How Long Must Zorch Push to Extend Earth's Rotation to 29 Hours?

[blimkie]

i need help with this problem


N.B. See eq. 13.26 and table 13.2 on p.382

11. [1pt] Zorch, an archenemy of Superman, decides to slow Earth's rotation to once per 29.0 h by exerting an opposing force at the equator and parallel to it. Although Superman knows that Zorch can only exert a force of 4.65×107 N (a little greater than a Saturn V rocket's thrust), he isn't sure if it is an immediate concern. Assuming that Earth's moment of inertia for such a process is 9.71×1037 kg·m2 and that its radius is 6.37×106 m, how long would Zorch have to push with this force to accomplish his nefarious goal?

Heres what i did so far..

here is the equation shown in the book a = T/I

a=angular acceleration
T=net torque
I=moment of inertia

so i determned the acceleration is 4.789x10^-31 m/s^2

then i used v=(2)(pi)(r)/T to find the veocity of Earth with the periods of rotation at 24 hours and 29 hours

v at 24h = 1.667x10^6 v at 29h = 1.379x10^6

so then i used the formula

Vf = Vf + a(t) i solved for t and my answer was 6.01 x10^35

when i entered my answer into the online homework grading system it was wrong and i have checked my work and ended up with the same answer twice so i guess I am just using the wrong approach

if anyone could help me that's great thanks

 

[durt]

When you calculated the angular acceleration, you forgot to consider the radius of the earth. And this is angular acceleration, so its units are radians/second^2. After you've found this, you can calculate the time it takes for the Earth to reach its new angular velocity by using the angular equivalent of the equation you selected: $$\omega=\omega_0 + \alpha t$$.

P.S. One way to make this problem (and others in the future) easier is to use symbols until you reach the final equation, into which you can plug in all the given information. That way, you won't be fumbling with big crazy numbers all over the place, and it will be easier to see mistakes.

 

[kyle8921]

I would like to provide some guidance on how to approach this problem. First, it is important to understand the concept of moment of inertia and how it relates to rotational motion. The moment of inertia is a measure of an object's resistance to changes in its rotational motion. In this case, we are dealing with Earth's rotation, so we need to consider its moment of inertia.

Next, we need to use the given equation a = T/I, where a is the angular acceleration, T is the net torque, and I is the moment of inertia. We are given the value for T, which is the opposing force exerted by Zorch, and we can calculate the moment of inertia for Earth using the given values.

Once we have calculated the angular acceleration, we can use the equation v = v0 + at (where v0 is the initial velocity, a is the acceleration, and t is the time) to calculate the final velocity of Earth after the force has been applied for a certain period of time. In this case, the final velocity would be the velocity at which Earth completes one rotation in 29 hours.

Finally, we can use the formula v = 2πr/T to calculate the velocity of Earth at its original rotation period of 24 hours. This will allow us to determine the change in velocity that needs to occur for the rotation period to increase to 29 hours.

Using the final velocity and the change in velocity, we can then use the formula Δv = aΔt to calculate the time it would take for the desired change in velocity to occur. This would be the amount of time that Zorch would have to push with the given force to slow down Earth's rotation to 29 hours.

It is important to carefully consider the units and conversions when solving this problem, as well as checking for any errors in calculations. I hope this helps in approaching this problem.