A 8.1 kg stone is at rest on a spring. The spring is compressed 12 cm by the stone. The stone is then pushed down an additional 27 cm and released. To what maximum height (in cm) does the stone rise from that position?
8.1 kg weighs M g = 73.4 N; so the spring constant must be k = 73.4/0.12 = 661.5 N/m
After compressing another X = 0.27 m, the releasable potential energy in the spring is (1/2)kX^2 = 24.1 J
Set 24.1 J = M g H to get the height H that the stone can rise.
H = 24.1 J/(8.1*9.8) = 0.30 m
To find the maximum height the stone rises from the released position, we need to use the conservation of mechanical energy.
The potential energy stored in the spring when it is compressed can be found using the formula:
E_spring = 0.5kx^2,
where k is the spring constant and x is the displacement of the spring from its equilibrium position.
Given that the stone compresses the spring by 12 cm and the stone is at rest, the initial potential energy stored in the spring is:
E_spring_initial = 0.5k(0.12)^2.
After the stone is pushed down an additional 27 cm, it gains potential energy equal to the work done on it:
E_gravitational = mgh,
where m is the mass of the stone, g is the acceleration due to gravity, and h is the height the stone rises.
To find the height the stone will rise to, we need to equate the initial potential energy stored in the spring to the maximum potential energy the stone will have. That is:
E_spring_initial = E_gravitational.
0.5k(0.12)^2 = mgh.
Substituting the known values:
0.5k(0.12)^2 = (8.1 kg)(9.81 m/s^2)h.
Simplifying the equation:
k(0.12)^2 = (8.1 kg)(9.81 m/s^2)h.
To find the spring constant, we need to use Hooke's Law, which states:
F = kx,
where F is the force exerted by the spring, x is the displacement of the spring, and k is the spring constant.
Given that the displacement is 12 cm and the force is the weight of the stone:
m * g = k * (12 cm).
Substituting the known values:
(8.1 kg)(9.81 m/s^2) = k * (0.12 m).
Solving for k:
k = [(8.1 kg)(9.81 m/s^2)] / (0.12 m).
Once we find k, we can substitute it back into the equation:
k(0.12)^2 = (8.1 kg)(9.81 m/s^2)h.
Substituting the known values and calculating:
[(8.1 kg)(9.81 m/s^2)] / (0.12 m) * (0.12)^2 = (8.1 kg)(9.81 m/s^2)h.
Simplifying:
h = [(8.1 kg)(9.81 m/s^2) * (0.12)^2] / [(8.1 kg)(9.81 m/s^2)].
Calculating:
h ≈ (0.12)^2 ≈ 0.0144 m = 1.44 cm.
Therefore, the stone will rise to a maximum height of approximately 1.44 cm from its released position.