It all follows from Newton's second law F = m a. The mass is rotating in a cricle with radius r = L sin(theta), so it's acceleration is:
a = v^2/r = v^2/[L sin(theta)]
which has a directon toward the center of the circle.
This means that a total force of
F = m a = m v^2/[L sin(theta)]
is acting on the mass, which points toward the center.
Then the source of this force is the string tension. If the string tension is T, then the horizontal component of this force is T sin(theta) and this must be equal to F:
T sin(theta) = m v^2/[L sin(theta)]
The total force in the vertical direction is zero, as the mass is not accelerating in that direction. Now, the string tension does have a comnponent in that direction of
T cos(theta). The gravitational force also acts in the mass, it acts with a foce of m g, directed downward. So, the total force in the vertical direction is
T cos(theta) - m g and thus has to be zero:
T cos(theta) - m g = 0 --->
T cos(theta) = m g
If you divide the two equations, you get:
T sin(theta)/[T cos(theta)] =
m v^2/[L sin(theta)] / (m g) --->
v = sin(theta)sqrt[L g /cos(theta)]
= 3.16 m/s
The string tension can be computed from e.g. T cos(theta) = m g, this gives T = 2.52 N
A ball of mass m = 0.2 kg is attached to a (massless) string of length L = 2 m and is undergoing circular motion in the horizontal plane (imagine a wave swinger ride at the amusement park).
What should the speed of the mass be for θ to be 39°? What is the tension in the string?
The answers are 3.16 m/s and 2.52 N. How do I solve for these answers?
2 answers
Thank you.