Question

Assume the Earth to be a sphere of uniform density. (a) Calculate its rotational kinetic energy. (b) Suppose that this energy could be harnessed for our use. For how long could the Earth supply $1.00\mathrm{kW}$ of power to each of the $6.17\times {10}^9$ persons on the Earth?

Short answer

The rotational kinetic energy of the Earth is 2.56x10^29 J. This energy could be harnessed to supply 1.00 kW of power to each of the 6.17 billion people on Earth for approximately 131 days.

Step-by-step solution

Calculate the Earth's rotational kinetic energy

First, we need to calculate the rotational kinetic energy of the Earth. The rotational kinetic energy ($K$) is given as $$K=0.5\times I\times {\omega }^2$$, where $I$ is the moment of inertia and $\omega$ is the angular speed. The moment of inertia for a sphere of uniform density is $$I=0.4\times M\times R^2$$, and the angular speed of the Earth is $$\omega =\frac{2\pi }{24\times 60\times 60}\text{ rad/s}$$, as it makes one rotation in one day, which is converted here to seconds. The mass of the Earth is $M=5.97\times {10}^{24}$ kg, and its radius is $R=6.38\times {10}^6$ m. Substitute these values in the formula to get $K$.

Calculate the total power

Next, we need to calculate the total power that needs to be supplied for every person on Earth. This will be $P=1.00\mathrm{kW}\times 6.17\times {10}^9$ (as there are 6.17 billion people). Convert the power from kilowatt to watt (since 1 kW = 1000 W).

Find the time

Finally, we need to solve for the time $t$ for which the rotational kinetic energy of the Earth can provide power to each person. We know that power is the rate at which energy is used, so $P=\frac{E}{t}$. Rearranging, we find $t=\frac{E}{P}$, where $E$ is the total energy (i.e., the rotational kinetic energy of the Earth) and $P$ is the total power. Substitute $E$ and $P$ into the equation to get $t$.

Key concepts

Moment of Inertia

The concept of moment of inertia is essential when understanding rotational motion, as it represents the distribution of mass in an object with respect to its axis of rotation. For a sphere with uniform density, the moment of inertia can be calculated using the formula:

• $I=0.4\times M\times R^2$

Where:

• $M$ is the mass of the sphere
• $R$ is the radius of the sphere

The factor 0.4 is specific to solid spheres, which have their mass evenly distributed throughout their volume. By knowing the moment of inertia, one can predict how much torque is needed to bring a rotating sphere, like Earth, to a certain rotational speed or to stop its rotation.
Unlike linear motion, where mass alone determines the inertia, rotational inertia also depends on how far the mass is from the rotation axis. This concept is crucial when calculating the rotational kinetic energy of any spherical object, particularly when assuming uniform density to simplify the calculations.

Angular Speed

Angular speed measures how fast something rotates around an axis. It is defined as the angle an object rotates through per unit of time. For Earth's rotation, the angular speed $\omega$ is calculated using:

• $\omega =\frac{2\pi }{24\times 60\times 60}$

Here, the division of $2\pi$ by the total seconds in a day (24 hours) translates a full rotation (360 degrees) into radians.This formula helps express Earth's rotation as $\omega$ in radians per second:

• Complete rotation in one day: 24 hours
• 360 degrees equivalent in radians: $2\pi$

Understanding angular speed is critical when calculating rotational kinetic energy, which uses $\omega$ to determine how much energy is stored in Earth's rotation. Angular speed varies widely in everyday objects, from the sweeping hands of a clock to high-speed rotating turbines.

Power Supply

Power refers to the rate at which energy is transferred or converted. In the context of Earth's rotational kinetic energy, we explore providing power to the planet's population. The problem asks us to consider how Earth's stored energy might supply power to 6.17 billion people, each needing 1.00 kW:

• Total power requirement: $P=1.00\text{kW}\times 6.17\times {10}^9$
• 1 kW equals 1000 watts

This simple calculation outlines the immense energy demand compared to the rotational energy the earth possesses.
Using the equation $P=\frac{E}{t}$, where $E$ represents Earth's total rotational kinetic energy and $t$ the time this energy could sustain the power demand, highlights the significant gap between global energy needs and the energy that Earth's rotation could potentially provide. Power supply is defined in terms of watts (W), connecting energy's relationship over time as an ongoing natural resource or mechanical output.

Uniform Density Sphere

A uniform density sphere means that the material making up the sphere is evenly distributed throughout its volume. This assumption simplifies calculations and is especially useful in physics when considering celestial bodies like planets. The notion of uniform density means that every part of the sphere, from the core to the surface, has the same mass concentration.Using this simplification, we calculate the moment of inertia with greater ease:

• $I=0.4\times M\times R^2$

The uniform distribution ensures that every unit volume of the sphere's material contributes equally to its inertia. This assumption helps in modeling problems involving Earth's rotation. However, it's an idealization, as real celestial bodies can have non-uniform density due to internal structures or varying materials.
Such simplification aids in estimating large-scale properties like Earth's rotational kinetic energy, necessary for hypothetical scenarios where this energy is harnessed.