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?
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)
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
Thank you Bob.
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
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
To find the tension in a string when it is attached to an object with mass "m" and length "r", you can use the following steps:
1. Identify the forces acting on the object: In circular motion, there are two forces acting on the object: the tension force in the string and the weight force acting vertically downward.
2. Resolve the forces: Separate the weight force into its vertical and horizontal components. The vertical component does not contribute to the tension, as it is balanced by the normal force exerted by the ground (if the object is swinging horizontally) or by the supporting structure (if the object is swinging vertically).
3. Analyze the horizontal component: Since the object is moving in a horizontal circle, you need to consider the horizontal component of the weight force. This component is responsible for providing the centripetal force required to keep the object in circular motion.
4. Equate the forces: The tension force in the string provides the centripetal force required for circular motion. Therefore, you can equate the horizontal component of the weight force to the tension force:
Tension = Horizontal component of weight force = m * g * cos(theta)
Where:
Tension is the tension in the string
m is the mass of the object
g is the acceleration due to gravity (approximately 9.8 m/s^2)
theta is the angle the string makes with the horizontal
cos(theta) is the cosine of the angle theta
Note: If the object is swinging vertically, the angle theta would be 90 degrees, and the tension in the string would be equal to the entire weight of the object.
By following these steps, you can calculate the tension in the string for both horizontal and vertical circular motion scenarios.