A spring (k=302 N/m) placed vertically on a surface has a 548 g ball placed on top of it. How far must the spring be compressed in order for the ball to be traveling at a speed of 2.8 m/s when it is located 438 cm above its initial position?
To solve this problem, we can apply the principle of conservation of mechanical energy. We'll consider the initial and final states of the system.
First, let's find the potential energy of the system when the ball is initially at rest on the compressed spring. The potential energy of a spring can be calculated using the formula:
Potential energy (PE) = (1/2)kx^2
where k is the spring constant and x is the displacement from the equilibrium position.
Here, the spring constant (k) is given as 302 N/m. Let's assume the spring is compressed by distance x.
Therefore, the potential energy of the system when the ball is at rest is:
PE_initial = (1/2)kx^2
Next, let's find the potential energy of the system when the ball reaches the final position at a height of 438 cm above the initial position. The potential energy of the ball at this height can be calculated using the formula:
PE_final = mgh
where m is the mass of the ball, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
Here, the mass of the ball (m) is given as 548 g. Converting it to kilograms:
m = 548 g ÷ 1000 = 0.548 kg
Now, let's calculate the potential energy at the final position:
PE_final = mgh = 0.548 kg × 9.8 m/s^2 × 4.38 m = 23.7748 J
According to the principle of conservation of mechanical energy, the total mechanical energy in the initial and final states of the system should be the same. Therefore, we can equate the two potential energies to find x:
PE_initial = PE_final
(1/2)kx^2 = 23.7748 J
Now, we can rearrange the equation and solve for x:
x^2 = (2 × 23.7748 J) ÷ k
x^2 = 47.5496 N m / 302 N/m
x^2 = 0.1574 m^2
Taking the square root of both sides, we get:
x = √0.1574 m^2
x ≈ 0.397 m
Therefore, the spring must be compressed by approximately 0.397 meters in order for the ball to be traveling at a speed of 2.8 m/s when it is located 438 cm above its initial position.