A 2 kg block has an initial velocity of 16 m/s at the bottom of the ramp (see
diagram below). The coefficient of kinetic friction between the block and the
ramp is 0.1. The spring has length 3 m and has a spring constant of 100
N/m. What maximum compression will the spring experience when the block
strikes it?
To find the maximum compression of the spring, we need to calculate the work performed by the block due to kinetic friction (which will be transferred into potential energy stored in the spring), and then use this value to calculate the maximum compression using Hooke's Law.
First, let's find the work performed by the block against friction. The work done against friction can be calculated using the formula:
Work = Force × Distance × cos(θ)
where:
- Force is the force of kinetic friction, which can be calculated using the formula: Force = coefficient of kinetic friction × Normal force
- Distance is the distance traveled by the block along the ramp
- θ is the angle of the ramp, which is equal to the angle whose cosine is equal to the ratio of the height of the ramp to the length of the ramp.
The normal force acting on the block is equal in magnitude and opposite in direction to the force of gravity, which can be calculated as follows:
Normal force = mass × gravitational acceleration
Next, we can calculate the distance traveled by the block along the ramp. This distance can be calculated using the formula:
Distance = length of the ramp × sin(θ)
Once we have the work done against friction, we can equate it to the potential energy stored in the spring, which is given by Hooke's Law:
Potential energy = (1/2) × spring constant × (maximum compression)^2
To find the maximum compression, rearrange the formula to solve for (maximum compression)^2:
(maximum compression)^2 = (2 × Work) / spring constant
Finally, take the square root of both sides to get the value of maximum compression.