Energy and Simple Harmonic Motion

[shawonna23]

In preparation for shooting a ball in a pinball machine, a spring (k = 675 N/m) is compressed by 0.0690 m relative to its unstrained length. The ball (m = 0.0540 kg) is at rest against the spring at point A. When the spring is released, the ball slides (without rolling) to point B, which is 0.300 m higher than point A. How fast is the ball moving at B?


I used this equation: v final=square root of k/m(x initial^2 - x final^2)
but I keep getting the wrong answer. Am I using the wrong equation? Can someone help me?

 

[Doc Al]

Instead of using that equation, think in terms of conservation of energy: Initially, the ball has only spring potential energy ([itex]1/2 k x^2[/itex]), which gets transformed into kinetic energy plus gravitational potential energy as it moves to position B. Set up that equation and solve for the speed.

 

[thepatientmental]

It seems like you are using the correct equation, but there may be an error in your calculations. Let's break down the problem and go through it step by step to see if we can identify the issue.

First, we need to find the potential energy stored in the spring when it is compressed by 0.0690 m. We can use the equation for potential energy in a spring, U = 1/2*k*x^2, where k is the spring constant and x is the distance the spring is compressed. Plugging in the values given, we get U = 1/2 * 675 N/m * (0.0690 m)^2 = 16.35 J.

Next, we can use the conservation of energy principle to find the kinetic energy of the ball at point B. At point A, the ball has zero kinetic energy since it is at rest. At point B, all of the potential energy stored in the spring is converted into kinetic energy. So, we can set the potential energy at point A equal to the kinetic energy at point B. This gives us the equation 16.35 J = 1/2 * m * v^2, where v is the velocity of the ball at point B. Plugging in the given mass of the ball, we get 16.35 J = 1/2 * 0.0540 kg * v^2. Solving for v, we get v = 7.26 m/s.

So, the ball will be moving at a speed of 7.26 m/s at point B. Make sure to double check your calculations and units to ensure you get the correct answer. It may also be helpful to draw a diagram and label all the given values to keep track of them. Keep practicing and you will become more confident in solving problems like this!