Janet wants to find the spring constant of a given spring,...

Janet wants to find the spring constant of a given spring, so she hangs the spring vertically and attaches a $0.40 \mathrm{kg}$ mass to the spring's other end. If the spring stretches $3.0 \mathrm{cm}$ from its equilibrium position, what is the spring constant?

Key Concepts

Gravitational Force

Gravitational force is the weight experienced by an object due to the gravitational pull of the Earth. It is calculated as the product of the mass of the object and the acceleration due to gravity. In a mass-spring system, this force is counteracted by the spring’s restoring force when the system is in equilibrium.

Static Equilibrium

In the context of a hanging spring, static equilibrium refers to the state where all forces acting on the system balance out so that there is no net acceleration. For a mass attached to a spring, this means the upward spring force exactly balances the downward gravitational force, allowing for the determination of the spring constant.

Spring Constant

The spring constant is a measure of a spring's stiffness. It quantifies how much force is needed to stretch or compress the spring by a certain distance. A larger spring constant implies a stiffer spring which requires more force to achieve the same displacement as a spring with a lower spring constant.

Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to the displacement from its equilibrium position. The relationship is often written as F = -kx, where k is the spring constant and x is the displacement. This principle is fundamental in understanding the behavior of elastic materials under applied forces.

Transcript

00:01 So janet wants to find out the spring constant of a spring.

00:07 And to do this, she hangs a mass that's known, a mass that's given to us vertically, and measures the distance by which this mass drops from the spring.

00:22 So what's given to us here is the mass of this block, which is 0 .4 kilograms.

00:39 And we're given the distance by which it drops.

00:44 So we're going to write this distance here, and that distance is 3 cm or 0 .03 meters.

01:11 So in order to solve this problem, we need to see what's going on with our mass and our spring system.

01:21 It's hanging vertically, and the best thing we can start with is to draw a force diagram on this vertical system.

01:46 Okay, so our force diagram should include the following here.

01:55 We have m .g, the force of gravity, pulling the mass down.

02:02 We also have the restoring force, or in this case, i would say the spring force, acting opposite to the force of gravity, and that force is minus kx.

02:22 Now this box, or this mass spring system, is not oscillating.

02:28 It's not in simple harmonic motion, nor is the system accelerating.

02:34 So we know that a here, our acceleration is zero.

02:43 Since a is zero, our f net, our net force on this object has to be zero, in accordance with newton's second law of motion.

02:54 So, in our second step, we're going to invoke the second law of motion, and we're going to put together what we know and solve for the spring constant...

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