A 10 kg block slides 4.0 m down a ramp (having a coefficient of kinetic friction of 0.4
everywhere along the ramp) as shown above, then collides with a spring whose spring
constant is 250 N/m. What is the maximum compression of the spring?
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To find the maximum compression of the spring, we need to consider the work done by the block due to friction and the work done by the block on the spring.
1. First, let's calculate the work done by the block due to friction. The work done by friction is given by the equation:
Work = friction force * distance
The friction force can be calculated using the coefficient of kinetic friction (μ) and the normal force (N). The normal force is equal to the weight of the block, which is given by:
N = mass * gravity
Using the given values, we can calculate the normal force:
N = 10 kg * 9.8 m/s^2 = 98 N
Now, we can calculate the friction force:
Friction force = μ * N = 0.4 * 98 N = 39.2 N
The work done by friction is then:
Work = 39.2 N * 4.0 m = 156.8 J
2. Next, let's calculate the work done by the block on the spring. The work done by the block on the spring is given by the equation:
Work = (1/2) * k * x^2
where k is the spring constant and x is the compression of the spring.
Rearranging the equation, we can solve for x:
x = sqrt(2 * Work / k)
Plugging in the values, we get:
x = sqrt(2 * 156.8 J / 250 N/m) = sqrt(1.2512) = 1.118 m
Therefore, the maximum compression of the spring is 1.118 meters.