A spring has its right end fixed and is installed on a horizontal table so that the free end, at equilibrium, is at x=3.00m. A 1.65kg block coming from the left slides along the table. When it passes the origin, it is moving at 5.58m/s. It strikes the spring, compresses it momentarily, and is then sent back toward the left, where it eventually comes to rest at the point x=1.5m. The coefficient of kinetic friction between the block and the table is 0.300. By what distance was the spring compressed?
The kicker is the energy lost to friction during compression and expansion which will be Ff(x) times 2.
So KE - 2Ff x - Ff(d) = 0
but Ff is mu mg
Note m is not really needed for the question.
.395
To find the distance by which the spring was compressed, we need to analyze the motion of the block when it comes into contact with the spring.
Step 1: Calculate the initial velocity of the block before it hits the spring:
The block is initially moving to the right with a velocity of 5.58 m/s.
Step 2: Calculate the deceleration of the block due to the friction force:
Friction force (f) can be determined using the equation: f = μ * m * g, where μ is the coefficient of kinetic friction (0.300), m is the mass of the block (1.65 kg), and g is the acceleration due to gravity (9.8 m/s^2).
Substituting the given values, we find f = 0.300 * 1.65 kg * 9.8 m/s^2 = 4.851 N.
The deceleration (a) of the block can be calculated using the equation: f = m * a, where m is the mass of the block.
Substituting the values, we find 4.851 N = 1.65 kg * a.
Rearranging the equation, we get a = 4.851 N / 1.65 kg ≈ 2.942 m/s^2.
Step 3: Calculate the time taken by the block to come to rest after hitting the spring:
Assuming the block travels a distance x when it comes to rest, we can use the equation of motion: v^2 = u^2 + 2 * a * x, where v is the final velocity (0 m/s), u is the initial velocity (5.58 m/s), a is the deceleration (-2.942 m/s^2), and x is the distance traveled.
Substituting the values, we have (0 m/s)^2 = (5.58 m/s)^2 + 2 * (-2.942 m/s^2) * x.
Simplifying the equation, we get 0 m^2/s^2 = 31.2164 m^2/s^2 - 5.884 m^2/s^2 * x.
Rearranging the equation, we find x = 31.2164 m^2/s^2 / (5.884 m^2/s^2) ≈ 5.307 m.
Therefore, the block travels a distance of 5.307 meters before coming to rest.
Step 4: Calculate the distance the spring is compressed:
Since the block initially comes to rest at x = 3.00 m and moves back until it stops at x = 1.5 m, the spring is compressed by the distance between these two points.
Distance compressed = 3.00 m - 1.5 m = 1.50 m.
Hence, the spring is compressed by 1.50 meters.