The volume of water in the Pacific Ocean is about 7.0 ×108 km 3 7.0 ×10^8 km^3. The density of

The volume of water in the Pacific Ocean is about 7.0 ×108...

The volume of water in the Pacific Ocean is about $7.0 \times 108 \mathrm{~km} 3$ $7.0 \times 10^{8} \mathrm{~km}^{3}$. The density of seawater is about $1030 \mathrm{~kg} / \mathrm{m}^{3}$. (a) Determine the gravitational potential energy of the Moon-Pacific Ocean system when the Pacific is facing away from the Moon. (b) Repeat the calculation when Earth has rotated so that the Pacific Ocean faces toward the Moon. (c) Estimate the maximum speed of the water in the Pacific Ocean due to the tidal influence of the Moon. For the sake of the calculations, treat the Pacific Ocean as a pointlike object (obviously a very rough approximation).

Key Concepts

Volume, Density, and Mass Calculations

In many physical problems, determining the mass of a substance from its volume and density is crucial. Volume and density calculations are used to obtain the mass of a fluid or object, which is a necessary parameter when computing gravitational forces and energies in large-scale systems.

Conservation of Energy

This principle states that energy cannot be created or destroyed, only converted from one form to another. In the context of gravitational interactions and tidal phenomena, the change in gravitational potential energy can be equated to the kinetic energy of moving water, thereby allowing an estimate of the maximum speed that the water can reach.

Approximations in Gravitational Calculations

Often in physics, simplified models are used to make complex problems tractable. Treating an extended distribution of mass as a point mass is one such approximation that simplifies the analysis of gravitational interactions. While not exact, these approximations provide a rough estimate of gravitational potential energy and related quantities in systems such as planetary or oceanic interactions.

Gravitational Potential Energy

This is the energy associated with the gravitational interaction between two or more masses. In problems involving celestial bodies, gravitational potential energy helps to quantify how much work is needed to move one body relative to another under the influence of gravity. It is often calculated using formulas that involve the masses, the gravitational constant, and the distance between the masses.

Tidal Forces

Tidal forces arise from the differential gravitational pull exerted by a celestial body (such as a moon) on different parts of another body (such as an ocean). These forces lead to stretching and squeezing, which can result in movements of large volumes of water. Understanding tidal forces allows one to estimate the magnitude of these effects, such as changes in water velocity.

Transcript

00:01 In this problem, we're going to be looking at energy consequences of the moon on the pacific ocean, potential energy, kinetic energy, and so on.

00:12 Before we get started, we're going to need to convert the volume of the pacific ocean from kilometers cube to meters cubed, and that's a very simple calculation, so we have that number now.

00:23 Now, looking at the diagram here, we're going to be looking at it in two positions, one where it's as far away from the moon, the pacific ocean, as it can be on the opposite.

00:32 Its side of the earth, effectively, and also when it's the nearest to the moon.

00:37 Those are two things.

00:38 Our om, the orbit of the moon, and mo is mass of the ocean.

00:47 So let us do part a.

00:49 We want the potential energy in the configuration i've drawn above.

00:54 On is g, mass of the moon, mass of the ocean.

00:58 Now, the distance from the center of the moon to the point particle, which is treating the ocean as a point particle, is rom plus re.

01:15 And we can put in our numbers, 6 .67 times 10 to the minus 11, newton, meter squared, kilogram squared, mass of the moon, 7 .35, times 10 to the 22nd.

01:42 Kilograms.

01:45 Now we need that volume, 7 times 10 to 17 cubic meters.

01:52 And they give us the density of 1030 kilogram cubic meter.

02:07 And then we just have to put in our 3 .84 times 10 to the 8 meters plus 6 .38 times 10 to the 6 .38 times 10 to the 6 meters.

02:27 And this works out to be minus 9 .1 times 10 to 24 joules.

02:37 So that is the potential energy of the moon ocean system.

02:42 Remember what that represents.

02:44 That represents you bringing the ocean from infinity to that position without changes kinetic energy.

02:52 That's what we mean by potential energy.

02:54 It's the work you do to bring it to that position.

03:02 Okay.

03:04 Now the second one is basically the same situation except now the the ocean is on the near side so here's the ocean now here's our e here's r zero m now the distance from the moon to the ocean is going to be r0m minus re.

03:40 So basically it's the same calculation, exactly the same calculation with just that one change.

03:47 Minus g, mass of the moon, mass of the ocean, r0m minus re.

03:59 And you can put in the numbers there and calculate that.

04:02 It's exactly the same thing just with a minus i in the denominator.

04:07 This becomes minus 9 .4 times 10 to 24th jewel.

04:15 So lesser using the minus sign that we are at a lower potential energy, our potential energy.

04:25 So that's like having a ball, certain height, and fall into the earth.

04:31 The height is got more potential energy than near the earth...