A 4.30 kg block starts from rest and slides down a frictionless incline, dropping a vertical distance of 2.60 m, before compressing a spring of force constant 2.20 104 N/m. Find the maximum compression of the spring.
To find the maximum compression of the spring, we can use the conservation of mechanical energy principle.
The initial energy of the block can be calculated using its gravitational potential energy formula:
E_initial = m * g * h
Where:
m = mass of the block (4.30 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = vertical distance the block slides down (2.60 m)
E_initial = (4.30 kg) * (9.8 m/s^2) * (2.60 m)
E_initial ≈ 106.004 Joules
At the highest point of compression, all the initial potential energy is converted into elastic potential energy of the spring.
According to the conservation of mechanical energy, the elastic potential energy of the spring can be calculated using the formula:
E_spring = 1/2 * k * x^2
Where:
k = force constant of the spring (2.20 * 10^4 N/m)
x = maximum compression of the spring (to be determined)
Setting the initial energy equal to the spring energy:
E_initial = E_spring
106.004 J = 1/2 * (2.20 * 10^4 N/m) * x^2
To solve for x, we rearrange the equation:
x = √(2 * (106.004 J) / (2.20 * 10^4 N/m))
x ≈ 0.139 m (rounded to three decimal places)
Therefore, the maximum compression of the spring is approximately 0.139 meters.