enter an expression for gravitational potential energy in terms of m, M, G, and R
Added by Diego M.
Step 1
Let's think step by step. Show more…
Show all steps
Your feedback will help us improve your experience
Khoobchandra Agrawal and 57 other Physics 101 Mechanics educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
The expression Ep = mgh applies only close to the surface of the Earth. The general expression for the potential energy of a mass m at a distance R from the center of the Earth (of mass me) is Ep = -Gmem/R. Write R = Re + h where Re is the radius of the Earth; show that, when h << Re, this general expression reduces to the special case, and find an expression for g. You will need the expansion (1+x)^-1 = 1 - x + ...
Khoobchandra A.
Gravitational potential energy at the surface of Earth, $$ \begin{aligned} E_{1} &=-\frac{\mathrm{GMm}}{R} \\ \text { G.P.E at } 3 R, E_{2} &=-\frac{\mathrm{GMm}}{(R+3 R)} \\ &=-\frac{\mathrm{GMm}}{4 R} \end{aligned} $$ Therefore, change in gravitational potential energy $$ \begin{aligned} &=E_{2}-E_{1} \\ &=-\frac{G M m}{4 R}+\frac{\mathrm{G} M m}{R} \\ &=\frac{3 \mathrm{GMm}}{4 R} \\ &=\frac{3}{4} m g R \end{aligned} $$
GRAVITATION
Gravitation
A shaft is drilled from the surface to the center of the earth (see Fig. 13.25). As in Example 13.10 (Section 13.6), make the unrealistic assumption that the density of the earth is uniform. With this approximation, the gravitational force on an object with mass $m$, that is inside the earth at a distance $r$ from the center, has magnitude $F_g = GmE mr/R_E{^3}$ (as shown in Example 13.10) and points toward the center of the earth. (a) Derive an expression for the gravitational potential energy $U(r)$ of the object$-$earth system as a function of the object's distance from the center of the earth. Take the potential energy to be zero when the object is at the center of the earth. (b) If an object is released in the shaft at the earth's surface, what speed will it have when it reaches the center of the earth?
Black Holes
Recommended Textbooks
University Physics with Modern Physics
Physics: Principles with Applications
Fundamentals of Physics
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD