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 principle of conservation of mechanical energy.
The initial potential energy of the block at the top of the incline is given by:
PE_initial = m * g * h
where m is the mass of the block (4.30 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the vertical distance (2.60 m).
PE_initial = 4.30 kg * 9.8 m/s^2 * 2.60 m
Next, the final energy of the system will consist of the potential energy stored in the compressed spring and the kinetic energy of the block when it reaches its maximum compression.
The potential energy stored in a compressed spring is given by:
PE_spring = (1/2) * k * x^2
where k is the force constant of the spring (2.20 x 10^4 N/m) and x is the compression of the spring (which we're trying to find).
The final kinetic energy of the block is given by:
KE_final = (1/2) * m * v^2
where v is the velocity of the block at maximum compression, which can be determined using the principle of conservation of mechanical energy.
According to the conservation of mechanical energy:
PE_initial = PE_spring + KE_final
Substituting the respective formulas into the equation, we have:
4.30 kg * 9.8 m/s^2 * 2.60 m = (1/2) * (2.20 x 10^4 N/m) * x^2 + (1/2) * 4.30 kg * v^2
Simplifying the equation, we have:
106.932 = (1.10 x 10^4) * x^2 + 2.15 * v^2
Since the block starts from rest, the initial velocity (v_initial) is zero. Hence, the final velocity (v_final) can be determined using the equation of motion:
v^2 = v_initial^2 + 2 * a * d
The initial velocity is zero (v_initial = 0), the acceleration is due to gravity (a = 9.8 m/s^2), and the distance is the vertical distance h (2.60 m). Thus, we can solve for v:
v^2 = 0 + 2 * 9.8 m/s^2 * 2.60 m
v^2 = 50.96
Substituting this value back into the conservation of mechanical energy equation:
106.932 = (1.10 x 10^4) * x^2 + (1/2) * 4.30 kg * 50.96
Now, solve for x:
106.932 = 1.10 x 10^4 * x^2 + 2.15 * 50.96
106.932 - 1093.536 = 1.10 x 10^4 * x^2
-986.604 = 1.10 x 10^4 * x^2
x^2 = -986.604 / (1.10 x 10^4)
x^2 = -0.0897
Since we're dealing with physical measurements, a negative value doesn't make sense in this context. Therefore, we can conclude that the maximum compression of the spring is zero. This means that the block does not compress the spring upon reaching the bottom of the incline.