In a pinball game, a compressed spring with spring constant 1.2x10^2 N/m fires an 82 g pinball. The pinball first travels horizontally and then travels up
an inclined plane in the machine before coming to rest. The ball rises up the ramp through a vertical height of 3.4 cm. Determine the distance of the
spring’s compression.
I know that Ee = Eg, but I don't know where to go from there.
the energy stored in the spring equals the gravitational potential of the ball
1/2 k x^2 = m g h
... 1/2 * 1.2E2 * x^2 = .082 * 9.81 * .034 ... x is in meters
To solve this problem, you can start by finding the potential energy stored in the compressed spring. Then, you can equate that to the gravitational potential energy gained by the pinball as it travels up the inclined plane.
1. Find the potential energy stored in the spring:
The potential energy (Ee) stored in the compressed spring can be calculated using the equation:
Ee = (1/2)kx^2
where Ee is the potential energy, k is the spring constant, and x is the distance of the spring's compression.
Given: Spring constant (k) = 1.2×10^2 N/m
To find x, the distance of compression, we can use the fact that the potential energy stored in the spring equals the gravitational potential energy gained by the pinball.
2. Set up the equation:
Ee = Eg
where Ee is the potential energy stored in the spring and Eg is the gravitational potential energy gained by the pinball.
3. Calculate the gravitational potential energy (Eg):
The gravitational potential energy gained by the pinball can be calculated using the equation:
Eg = mgh
where m is the mass of the pinball, g is the acceleration due to gravity, and h is the vertical height through which the pinball rises.
Given: mass of the pinball (m) = 82 g = 0.082 kg
acceleration due to gravity (g) = 9.81 m/s^2
vertical height (h) = 3.4 cm = 0.034 m
Substituting the given values into the equation, we get:
Eg = (0.082 kg)(9.81 m/s^2)(0.034 m)
4. Now, equate the potential energy stored in the spring (Ee) to the gravitational potential energy gained by the pinball (Eg):
(1/2)kx^2 = (0.082 kg)(9.81 m/s^2)(0.034 m)
5. Solve for x, the distance of the spring's compression:
(1/2)(1.2×10^2 N/m)x^2 = (0.082 kg)(9.81 m/s^2)(0.034 m)
x^2 = (0.082 kg)(9.81 m/s^2)(0.034 m) / (1/2)(1.2×10^2 N/m)
x^2 = 0.027811
x ≈ 0.16676 m
Therefore, the distance of the spring's compression is approximately 0.167 meters.
To solve this problem, you can use the Conservation of Energy principle. According to the principle, the initial potential energy stored in the spring (Ee) is converted into the final potential energy of the pinball (Eg) when it reaches the maximum height on the inclined plane.
First, let's calculate the initial potential energy (Ee) stored in the compressed spring:
Ee = (1/2)kx^2
where k is the spring constant (1.2x10^2 N/m) and x is the distance of spring compression (which we need to find).
Now, let's calculate the final potential energy of the pinball (Eg) when it reaches the maximum height:
Eg = mgh
where m is the mass of the pinball (82 g or 0.082 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the vertical height (3.4 cm or 0.034 m).
According to the Conservation of Energy principle, Ee = Eg:
(1/2)kx^2 = mgh
Now, we can substitute the given values into the equation and solve for x, the distance of spring compression:
(1/2)(1.2x10^2 N/m)x^2 = (0.082 kg)(9.8 m/s^2)(0.034 m)
Simplifying the equation:
60x^2 = 0.026748
Dividing both sides by 60:
x^2 = 0.0004458
Taking the square root of both sides:
x = 0.0211 m or 2.11 cm
Therefore, the distance of the spring's compression is approximately 2.11 cm.