A block of mass, 10 kg, collides with a spring with a spring constant of 200 N/m at a speed of 12 m/s compressing the spring. When the block stops moving, how far is the spring compressed?
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To find the distance by which the spring is compressed when the block stops moving, we can use the principle of conservation of mechanical energy.
Here's the step-by-step process to calculate the compression distance:
Step 1: Find the initial kinetic energy of the moving block.
The initial kinetic energy (KEi) can be calculated using the formula:
KEi = 0.5 * mass * velocity^2
Given:
Mass of the block (m) = 10 kg
Initial velocity (vi) = 12 m/s
Substituting the values into the formula:
KEi = 0.5 * 10 kg * (12 m/s)^2
Compute the value of KEi.
Step 2: Find the potential energy of the compressed spring.
The potential energy (PE) stored in a spring can be calculated using the formula:
PE = 0.5 * k * x^2
Given:
Spring constant (k) = 200 N/m (from the question)
Compression distance (x) = ?
Substituting the values into the formula:
PE = 0.5 * 200 N/m * x^2
Step 3: Equate the initial kinetic energy to the potential energy of the spring.
At the point when the block stops moving, all of its kinetic energy will be converted into potential energy stored in the spring. Therefore, we can equate the two energies:
KEi = PE
Substitute the values of KEi and PE into the equation:
0.5 * 10 kg * (12 m/s)^2 = 0.5 * 200 N/m * x^2
Step 4: Solve for the compression distance (x).
Rearrange the equation to isolate x:
x^2 = (10 kg * (12 m/s)^2) / (200 N/m)
Simplify and compute the value of x by taking the square root of both sides.