Angle of twist due to linear varying torque

[Dell]

in the following question
http://stommel.tamu.edu/~esandt/Teach/Summer00/CVEN305/Examples/Set3/prob3413.jpg

while G and J are parameters and the moment changes by the function M(x)=t*x
i am asked to find the angle of twist from one end to another

i know

dθ/dx=M/(GJ)

therefore

dθ=M/(GJ)dx

dθ=tx/(GJ)*dx

since t,G,J are independant of X

$$\int$$dθ=t/(GJ)$$\int$$xdx from 0 to L

θ=tL^2/(2GJ)


but the correct answer
θ=tL^3/(6GJ)

i can see that they have integrated twice, i can only imagine that they integrated once to find the total torque and the second time to find the angle,,,
but surely this should come from the differential equation? surely if M is a function of X and i integrate dx i souldnt need to sum up the torqe seperately??

 

[Valkarie]

The equation you have written down is the differential equation for the angle of twist. By integrating the equation twice, you can calculate the angle of twist from one end to another. The first integration will give you the total torque (M(x)) and the second integration will give you the angle of twist (θ).

 

[Mene]

Your understanding is correct. In order to find the angle of twist from one end to another, you would need to integrate the function M(x) with respect to x to find the total torque. This would be the first integration. Then, in order to find the angle of twist, you would need to integrate the differential equation dθ/dx=M/(GJ) with respect to x. This would be the second integration, and it would result in the correct answer of θ=tL^3/(6GJ).

It is important to note that in this case, the torque is a function of x, so the torque at any point along the length of the object is different. This is why the first integration is necessary in order to find the total torque.

In summary, the correct approach is to first find the total torque by integrating M(x) with respect to x, and then use this result to integrate the differential equation for the angle of twist. This will result in the correct answer of θ=tL^3/(6GJ).