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 need to use the conservation of mechanical energy.
1. Calculate the gravitational potential energy:
The block drops a vertical distance of 2.60 m, so the change in height is Δh = 2.60 m.
The gravitational potential energy of the block is given by: PE_grav = m * g * Δh.
Here, m is the mass of the block (4.30 kg) and g is the acceleration due to gravity (9.8 m/s²).
PE_grav = 4.30 kg * 9.8 m/s² * 2.60 m = 113.604 J.
2. Calculate the spring potential energy:
When the block compresses the spring, the potential energy stored in the spring is given by: PE_spring = (1/2) * k * x².
Here, k is the force constant of the spring (2.20 * 10^4 N/m) and x is the maximum compression of the spring (what we want to find).
3. Apply conservation of mechanical energy:
According to the conservation of mechanical energy, the initial potential energy (PE_grav) is converted into the potential energy stored in the spring (PE_spring).
Thus, we have:
PE_grav = PE_spring
113.604 J = (1/2) * (2.20 * 10^4 N/m) * x².
4. Solve for x:
To find x, rearrange the equation:
x² = (2 * 113.604 J) / (2.20 * 10^4 N/m)
x² = 2589.27 m²/N
x ≈ ±50.89 m.
Since the question asks for the maximum compression, we take the positive value:
The maximum compression of the spring is approximately 50.89 m.