Solving for Horizontal Displacement: Ball on a String

In summary, a ball on a string moves in a vertical circle of radius R with a constant speed and centripetal acceleration a. When the string breaks at the highest point, the ball travels horizontally before reaching the ground. It was 2R off the ground when released. The horizontal displacement can be found using the kinematic equation v = √acR.
  • #1
aquapod17
11
0

Homework Statement


A ball on a string moves in a vertical circle of radius R at a constant speed with a centripetal acceleration a. At its
lowest point, the ball is a negligible height above the ground. After several revolutions, the string breaks at the highest
point in the motion. In terms of R, a, and g, find the horizontal displacement of the ball from the time the string breaks.


Homework Equations


ac=v2/r


The Attempt at a Solution


I found v = √acr but I don't know where to go from here.
 
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  • #2
If it breaks at it's highest point, what direction will the ball be traveling?

How high off the ground was it when it was released?

Doesn't it just become a simple kinematic problem?
 
  • #3
If it breaks at its highest point it will travel horizontally first, but then reach the ground.

It is 2R off the ground when it was released.

OMG I GOT IT! Thanks so much! I just had trouble manipulating the equation. Embarrassing.
 

1. What is the equation for finding the horizontal displacement of a ball on a string?

The equation for finding the horizontal displacement of a ball on a string is x = R * sin(θ), where x is the horizontal displacement, R is the length of the string, and θ is the angle between the string and the vertical.

2. How do you determine the angle θ in the equation?

The angle θ can be determined by using a protractor to measure the angle between the string and the vertical. Alternatively, if the length of the string and the height of the ball are known, the angle can be calculated using the formula θ = sin-1(x/R), where x is the horizontal displacement and R is the length of the string.

3. Can the equation be used for any ball on a string or are there specific conditions?

The equation can be used for any ball on a string as long as the string is taut and the ball is moving in a circular motion. Additionally, the equation assumes that there is no air resistance or friction acting on the ball.

4. How is the horizontal displacement affected by changes in the length of the string?

The horizontal displacement is directly proportional to the length of the string. This means that as the length of the string increases, the horizontal displacement also increases, and vice versa.

5. Are there any real-world applications for solving for horizontal displacement in a ball on a string?

Yes, this equation has real-world applications in fields such as physics and engineering. For example, it can be used to calculate the horizontal displacement of a pendulum or the movement of a satellite in orbit. It can also be used to determine the horizontal distance covered by a ball during a sport such as golf or baseball.

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