a ball of mass m is attached to a rope of negligible mass and is swung around in a vertical circle. find the tension in the string at any given time in terms of m, v, r, g, theta. theta is the angle that the string makes with the vertical.
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Consider the forces acting upon the ball: the force of gravity on the ball and the tension of the rope. Assuming that v is the velocity of the ball, then the net force on the bottom with respect to a radial line is the centripetal force, mv^2/r.
The component of gravity along the radial line is mg*cosθ and points away from the center. Tension acts towards the center of the circle, so the net force on the ball is:
ΣF = F(t) - mg*cosθ.
But ΣF = mv^2/r, so:
mv^2/r = F(t) - mg*cosθ ==> F(t) = mv^2/r + mg*cosθ.
Just a note here: if superimpose the xy-plane on the vertical circle such that the positive y-axis points upward, θ will be the angle between the rope and the positive y-axis. So, at the bottom of the circle, θ = 180 degrees.
I hope this helps!
The component of gravity along the radial line is mg*cosθ and points away from the center. Tension acts towards the center of the circle, so the net force on the ball is:
ΣF = F(t) - mg*cosθ.
But ΣF = mv^2/r, so:
mv^2/r = F(t) - mg*cosθ ==> F(t) = mv^2/r + mg*cosθ.
Just a note here: if superimpose the xy-plane on the vertical circle such that the positive y-axis points upward, θ will be the angle between the rope and the positive y-axis. So, at the bottom of the circle, θ = 180 degrees.
I hope this helps!