General Rule for Spring Compression?

[FrenchAtticus]

Though this is part of my homework I posted it here because it's a general rule that applies to all physics and doesn't contain specific values from my homework.

A block of mass m slides on a horizontal frictionless table with an initial speed. It then compresses a spring of force constant k and is brought to rest. How much is the spring compressed x from it's natural length.

*also* Is k the same K as in Force of spring = -Kx

 

[tanker]

(Yes, the k in Hooke's law is lower case).

For this problem you must realize that as the block hits the spring, it moves (covers distance) against the force kx. As a result, the spring does work on the block. Now, how much work it takes to slow the block down as a function of compressed spring distance is essentially what they're asking in this question.

I can't tell you the formula, as that would be doing your homework for you. But...think about how much work the spring does for a small distance assuming the force is constant over that distance - and use that to figure out how much work is done for any compression length x.

 

[astrokreng]

The general rule for spring compression is that the amount of compression, x, is directly proportional to the force applied to the spring and inversely proportional to the spring's stiffness, represented by the force constant k. This can be expressed mathematically as x = F/k, where F is the applied force and x is the resulting compression.

In this scenario, the block's initial speed is converted into kinetic energy, which is then transferred to the spring as potential energy when the block compresses it. This potential energy is equal to the work done on the spring, which can be calculated using the equation W = (1/2)kx^2. Setting this equal to the initial kinetic energy of the block, we can solve for x to determine the amount of compression.

To answer your second question, yes, the k in the equation for the force of a spring, F = -kx, is the same k as in the force constant of the spring. This equation represents Hooke's law, which states that the force exerted by a spring is directly proportional to the amount of compression or extension from its natural length. So, the greater the force constant, the stiffer the spring and the more force it takes to compress or extend it.