A wooden block with a density of 710 kg / m^3 and a volume of 0.012 m^3 is attached to the top o

A wooden block with a density of 710 kg / m^3 and a volume...

A wooden block with a density of 710 $\mathrm{kg} / \mathrm{m}^{3}$ and a volume of 0.012 $\mathrm{m}^{3}$ is attached to the top of a vertical spring whose force constant is $k=540 \mathrm{N} / \mathrm{m} .$ Find the amount by which the spring is stretched or compressed if it and the wooden block are (a) in air or (b) completely immersed in water. [The density of air may be neglected in part ( $($ a). $]$ .

A wooden block with a density of 710 $\mathrm{kg} / \mathrm{m}^{3}$ and a volume of
0.012 $\mathrm{m}^{3}$ is attached to the top of a vertical spring whose force
constant is $k=540 \mathrm{N} / \mathrm{m} .$ Find the amount by which the spring is stretched or compressed if it and the wooden block are (a) in air
or (b) completely immersed in water. [The density of air may be
neglected in part ( $($ a). $]$ .

Key Concepts

Buoyant Force and Archimedes’ Principle

Archimedes’ Principle explains that a body immersed in a fluid experiences an upward force equal to the weight of the fluid displaced. This buoyant force effectively reduces the net weight of the object when it is submerged, which in turn affects the equilibrium displacement of the spring. This principle is essential in problems where the effects of immersion in a fluid must be considered.

Gravitational Force (Weight)

Gravitational force, or weight, is the force exerted on a mass by gravity, and is given by the product of mass and the acceleration due to gravity. In problems involving springs and attached masses, this force is a key factor in determining the amount the spring stretches or compresses under load.

Hooke's Law

Hooke’s Law states that the force exerted by a spring is proportional to the displacement of the spring from its equilibrium position, expressed by the formula F = -kx, where k is the spring constant and x is the displacement. This relationship is fundamental in determining how much a spring stretches or compresses under various loads.

Static Equilibrium in Spring-Mass Systems

In a static equilibrium scenario, the net force acting on the system is zero; this means that the forces due to gravity (or effective weight) and the restoring force of the spring balance each other. Understanding this balance is critical for calculating the displacement of the spring when a mass is attached to it.

Transcript

00:01 Question 86.

00:03 A wooden block with a density of 710 kilograms per meter cubed and a volume of 0 .0 .12 meters cubed is attached to the top of vertical spring whose force constant is k equal to 540 newtons per meter.

00:21 Find the amount by which the spring is stretched or compressed if it and the wooden block are a in air or be completely immersed in water.

00:31 And of course, in part a, the density of air may be neglected.

00:36 And that's because it's so small in comparison to all.

00:42 So we have a very simple equation here that in part a, the force of the spring, must be equal to the weight in the block for it to be in equilibrium.

01:02 And so minus kx is equal to, and we're just a fine weight to be the density of the block, row b, times the volume of the block.

01:15 Multiplied by g.

01:18 And thus any compression or extension can be taken to be minus row b, v, g over k.

01:34 And we just substitute our values.

01:42 710 for a density multiplied by 0 .0 .122.

01:51 For our volume, i multiply by 9 .8 .1.

01:59 Divide this by 540.

02:04 And we get an answer of minus 0 .1 .5 meters.

02:12 This involves a compression of the spring, but let's just quickly look at a diagram of the situation, where we have a spring here and a block onto it, and of course naturally in air, the spring compresses.

02:30 And so in the second situation, you consider taking the spring and then submerging it in water to some level.

02:46 And we can see that actually, before we even do any calculations, there's going to be an extension.

02:51 And why is this? well, we know with it a density of 710 kilograms per meter cubed, the block is less dense than water, and so it's going to want to float.

03:03 So the buoyancy force is going to pull it up, and so this tells us very much that the spring is going to be stretched...

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