Respuesta :
Answer:
If both the radius and frequency are doubled, then the tension is increased 8 times.
Explanation:
The radial acceleration ([tex]a_{r}[/tex]), measured in meters per square second, experimented by the moving end of the string is determined by the following kinematic formula:
[tex]a_{r} = 4\pi^{2}\cdot f^{2}\cdot R[/tex] (1)
Where:
[tex]f[/tex] - Frequency, measured in hertz.
[tex]R[/tex] - Radius of rotation, measured in meters.
From Second Newton's Law, the centripetal acceleration is due to the existence of tension ([tex]T[/tex]), measured in newtons, through the string, then we derive the following model:
[tex]\Sigma F = T = m\cdot a_{r}[/tex] (2)
Where [tex]m[/tex] is the mass of the object, measured in kilograms.
By applying (1) in (2), we have the following formula:
[tex]T = 4\pi^{2}\cdot m\cdot f^{2}\cdot R[/tex] (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:
[tex]\frac{T_{2}}{T_{1}} = \left(\frac{f_{2}}{f_{1}} \right)^{2}\cdot \left(\frac{R_{2}}{R_{1}} \right)[/tex] (4)
[tex]\frac{T_{2}}{T_{1}} = 4\cdot 2[/tex]
[tex]\frac{T_{2}}{T_{1}} = 8[/tex]
If both the radius and frequency are doubled, then the tension is increased 8 times.
If both the radius and frequency are doubled, then the tension will be increased 8 times.
From kinematic formula:
[tex]\bold {a_r = 4\pi r^2 \times f^2 \times R^2}[/tex] .............................. (1)
Where:
f - Frequency,
R - Radius of rotation,
From Second Newton's Law,
[tex]\bold {\sum F = T = m\times a_r }[/tex] ......................... (2)
Where
m is the mass of the object,
From equation 1 and 2,
[tex]\bold {T = m\times 4\pi r^2 \times f^2 \times R^2}\\[/tex] .................... (3)
From equation, 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:
[tex]\bold {\dfrac {T_2}{T_1} = (\dfrac {F_2}{F_1})^2 \times \dfrac {R_2}{R_1} = 4 \times 2 = 8 }[/tex]
Therefore, If both the radius and frequency are doubled, then the tension will be increased 8 times.
To know more about frequency,
https://brainly.com/question/24864018
