In a new scenario, the block only makes it (exactly) halfway through the rough spot. How far was the spring compressed from its unstretched length?
1.123
To determine how far the spring was compressed from its unstretched length, we need to gather some information and make a few calculations.
Let's assume the following variables:
- L: Unstretched length of the spring
- d: Distance the block travels before the rough spot
- x: Compression distance of the spring
Since the block makes it exactly halfway through the rough spot, we can conclude that the distance traveled after passing the rough spot is also equal to d.
Therefore, the total distance traveled by the block is:
Total distance = d + d = 2d
Now, we know that the block exerts a force on the spring and compresses it. The spring force is given by Hooke's Law:
Spring force = k * x
In this case, since the spring is compressed, the spring force acts against the motion and slows down the block. The frictional force on the block is the equal and opposite force to the spring force, so we can say:
Frictional force = -k * x
The work done by the frictional force is given by the negative integral of the frictional force over the distance x. Since the frictional force is constant, we can simplify this to:
Work done = -Frictional force * x = -(-k * x) * x = k * x^2
The work done by the frictional force is equal to the change in kinetic energy (KE) of the block. Let's assume the block initially has no kinetic energy, so the change in KE is equal to the final KE, given by:
Change in KE = 1/2 * m * v^2
In this case, the block makes it halfway through the rough spot, so its final velocity v is zero.
Change in KE = 1/2 * m * v^2 = 1/2 * m * 0^2 = 0
Since the work done by the frictional force is zero, we can equate it to the change in KE:
k * x^2 = 0
Since the spring force depends on the compression distance x, we can conclude that x = 0.
Therefore, if the block only makes it halfway through the rough spot, the spring is not compressed, and its compression distance from its unstretched length is zero.