If a fish is attached to a vertical spring and slowly...

If a fish is attached to a vertical spring and slowly lowered to its equilibrium position, it is found to stretch the spring by an amount $d$. If the same fish is attached to the end of the unstretched spring and then allowed to fall from rest, through what maximum distance does it stretch the spring? (Hint: Calculate the force constant of the spring in terms of the distance $d$ and the mass $m$ of the fish.)

Key Concepts

Hooke's Law

Hooke's Law is a fundamental principle stating that the force exerted by a spring is directly proportional to the displacement from its equilibrium position, mathematically expressed as F = -kx. This relationship is essential in determining how much a spring stretches or compresses under an applied force.

Equilibrium in Spring-Mass Systems

In a vertical spring-mass system, equilibrium is reached when the upward spring force equals the downward gravitational force acting on the mass. This balance allows for the calculation of the spring constant and the static displacement, which serves as a reference point for analyzing dynamic behavior.

Spring Constant

The spring constant, denoted as k, is a measure of a spring's stiffness, defined by the ratio of the force applied to the displacement produced in the spring. Determining the spring constant through equilibrium conditions is crucial for analyzing both static and dynamic situations in spring systems.

Elastic Potential Energy

Elastic potential energy is the energy stored in a spring when it is deformed, either stretched or compressed, and is given by the equation (1/2)kx². This concept is central when using energy conservation methods to analyze the transformation of energy in a mass-spring system.

Energy Conservation in Mechanical Systems

The principle of energy conservation states that the total energy in an isolated system remains constant, allowing for the conversion between different forms of energy such as gravitational and elastic potential energy. This principle is applied to determine the maximum displacement in dynamic spring problems by equating the initial gravitational potential energy with the stored elastic potential energy at maximum stretch.

Transcript

00:02 Problem 764, we have a fish that's attached to a vertical spring and slowly lowered to the equilibrium position.

00:06 So the spring initially was just this length here, but then a fish is attached to it and it stretches the spring long distance deep.

00:15 So then we then say if this fish is attached to the end of the unsetched being allowed to fall from rest, what maximum distance will stretch? so now that we know that the even position is at d, we basically push it back up to the to this position and we let it fall how far does it go and so we're going to use this in here calculate the force constant of the spring in terms of distance d and the mass m so if we create a free by diagram for the fish right there at its point we'll note that we it has a force to do gravity on it but what's holding it up is that we have a spring force k x and that's how far it's uh where x is distance has been stretched d and so since those sum of forces is equal to zero in that case, writing faster than i speak, we have then that if we define this distance as positive, we have kx minus mg is equal to zero.

01:19 So then you can solve and get k is equal to mg over x.

01:25 But since x is the distance d here, we get mg over d.

01:32 Okay.

01:34 So now, we have to write down our conservation energy formula for that release from rest here and see how far it falls.

01:41 So we have potential energy and it's going to then release and it's all going to come down to some bottom here.

01:54 That's a fish and that's going to have a zero velocity there when it's come in when it's at a stopping point and then we'll start to go back up...

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