Question

We are learning about circular motion, uniform and non uniform. how do you find tension in a string, when the string is attached to a object with mass m and length r? What about if you are swinging it around in a horizontal circle and the string makes a angle theata with the horizontal?

Answer 1

If we ignore gravity, and calculate the tension in the string due solely to the circular motion, then
Tc=mrω²
where
Tc=tension due to circular motion
m=mass attached to the end of the string
r=radius
&omega=angular velocity, in radians / second
For a horizontal circular motion, the above value of Tc should be vectorially added to the weight, i.e.
Total Tension, T
= &radic(T²+(mg)^2)
The angle
tan^-1(mg/Tc)
is the angle with the horizontal.

For a vertical circular motion, Tc must be added to the vertical component of weight, -mg (positive upwards)
The vertical component can be obtained by -mg*sin(θ),
where θ=angle above horizontal, equals zero when mass is horizontal and going up.

Tension T
=Tc - mg*sin(θ)

Answer 2

The only thing I would add to the above, is that the r in the formula for centripetal force is the radius of the horizonal circle, not the length of the string. if l is the length of the string, then r=lcosTheta. This makes Tc = mw^2 l cosTheta

Answer 3

Thank you Bob.

Answer 4

In the total tension, what does &radic mean?
Total Tension, T
= &radic(T²+(mg)^2)
The angle
tan-1(mg/Tc)
is the angle with the horizotal

Answer 5

Total Tension, T
= √(T²+(mg)^2)
The angle
tan-1(mg/Tc)
is the angle with the horizotal

√ is the equaivalent of square-root.
The formula is a typical application of Pythagoras theorem while adding the vertical and horizontal components of force. I could rewrite it as:

Total Tension, T
= sqrt(T²+(mg)^2)
The angle
tan^-1(mg/Tc)
is the angle with the horizotal