Elevator Torque and Angular Momentum

[usnberry]

Homework Statement

A 22,500N elevator is to be accelerated upward by connecting to it a counterweight using a light cable passing over a solid uniform disk-shaped pulley. There is no appreciable friction at the axle of the pulley, but its mass is 875kg and it is 1.50 m in diameter. (a) How heavy should the counterweight be so that it will accelerate the elevator upward through 6.75 m in the first 3.00 s, starting from rest? (b) Under these conditions, what is the tension in the cable on each side of the pulley?

Homework Equations

I=1/2*875*.75$$^{2}$$=246.1 kg*m$$^{2}$$
a=(2*6.75)/(3$$^{2}$$)=1.5m/s$$^{2}$$ from x=(x$$_{i}$$)+(.5)(a)(t$$^{2}$$)
$$alpha$$ = a/r = 2 rad/s^2 from a = R * $$alpha$$
$$\Sigma$$$$\tau$$=I*a

The Attempt at a Solution

So I get from this is that for the elevator to rise the 6.75 feet in 3.00 seconds the total torque must be 492.2 N*m.

T$$_{1}$$=m$$_{elev}$$*g-m$$_{elev}$$*a
T$$_{2}$$=m$$_{counter}$$*g+m$$_{counter}$$*a

From this I get T$$_{1}$$ = 19056 N

$$\Sigma$$$$\tau$$ = T$$_{1}$$*r-T$$_{2}$$*r

So...

T$$_{2}$$ = 18400 N

So should be...
m$$_{counter}$$ = 1628 kg * 9.8 = 1.6*10^4 N

Unfortunately, according to the book I am wrong on all three counts, the tensions and the mass. The answers are supposed to be 3.16*10^4 N mass counter, 2.60 * 10^4N Tension 1 and 2.67 * 10^4N for Tension 2.

Any help would be greatly appreciated. Thanks in advance.

 

[learningphysics]

The directions for your forces are wrong I think... the elevator is accelerating upward, so what's the sum of the forces in the vertical direction for the elevator?

Remember that a is upward for the elevator.

 

[m2003]

I would say that your calculations and reasoning seem to be mostly correct. However, it is possible that the book's answers are taking into account some other factors that were not mentioned in the problem statement. For example, the book's answers may be taking into account the weight of the cable itself, or the weight of the pulley, which could affect the tension in the cable and the required mass of the counterweight.

Additionally, the book's answers may be using slightly different values for the acceleration due to gravity (9.8 m/s^2 instead of 9.81 m/s^2) or the mass of the elevator (22,500 N instead of 22,500 kg). These small differences could result in slightly different values for the tension and mass.

In any case, it would be a good idea to double-check your calculations and make sure you are using the correct values for all the variables. If you are still unable to arrive at the book's answers, you could also try reaching out to the author or your professor for clarification.