What Are the Dynamics of a Motor Tethered to a Tower?

[ataglance05]

Homework Statement

A 50 meter tower is on a flat plane. A 10 kg motor is tethered to the top of the tower by a 100 meter wire. The motor fires with sufficient force that has a constant angular velocity of .75 rad/sec.

1) What is the tension in Newtons on the tether? I think one uses F=mω2r2/r
2) How high off the ground is the object as it spins around the tower?
3) How does the angle that the tether makes with the horizontal vary with the angular velocity? (In other words: What is the equation that relates the angle to the angular velocity?)

Homework Equations

The Attempt at a Solution

Not sure if any of these are right:

1) F=m^2r^2/r
=10(.75)(100)
=750 N
2) Highest point off the ground is 50m because that's how high the tower is.
3)The angle is directly proportional to the angular velocity. (The equation is theta=ωi(T)+1/2(alpha)(T)^2

 

[Dick]

You mean that a 10kg object is tethered to a motor atop a tower, right? You need to draw a force (and geometry) diagram and split into components. Both the tension AND GRAVITY are acting on the circling object. That's why the angle of the tether isn't horizontal.

 

[m2003]

)

I would like to clarify a few points before providing a response to the questions. First, it is important to note that the motor itself does not have a constant angular velocity of .75 rad/sec. It is the object attached to the motor, such as a pendulum or another rotating mass, that has this velocity. Additionally, the equation F=mω2r2/r is incorrect. The correct equation for the tension in the tether is T=mrω2, where T is the tension, m is the mass of the object, r is the length of the tether, and ω is the angular velocity.

Now, to answer the questions:

1) The tension in the tether can be calculated using the equation T=mrω2. Plugging in the given values, we get:
T=(10 kg)(.75 rad/sec)^2(100 m)=562.5 N. Therefore, the tension in the tether is 562.5 Newtons.

2) The object attached to the motor will be at a constant height of 50 meters off the ground, as it is attached to the top of the tower.

3) The angle that the tether makes with the horizontal does not vary with the angular velocity. This angle is determined by the length of the tether and the height of the tower, and is not affected by the angular velocity of the object attached to the motor. Therefore, there is no equation that relates the angle to the angular velocity.

In conclusion, the tension in the tether is 562.5 N, the object will remain at a constant height of 50 meters, and the angle of the tether with the horizontal is not affected by the angular velocity.