How Do You Calculate Spring Length and Ball Speed in Circular Motion?

In summary, the conversation discusses a problem involving a ball of mass 4kg suspended on a spring with a constant of 850n/m. The ball is spun in a circular orbit at a constant speed, making an angle of 22 degrees from the vertical. The conversation also includes a diagram and calculations for finding the length of the spring and the speed of the ball. There is some discussion about the accuracy of the calculations, but in the end, it is determined that the method used is correct. The conversation also considers the possibility of an unstretched length for the spring and its impact on the calculations. Overall, the conversation focuses on finding the length of the spring and the speed of the ball in the given scenario.
  • #1
mathzing
9
0
Hi, i am kind of at a loss here. I can't seem to figure out this problem, not sure if i am doing it right or not. All help is greatly appreciated.

A ball of mass 4kg is suspended on a light spring with a constant of 850n/m. the ball is spun in a circular orbit at a constant speed so the string makes an angle of 22 degrees from the vertical. find the length of the spring and the speed of the ball.

Here is what i believe the diagram would look like.

http://img212.imageshack.us/my.php?image=22me2.png

alright so here is what i did right/wrong.

fg=mg
= 4(9.8)
fg =39.2N

fg=fspring
mg=kxcos22
39.2=850xcos22
x=aprox 5cmthe length of the spring is 5cm . That sounds pretty wrong...hope in right :D

ok onto the speed.

fc=(mv^2)/r
T=39.2/cos22=42.3

tsin22=fc
42.3sin22=(m(v^2))/r
42.3sin22=(4(v^2))/5sin22
v=2.7m/s

speed = 2.7m/s

so did i do anything wrong..?
 
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  • #2
mathzing said:
alright so here is what i did right/wrong.

fg=mg
= 4(9.8)
fg =39.2N

fg=fspring
mg=kxcos22
39.2=850xcos22
x=aprox 5cm


the length of the spring is 5cm . That sounds pretty wrong...hope in right :D
You calculated the amount the spring stretches, but that's not its length.
 
  • #3
Hmm, i could be wrong, but i think my teacher intended it as a stretching question. Just incase, how do i find the length, iv never had any lessons relating to the actual length? is my speed correct, or will it need editing?
 
  • #4
Generally a spring has some non-zero unstretched length. But if your teacher wanted you to assume that the spring has an unstretched length of zero, then your method is fine. (I didn't check your arithmetic.)

(You would need additional information to find the unstretched length of the spring.)
 
  • #5
yeah that explains it, then he wanted us to find the amount it stretched by, how about the speed of the object, is that correct?
 
  • #6
mathzing said:
yeah that explains it, then he wanted us to find the amount it stretched by, how about the speed of the object, is that correct?
Yes, your method--for both parts--is correct (assuming, as I said, the unstretched length of the spring is 0).

In the first part, you clearly are finding the amount the spring stretches. But when you use that to find the radius for computing the centripetal force, that's when you are assuming the unstretched length is zero. (Say the unstretched length were 10 cm instead of 0. Then you would find its stretched length to be 10 + 5 = 15 cm. Which would give you a completely different speed. Make sense?)
 
  • #7
yup :D thanks
 

FAQ: How Do You Calculate Spring Length and Ball Speed in Circular Motion?

What is the relationship between spring length and speed?

The relationship between spring length and speed is described by Hooke's law, which states that the force required to stretch or compress a spring is directly proportional to the displacement of the spring from its equilibrium position. In other words, the longer the spring is stretched or compressed, the greater the speed at which it will oscillate.

How does the stiffness of a spring affect its length and speed?

The stiffness of a spring, also known as its spring constant, determines how much force is required to stretch or compress the spring. A stiffer spring will have a higher spring constant and will require more force to change its length, resulting in a faster speed of oscillation. Conversely, a less stiff spring will have a lower spring constant and a slower speed of oscillation.

Can the length of a spring affect the period of its oscillation?

Yes, the length of a spring can affect the period of its oscillation. According to the equation T = 2π√(m/k), where T is the period, m is the mass attached to the spring, and k is the spring constant, the period of a spring's oscillation is inversely proportional to the square root of its spring constant. Therefore, a longer spring with a lower spring constant will have a longer period of oscillation compared to a shorter spring with a higher spring constant.

How does gravity affect the length and speed of a spring?

Gravity does not directly affect the length and speed of a spring. However, it can impact the period of oscillation if the spring is vertical and the mass attached to it is not negligible. In this case, the weight of the mass will affect the spring's equilibrium position and therefore the length and speed of its oscillation.

Why do springs exhibit simple harmonic motion?

Springs exhibit simple harmonic motion because they follow Hooke's law, which is a simple harmonic oscillator equation. This means that the force exerted by the spring is directly proportional to the displacement from its equilibrium position, resulting in a sinusoidal motion. Additionally, the restoring force of the spring is always directed towards its equilibrium position, causing it to oscillate back and forth in a predictable pattern.

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