# Tangential Acceleration Problem, UCM

• stark3000
In summary, the problem involves a student swinging a ball on a string with initial length l0 and initial speed v0. The student increases the ball's speed with a constant tangential acceleration at, while simultaneously allowing the string to lengthen in order to maintain constant tension. The goal is to find expressions for the length of the string and the total acceleration (magnitude and direction) as a function of time, l0, v0, and at. To solve this, the key equation is Tension = (mv^2)/r, where r is the length of the string. By using F=ma and m=F/a, the tension equation can be rewritten as Tension = (Fv^2/ar). The length of the
stark3000
Homework Statement
A student is swinging a ball on a string overhead in a horizontal plane. The string initially has length đť‘™0 and the ball is moving with an initial speed đť‘Ł0 . 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 đť‘Ž_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, đť‘™0 , đť‘Ł0 , and đť‘Ž_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 đť‘™0 and the ball is moving with an initial speed đť‘Ł0 . 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 đť‘Ž_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, đť‘™0 , đť‘Ł0 , and đť‘Ž_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)

Welcome to the PF.
stark3000 said:
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?

berkeman said:
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.

## 1. What is tangential acceleration?

Tangential acceleration is the rate of change of an object's tangential velocity in a circular motion. It is measured in meters per second squared (m/s^2).

## 2. What is a tangential acceleration problem in UCM?

A tangential acceleration problem in UCM (Uniform Circular Motion) involves calculating the tangential acceleration of an object moving in a circular path with a constant speed.

## 3. How do you calculate tangential acceleration in UCM?

The formula for tangential acceleration in UCM is a = v^2/r, where v is the tangential velocity in meters per second (m/s) and r is the radius of the circular path in meters (m).

## 4. What are some real-life examples of tangential acceleration in UCM?

One example is the motion of a car driving around a circular track at a constant speed. Another is the motion of a satellite orbiting the Earth.

## 5. How does tangential acceleration affect the motion of an object in UCM?

Tangential acceleration in UCM causes a change in the object's direction, as it is constantly changing its velocity vector. It also affects the object's centripetal acceleration, which keeps the object moving in a circular path.

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