A 2kg mass is released from rest and slides down an inclined plane of 60 degrees a distance of 1 eter. It strikes a 250 N/m spring. What is the maximum compression of a spring?
To find the maximum compression of the spring, we need to apply the principle of conservation of mechanical energy. Here's how you can do it step by step:
1. Start by calculating the potential energy at the top of the inclined plane. The potential energy (PE) of an object at height h is given by the formula PE = m * g * h, where m is the mass and g is the acceleration due to gravity (9.8 m/s^2). In this case, the mass (m) is 2 kg and the height (h) is the vertical distance the mass has moved down the inclined plane. Since the inclined plane makes an angle of 60 degrees with the horizontal, the vertical distance can be found using the formula h = d * sin(theta), where d is the distance (1 meter) and theta is the angle in radians (60 degrees = pi/3 radians).
h = d * sin(theta)
h = 1 * sin(pi/3)
h = sqrt(3)/2 meters
PE = m * g * h
PE = 2 * 9.8 * sqrt(3)/2
PE = 9.8 * sqrt(3) Joules
2. Next, we calculate the maximum potential energy stored in the spring. The maximum potential energy (PE') is given by the formula PE' = (1/2) * k * x^2, where k is the spring constant and x is the maximum compression of the spring.
PE' = (1/2) * k * x^2
PE' = (1/2) * 250 * x^2
PE' = 125 * x^2 Joules
3. Since mechanical energy is conserved, we can equate the potential energy at the top of the inclined plane (PE) to the maximum potential energy stored in the spring (PE') and solve for x:
PE = PE'
9.8 * sqrt(3) = 125 * x^2
Solve this equation to find the value of x, which represents the compression of the spring.
x ≈ 0.337 meters
So, the maximum compression of the spring is approximately 0.337 meters.