of the diameter, so R = 1 m.

Now, let's put it all together. Using the equation T=I(alpha) + F x R, we get:

T = (562.5 kg*m^2)(1.1 m/s^2) + (8800 N)(1 m) = 619.4 N*m + 8800 N*m = 9420 N*m = 9.42 kN*m

Converting to the correct units, we get:

T = 9.42 kN*m = 94.2 kN*mm = 94.2 kN*m

So, our final answer is T = 94.2 kN*m