Answer:
If both the radius and frequency are doubled, then the tension is increased 8 times.
Explanation:
The radial acceleration (
), measured in meters per square second, experimented by the moving end of the string is determined by the following kinematic formula:
(1)
Where:
- Frequency, measured in hertz.
- Radius of rotation, measured in meters.
From Second Newton's Law, the centripetal acceleration is due to the existence of tension (
), measured in newtons, through the string, then we derive the following model:
(2)
Where
is the mass of the object, measured in kilograms.
By applying (1) in (2), we have the following formula:
(3)
From where we conclude that tension is directly proportional to the radius and the square of frequency. Then, if radius and frequency are doubled, then the ratio between tensions is:
(4)
![\frac{T_{2}}{T_{1}} = 4\cdot 2](https://tex.z-dn.net/?f=%5Cfrac%7BT_%7B2%7D%7D%7BT_%7B1%7D%7D%20%3D%204%5Ccdot%202)
![\frac{T_{2}}{T_{1}} = 8](https://tex.z-dn.net/?f=%5Cfrac%7BT_%7B2%7D%7D%7BT_%7B1%7D%7D%20%3D%208)
If both the radius and frequency are doubled, then the tension is increased 8 times.