# Torque to lift object by cable on a drum accelerating

• swotty
In summary: I made a mistake when I did it quickly. So the book was wrong to the tune of 1.6 kNm. :)In summary, there is a discrepancy in the given answer for the torque required to lift an object using a steel cable and a drum mounted on a horizontal shaft. The correct torque is calculated to be 13.8 kNm, while the textbook gives an answer of 11.2 kNm. It is possible that the textbook used the radius of gyration instead of the actual drum radius, resulting in a miscalculation.
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:
$$T_n=T_1+T_2$$
$$T_n=13.2\times10^3+600$$
$$T_n=13.8kNm$$

berkeman
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.

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

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.

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

TomHart said:
they forgot to multiply the tension in the rope by the radius.

TomHart
haruspex said:
I didn't think about that, but that seems more likely.

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

swotty said:
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.

## 1. What is torque?

Torque is a measure of the rotational force applied to an object. It is typically measured in units of Newton-meters (Nm) or foot-pounds (ft-lb).

## 2. How does torque affect lifting an object with a cable on a drum?

The torque applied to the drum affects the speed and power at which the cable can lift an object. A higher torque will result in a faster and stronger lift, while a lower torque will result in a slower and weaker lift.

## 3. How is torque calculated for lifting an object with a cable on a drum?

Torque can be calculated by multiplying the radius of the drum by the force applied to the cable. This force can be determined by the weight of the object being lifted and the acceleration of the drum.

## 4. What are the factors that affect torque in this situation?

The factors that affect torque in this situation include the radius of the drum, the weight of the object being lifted, the acceleration of the drum, and any friction or resistance in the cable or drum system.

## 5. How can torque be increased for lifting heavier objects?

To increase the torque for lifting heavier objects, the radius of the drum can be increased, a larger force can be applied to the cable, or the acceleration of the drum can be increased. It is important to consider the capabilities and limitations of the cable and drum system when making these adjustments.

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