Respuesta :
Answer:
[tex]T = 37.5 N[/tex]
Explanation:
As the pendulum reached to the lowest position then we will have
[tex]T - mg = \frac{mv^2}{L}[/tex]
[tex]14.2 - m(9.81) = \frac{m(2.9^2)}{L}[/tex]
now when it will reach to the height of the peg then its speed is given as
[tex]v_f^2 - v_i^2 = 2 a d[/tex]
so we will have
[tex]v_f^2 - 2.9^2 = 2(-9.81)(\frac{L}{5})[/tex]
[tex]v_f^2 = 2.9^2 - 3.924L[/tex]
also we know that
[tex]2.9^2 - 0 = 2(9.81)(L)[/tex]
[tex]L = 0.43 m[/tex]
[tex]m = 0.48 kg[/tex]
now we have speed of the pendulum when it reach the same height is given as
[tex]v_f^2 = 2.9^2 - (3.924(0.43)[/tex]
[tex]v_f = 2.6 m/s[/tex]
Now the tension in the string is given as
[tex]T = \frac{mv_f^2}{\frac{L}{5}}[/tex]
[tex]T = \frac{0.48(2.6)^2}{\frac{0.43}{5}}[/tex]
[tex]T = 37.5 N[/tex]
Centrifugal forces act outwards from the center. The tension in the string at the same vertical height as the peg is 37.5 N.
What is centrifugal force?
Centrifugal force can be defined as the outward force that is applied to an object of mass m when it is rotated about an axis. It is given by the formula,
[tex]F= \dfrac{mv^2}{R} = m\omega^2 R[/tex]
Given to us
Velocity at the bottom of the path, v = 2.9 m/s
Tension in the string, T = 14.2 N
Peg placement, d = 4/5
When the pendulum will be at the bottom, then according to the third equation of motion,
[tex]v^2 - u^2 = 2as\\\\2.9^2 - 0^2 = 2(9.81)(L)\\\\L = 0.429\ m[/tex]
When the pendulum will be at the bottom, there will be 3 forces that will be acting on the pendulum,
- Centrifugal force, [tex]F_c[/tex]
- Tension in the string, T
- Weight of the pendulum, W
Calculating all the vertical forces,
[tex]\sum F_y = 0\\\\T = W + F_c\\\\14.2 = mg + (\dfrac{mv^2}{L})\\\\14.2 = m(g + \dfrac{v^2}{L})\\\\14.2 = m(9.81 + \dfrac{2.9^2}{0.429})\\\\m = 0.483\ kg[/tex]
Also, the speed of the pendulum when it reaches the height of the peg, According to the third equation of the motion,
[tex]v^2 - u^2 = 2as[/tex]
Substitute the values,
[tex]v_p^2 -v^2 = 2(g)(\dfrac{4L}{5})\\\\v_p^2 -(2.9)^2 = 2(9.81)(\dfrac{4\times 0.429}{5})\\\\v_p = 2.595\rm\ m/s[/tex]
At the point when it reaches the height of the peg, the tension in the string will be equal to centrifugal force,
[tex]F = \dfrac{mv^2}{R}\\\\F = 37.5 \rm\ N[/tex]
Hence, the tension in the string at the same vertical height as the peg is 37.5 N.
Learn more about Centrifugal Force:
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