How Do You Calculate Velocity in a Horizontal Slingshot Setup?

In summary, we are trying to find the velocity of a mass being released from a horizontal slingshot consisting of two identical springs with a spring constant of 74 N/m and an equilibrium length of 49 cm. The cup containing the mass is pulled to 0.6 m to the left of the vertical and then released. Using the energy approach, we can calculate the energy stored in the stretched springs by finding the elongation of the springs from their equilibrium length. This can be done using the Pythagorean theorem, giving us a total energy of 5.99 J for both springs.
  • #1
J.live
95
0

Homework Statement


A horizontal slingshot consists of two light, identical springs (with spring constants of 74 N/m) and a light cup that holds a 1-kg stone. Each spring has an equilibrium length l0 = 49 cm. When the springs are in equilibrium, they line up vertically. Suppose that the cup containing the mass is pulled to x = 0.6 m to the left of the vertical and then released.




The Attempt at a Solution



ME= PE +KE

Initially PE = 1/2kx^2 KE= 0 --> ME = 1/2kx^2 +0 ---> 1/2(74)(.6)^2 +0 ?

Idk how to find the velocity. I am guessing we'll just use ME= 0+1/2mv^2 ?

Or, do I have to incorporate the equilibrium length into the equation somehow?

Any help will be appreciated.
 

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  • #2
anyone ?
 
  • #3
J.live said:
Or, do I have to incorporate the equilibrium length into the equation somehow?
Definitely. Figure out how much the springs are stretched from their equilibrium lengths when the cup is moved. (It's not simply 0.6 m.) Then your energy approach will work.
 
  • #4
How do I calculate that ?:/

Edit: Pythagorean Theorem ?
 
  • #5
J.live said:
How do I calculate that ?:/
How long are the stretched springs? How does that compare with their unstretched length?

Yes, Pythagorean theorem!
 
  • #6
This is what I did

C^2 = .6^2 + .49 ^2 = .77 m

Total ME = 1/2(74)(.77)^2 +0 = 21. 97 J ?

That seems off. Am I suppose to subtract and add the x and y components before applying the Pythagorean Theorem ?

If so then isn't y = 0 since they are in opposite directions? x = .8 +.8 = 1.6 ?

I am still kind of confused.

Edit- I just tried it that makes no sense whatsoever. :/ What am i doing wrong ?
 
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  • #7
When I subtract the values I am getting the right answer. C^2= .64^2 - .48^2 ?
 
  • #8
J.live said:
This is what I did

C^2 = .6^2 + .49 ^2 = .77 m
You correctly found the stretched length of the springs. So how much did they stretch compared to their unstretched length? That elongation will tell you the amount of stored elastic potential energy.
 
  • #9
I don't understand which value you are referring to exactly as unstretched length?

.49 m?
 
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  • #10
J.live said:
I don't understand which value you are referring to exactly as unstretched length?

.49 m?
Yes, what they are calling the equilibrium length or l0. To find the energy stored in the stretched springs, you need to know how far they are stretched from their equilibrium positions.
 
  • #11
So, will it be .77 -.49 = .28 ?
 
  • #12
J.live said:
So, will it be .77 -.49 = .28 ?
Good.
 
  • #13
Now I plug it in the equation

ME = 1/2kx^2 +0 --> 1/2 74 (.28)^2= 2.9 J ?
 
  • #14
J.live said:
Now I plug it in the equation

ME = 1/2kx^2 +0 --> 1/2 74 (.28)^2= 2.9 J ?
That's the energy stored in each spring.
 
  • #15
ME= 2*(1/2 ) 74 (.28)^2 = 5.99 j
 
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Related to How Do You Calculate Velocity in a Horizontal Slingshot Setup?

What is mechanical energy?

Mechanical energy is the sum of potential and kinetic energy of an object. Potential energy is the energy an object has due to its position or state, while kinetic energy is the energy an object has due to its motion.

What are the types of mechanical energy?

The two types of mechanical energy are potential energy and kinetic energy. Potential energy can further be categorized into gravitational potential energy, elastic potential energy, and chemical potential energy.

What is the principle of conservation of mechanical energy?

The principle of conservation of mechanical energy states that in a closed system, the total amount of mechanical energy remains constant. This means that energy cannot be created or destroyed, but can only be converted from one form to another.

How is mechanical energy calculated?

Mechanical energy can be calculated by adding the potential energy and kinetic energy of an object. The formula for mechanical energy is E = PE + KE.

What are some examples of mechanical energy?

Some examples of mechanical energy are a swinging pendulum, a moving car, a stretched spring, a spinning top, or a bouncing ball. These objects all have both potential and kinetic energy, making up their total mechanical energy.

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