# Tangential Acceleration Problem, UCM

Homework Statement:
A student is swinging a ball on a string overhead in a horizontal plane. The string initially has length 𝑙0 and the ball is moving with an initial speed 𝑣0 . The student decides to see how fast they
can spin the ball, so they begin moving it faster and faster with a constant tangential acceleration 𝑎_t. However, they know that the string is close to breaking, so they decide to keep letting the string get longer and longer in order to keep the tension in the string constant. Find expressions for the length of the string and the total acceleration (magnitude and direction) as a function of time, 𝑙0 , 𝑣0 , and 𝑎_t . You may ignore the effect of gravity on the ball.
Relevant Equations:
F_R= m(a_R)
Tension = (mv^2)/r
Tension = m(a_R)
Homework Statement: A student is swinging a ball on a string overhead in a horizontal plane. The string initially has length 𝑙0 and the ball is moving with an initial speed 𝑣0 . The student decides to see how fast they
can spin the ball, so they begin moving it faster and faster with a constant tangential acceleration 𝑎_t. However, they know that the string is close to breaking, so they decide to keep letting the string get longer and longer in order to keep the tension in the string constant. Find expressions for the length of the string and the total acceleration (magnitude and direction) as a function of time, 𝑙0 , 𝑣0 , and 𝑎_t . You may ignore the effect of gravity on the ball.
Homework Equations: F_R= m(a_R)
Tension = (mv^2)/r
Tension = m(a_R)

I will need 2 F net equations.
F_R= m(a_R)
Tension = (mv^2)/r
Tension = m(a_R)

berkeman
Mentor
Welcome to the PF. Tension = (mv^2)/r
It seems like that is the key equation, no? Can you say more about how to use that one equation and the variables they mention to accomplish the goal?

Welcome to the PF. It seems like that is the key equation, no? Can you say more about how to use that one equation and the variables they mention to accomplish the goal?
Since I know F=ma, m=F/a. Then, can I plug in this m into the Tension equation to get Tension = (Fv^2/ar)? I am unsure about how to relate the length of the string though.