Torque to lift object by cable on a drum accelerating

[swotty]

Homework Statement

An object of mass 1000 kg is lifted by means of a steel lifting cable being wound round a drum of diameter 2.5m mounted on a horizontal shaft. The drum and shaft have a mass of 1000 kg and a radius of gyration of 1.0 m. What is the torque required to give the object an upward acceleration of 0.75 m/s^2

Homework Equations

Tension in cable $Fc=mg + ma$
Moment of inertia $I=mk^2$
Torque $T=I\alpha=Fr$
Linear acceleration $a=r\alpha$

The Attempt at a Solution

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My problem is I don't get the answer given in the textbook. Which is 11.2 kNm. I always assume at first that I have missed something. But I've tried various tacks and still do not get this answer. Am I missing something?

Here is my calculations:

Tension in cable due to mass of object and acceleration upwards:

$$F_{c}=1000\times9.81 + 1000\times0.75$$
$$=10560N$$

Torque on rim of drum to create tension:
$$T_1=Fc\times r $$$(r=radius of drum)$
$$T_1=10560\times 1.25$$
$$=13.2\times10^3 Nm$$
Torque to accelerate drum:
$$T_2=I\times\alpha$$
$$I=1000\times 1^2=1000 kg m^2$$
acceleration Angular
$$\alpha = a/r$$
$$ \alpha =0.75/1.25$$
$$\alpha = 0.6 rad/s^2$$
$$T_2=1000\times0.6$$
$$T_2=600Nm$$
Total Torque to accelerate drum and object:
$$T_n=T_1+T_2$$
$$T_n=13.2\times10^3+600$$
$$T_n=13.8kNm$$

 

[TomHart]

I can't find any problem with what you did. I think they came up with 11.2 because they forgot to multiply the tension in the rope by the radius. In other words, they used tension instead of torque when summing torques. Then again, I ran through this pretty quickly myself.

 

[swotty]

Many thanks Tom for quick reply. I thought I was right but human nature being what it is you expect someone who has written a textbook will get it write. I should learn though it's happened to me before.
Regards
Glyn

 

[TomHart]

Yes, they do make mistakes sometimes. But I try to not be too hard on them. I make lost of mistakes. :)

By the way, welcome to Physics Forums.

 

[swotty]

Thanks for the welcome and yes I too make loads of them hence write above instead of right
This question also does not specify information about the construction of the drum i.e. whether the drum is solid or a thin walled cylinder which changes the answer of course. I do a bit of part time teaching and I wanted to use this as a homework exercise. I think I will but with some modification

 

[haruspex]

they forgot to multiply the tension in the rope by the radius.

Or perhaps they used the radius of gyration instead of the drum radius.

 

[TomHart]

Or perhaps they used the radius of gyration instead of the drum radius.

I didn't think about that, but that seems more likely.

 

[swotty]

Yes that's a possibility, the numbers give 11.16kNm but the radius of gyration is used to find the torque to accelerate the drum; the tension acts at the rim and therefore the couple is tension times the radius of the drum i.e. 1.25m

 

[haruspex]

Yes that's a possibility, the numbers give 11.16kNm but the radius of gyration is used to find the torque to accelerate the drum; the tension acts at the rim and therefore the couple is tension times the radius of the drum i.e. 1.25m

Oh yes, yours is definitely the right answer.