The gravitational attraction F on a body a distance r from...

The gravitational attraction $F$ on a body a distance $r$ from the center of Earth, where $r$ is greater than the radius of Earth, is a function of its mass $m$ and the distance $r$ as follows: $$F=\frac{m g R^{2}}{r^{2}}$$ where $R$ is the radius of Earth and $g$ is the force of gravity $-$ about 32 feet per second per second (ft per $\sec ^{2} ) .$ a. Find and interpret $F_{m}$ and $F_{r}$ . b. Show that $F_{m}>0$ and $F_{r}<0 .$ Why is this reasonable?

The gravitational attraction $F$ on a body a distance $r$ from the center of Earth, where $r$ is greater
than the radius of Earth, is a function of its mass $m$ and the distance $r$ as follows:
$$F=\frac{m g R^{2}}{r^{2}}$$
where $R$ is the radius of Earth and $g$ is the force of gravity $-$ about 32 feet per second per second (ft per $\sec ^{2} ) .$
a. Find and interpret $F_{m}$ and $F_{r}$ .
b. Show that $F_{m}>0$ and $F_{r}<0 .$ Why is this reasonable?

Key Concepts

Inverse-Square Law

This concept explains that the magnitude of a force, such as gravity, decreases in proportion to the square of the distance from the source. It underlies the mathematical relationship where the force diminishes rapidly as distance increases, and is a common principle in fields like gravity and electrostatics.

Partial Derivatives

Partial derivatives represent the rate at which a function changes as one of its variables is varied while all other variables are held constant. They are essential in understanding how changes in each independent variable individually affect the overall function.

Interpreting the Signs of Partial Derivatives

The sign of a partial derivative indicates the direction of change of the function with respect to that variable. A positive partial derivative means the function increases as the variable increases, while a negative derivative signifies that the function decreases as the variable increases. In the context of gravitational force, a positive derivative with respect to mass implies that as mass increases, the force increases, whereas a negative derivative with respect to distance indicates that as the distance increases, the gravitational force decreases.

Transcript

00:01 Okay, so this problem is about a gravitational force as a function of m and r, where m is mass and r is the distance from the center.

00:09 We're talking about the earth, and here's a formula for the function of gravitational force on earth, where r is, where big r is the radius of the earth.

00:23 G is a gravitational constant, which is 32 p .3 p.

00:28 Per second squared.

00:30 M and r values are the mass and radius of a body.

00:40 So this problem has two parts, a and b.

00:43 Part a wants you to find the partial derivatives with respect to m and with respect to r.

00:49 So if we took the partial derivative with respect to m, if we do the partial derivative with respect to m, this is essentially just taken the derivative of m, which is one, and everything else is constant.

01:01 So our equation becomes g times big r squared over small r squared.

01:09 All right, so this is formula.

01:12 Now they also want us to interpret what this is, what means.

01:16 So we're going to go ahead and type it in this text box.

01:20 And this is the rate of change, rate of change, of in the force, in the force per unit, the force per unit, the force per unit change in mass while keeping the distance from the center.

01:57 All right, so this is our interpretation for the partial derivative with respect to m.

02:03 Now they also want us to take the partial derivative with respect to r.

02:07 If we do that, well, there is an r squared in the denominator, so using the power rule, we'd have to bring down a negative 2.

02:15 So in the numerator, we would have negative 2mg, big r squared, divided by r cubed, since we have to subtract the exponent by 1.

02:28 Because of power rule, now to interpret this, we're going to say this is the rate of change in the force.

02:41 Per unit, per unit change in distance while keeping the mass of the body constant.

02:59 So this is our interpretation for the partial derivative with respect to r...

Please give Ace some feedback