Initially sliding with a speed of 2.3 m/s , a 1.8 kg block collides with a spring and compresses it 0.36 m before coming to rest. WHat is the force constant of the spring?
To find the force constant of the spring, we need to use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. The formula for Hooke's Law is:
F = k * x
Where:
F is the force exerted by the spring,
k is the force constant of the spring, and
x is the displacement of the spring.
In this case, the block is coming to rest after compressing the spring, so the final velocity of the block is zero. We can use the work-energy principle to find the force constant.
The work done on the block by the spring is equal to the change in its kinetic energy:
0.5 * m * v^2 = 0.5 * k * x^2
Where:
m is the mass of the block, which is 1.8 kg,
v is the initial velocity of the block, which is 2.3 m/s,
k is the force constant of the spring that we want to find, and
x is the displacement of the spring, which is 0.36 m.
Plugging in the given values, we can solve for k:
0.5 * 1.8 kg * (2.3 m/s)^2 = 0.5 * k * (0.36 m)^2
Calculating the expression on the left side:
0.5 * 1.8 kg * (2.3 m/s)^2 ≈ 4.4414 N
Now we can rearrange the equation to solve for k:
k = (4.4414 N) / (0.5 * (0.36 m)^2)
Calculating the expression on the right side:
k ≈ (4.4414 N) / (0.5 * 0.1296 m^2)
k ≈ 4.4414 N / 0.0648 m^2
Simplifying the expression:
k ≈ 68.424 N/m
Therefore, the force constant of the spring is approximately 68.424 N/m.