What is Solid sphere: Definition and 100 Discussions

In mathematics, a ball is the volume space bounded by a sphere; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball or hyperball in n dimensions is called an n-ball and is bounded by an (n − 1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment.
In other contexts, such as in Euclidean geometry and informal use, sphere is sometimes used to mean ball.

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  1. mncyapntsi

    E-field of solid sphere with non-uniform charge density

    Hi! I've been trying to attempt this problem over here but the solutions state that the solution is this below? However, from integrating the density and then plugging it into Gauss's law, I get the exact same thing, except a 15 instead of a 5. Could any please help point out if there is an...
  2. samy4408

    I don't understand this problem -- Surface area of a section of a solid sphere

    Sorry i don't understand English very well , if someone want to explain to me this problem?
  3. H

    Comparing the kinetic energies among a solid sphere, a cylinder and a hoop

    I did the question as attached, so I think A and D are correct but the given answer is E. Where am I wrong? Thanks.
  4. G

    Polarization of a solid sphere of nonconducting material

    a) Just using the equations gives us: surface charge density = ## \rho_{\rho s} = kR^2 ## volume charge density = ## \rho_\rho = -4kR ## b) I am not sure here but the Q on the shell is the same as within. If that's the case we can use gauss law to find Q which I guess is the total charge. ##...
  5. Jojo-11

    Why does acceleration increase by less as angle of slope increases?

    My only guess is that this is due to air resistance. Below an example of the predicted graph:
  6. agnimusayoti

    Potential inside a uniformly charged solid sphere

    Well, in this problem, I try to use $$d \tau '= \mu ^2 \sin {\theta} {d\mu} {d\theta} {d\phi}$$ With these domain integration: $$0<\mu<r$$ $$0<\theta<\pi$$ $$0<\phi<2\pi$$ , I get $$V=\frac{1}{4\pi \epsilon_0} \frac{3Qr^2}{2R^3}$$ This result is wrong because doesn't match with Prob 2.21, which...
  7. rxh140630

    Drilling a hole through the center of a solid sphere

    The volume of the sphere = \frac{32{\pi}r^{3}}{3} The answer given at the back of the book is (\frac {32}{3} - 4\sqrt{3}){\pi}r^3 To drill a hole completely through the sphere, the hole would have to have a length of 4r. To get the answer in the back of the book, it requires setting the...
  8. A

    Rotational Mechanics -- A solid sphere is rolled on a rough surface

    I found out the time when rotation ceases to be 4 ##v_0## /5*mew*g, where mew=coefficent of friction of surface but I am unable to plot the graph post that time
  9. Arman777

    Electrostatic Energy of a solid sphere with a cavity

    I tried to use ##W = ε_0/2 \int E^2d\tau## for all space. So I find that ##E = \frac{(R^3 - b^3)\rho}{3ε_0r^2}## where ##\rho## is the charge denisty. So from here when I plug the equation I get something like $$W = \frac{(R^3 - b^3)^2\rho^2 4 \ pi}{18ε_0} \int_{?}^{\inf}1/r^2dr$$ Is this...
  10. L

    Pressures distribution: solid sphere on a flat surface

    In a real case (not ideally rigid bodies), a (e. g.) hard metal sphere is on a flat (e. g.) hard metal surface (a table) and the sphere is "charged" vertically on the table by a vertical force directed downward. In this situation, an engineer told me that the maximum pressure on the table is not...
  11. J

    Solid sphere rolling along a track

    Homework Statement Please see the attached file. Homework Equations Ei = Ef The Attempt at a Solution I don't have an answer key provided, but I'd really like to verify that I'm right (or if I'm wrong, why). I think ti'd be (c) because assuming that due to inertia, B will continue going...
  12. Krushnaraj Pandya

    Ring and solid sphere rolling down an incline - rotation problem

    Homework Statement A solid sphere, hollow sphere, disk and ring are released simultaneously from top of a incline. Friction is sufficient to prevent slipping of hollow sphere- what will reach the bottom first? Homework Equations a in pure rolling down an incline=gsinθ/(1 + I/mR^2) The Attempt...
  13. Math_Maniac

    Moment of Inertia of Solid Sphere - Proof

    So I have been having a bit of trouble trying to derive the moment of inertia of a solid sphere through its center of mass. Here is my working as shown in the attached file. The problem is, I end up getting a solution of I = (3/5)MR^2, whereas, in any textbook, it says that the inertia should...
  14. V

    Acceleration of a uniform solid sphere rolling down incline

    Homework Statement Find the acceleration of a uniform solid sphere (of mass ##m## and radius ##R##) rolling without slipping down an incline at angle ##\alpha## using the Lagrangian method. Homework Equations Euler-Lagrange equation which says, $$\frac{\partial\mathcal{L}}{\partial...
  15. V

    Electric Field of a solid sphere of non-uniform surface density

    A solid sphere has surface charge density, Rho (r) Rho(r) = k 1 ( 0 < r < a) k2 x ( a < r < R) 2) Find the electric field in all region i.e 1) r < a and 2) a < r < R and 3 ) R < The attempted solution and the question with the diagram is attached below Could the answer be verified...
  16. starstruck_

    Calculating voltage within and outside of a solid sphere

    Homework Statement A solid sphere with radius R=12 m has charge Q=3 nC distributed uniformly throughout its volume. (a) Calculate the potential difference between a location at infinity and a location on the sphere’s surface. (b) Calculate the potential difference between a location on the...
  17. starstruck_

    Kinetic energy of solid sphere that is rolling

    Homework Statement A solid sphere of mass m=2.5 kg is rolling at v=5.3 m/s. Calculate the transitional kinetic energy, rotational kinetic energy, and the ratio of the two (Rotational/ Transitional). Homework Equations [/B] Inertia of solid sphere = 2/5 mR^2 (where R is the radius and m is the...
  18. S

    Solid sphere rolling down a house roof.... angular speed

    Homework Statement A solid sphere of radius 16cm and mass 10kg starts from rest and rolls without slipping a distance of 9m down a house roof that is inclined at 43 degrees. What is the angular speed about its center as it leaves the house roof? The height of the outside wall of the house is...
  19. Marcus Nielsen

    Electric potential due to a solid sphere

    Hello Guys! This is my first post so bear with me. I am currently studying the basics of electrostatics using the textbook "Introduction to electrodynamics 3 edt. - David J. Griffiths". My problem comes when i try to solve problem 2.21. Find the potential V inside and outside a uniformly...
  20. Arisylia

    Deriving the moment of inertia of solid sphere

    So i was going through derivations of moments of inertia of objects. For objects like the disk and rod, i was able to assume a relationship between mass and volume and integrate From there like $$ \frac{d_m}{m} = \frac{dl}{l} \\ d_m = \frac{dl*m}{l} \\ \int_{0}^{L}r^2\frac{dl*m}{l} \\...
  21. R

    Calculating the moment of inertia of a solid sphere

    Homework Statement To calculate moment of inertia of a solid sphere of uniform density[/B]Homework Equations $$ I = \int r^2 dm$$ The attempt at a solution I consider an elemental disk of small thickness ##d\theta## ##dm = \frac{M}{4/3 \pi R^3}*\pi R^2\cos^2\theta* Rd\theta##...
  22. Andy SV

    B Gravity of hollow sphere vs. solid sphere of same mass

    Would a hollow sphere measure the same gravitationally as a solid sphere if it was the same mass? Just sharing an interesting question
  23. S

    A small solid sphere of mass M0, of radius R0, and of unifor

    Homework Statement A small solid sphere of mass M0, of radius R0, and of uniform density ρ0 is placed in a large bowl containing water. It floats and the level of the water in the dish is L. Given the information below, determine the possible effects on the water level L, (R-Rises, F-Falls...
  24. M

    Static structural analysis on a solid sphere?

    Could you perform Static structural analysis on this solid sphere? No contacts to be used. I do not have any specific boundary conditions(unable to apply constraints to a complete solid spherical body) All i know is the pressure that needs to be applied. You could consider a smiley ball on...
  25. J

    Derivation of moment of inertia of a solid sphere

    Homework Statement I need to know why my derivation does not work. I am attempting to derive I=2/5 mR^2Homework Equations I have seen people derive it using disks but my question is why do the shells not work? Where in my set up did I go wrong? Thanks The Attempt at a Solution I am attempting...
  26. S

    How Does Replacing a Sphere Affect Water Level in a Bowl?

    A small solid sphere of mass M0, of radius R0, and of uniform density ρ0 is placed in a large bowl containing water. It floats and the level of the water in the dish is L. Given the information below, determine the possible effects on the water level L, (R-Rises, F-Falls, U-Unchanged), when that...
  27. imadrea

    How to devise moment of inertia formula of solid sphere?

    Homework Statement how to divide moment of inertia of solid sphere about its central axis?. Solid sphere has radius R, mass M. Homework Equations I=∫r2dm 2/5 MR^2 The Attempt at a Solution https://photos.google.com/search/_tra_/photo/AF1QipPoXyad0q1Y3yisc0LeeJHGApkIrGbitK6kAk5p i try to...
  28. Y

    Solid sphere and hollow sphere rolled down an inclined plane

    A solid sphere and a hollow one of same mass and radii are rolled down a rough inclined plane. Which of the following is true? (A) solid sphere reaches bottom with greater speed. (B) solid sphere reaches the bottom with greater kinetic energy. ( I know that option A is correct.Thus I felt that...
  29. H

    Moment of inertia of a solid sphere

    Homework Statement Find the moment of inertia of a solid sphere of uniform mass density (like a billiard ball) about an axis through its center Homework Equations I = ∫rρdV The Attempt at a Solution I =ρ ∫r4πr2dr = ρ4π∫r4 Then I integrate this from 0 (the center) to R, so I = (ρ4π)*(R5/5) And...
  30. C

    Moment of Inertia of a solid sphere

    Homework Statement Taylor, Classical Mechanics Problem 10.11 ** a) Use the result of problem 10.4 (derivation of the general integral for a moment of inertia of a continuous mass distribution in spherical coordinates, using point particles) to find the moment of inertia of a uniform solid...
  31. Q

    Help I'm feeling stupid. :( ("A solid sphere rolls down an incline)

    Homework Statement "A solid sphere rolls down an incline plane without slipping. If the center of mass of the sphere has a linear acceleration of 1.21 m/s2, what is the angle of the incline to the horizontal?" Homework Equations a = g * sin(θ) The Attempt at a Solution I got home...
  32. G

    Effects of New Solid Sphere on Water Level

    Homework Statement A small solid sphere of mass M0, of radius R0, and of uniform density ρ0 is placed in a large bowl containing water. It floats and the level of the water in the dish is L. Given the information below, determine the possible effects on the water level L, (R-Rises, F-Falls...
  33. C

    Potential for a system of a solid sphere and spherical shell

    Homework Statement A metal sphere with radius a is supported on an insulating stand at the center of a hollow, metal spherical shell with radius b. There is charge +Q on the inner sphere and charge -Q on the outer shell. Take the potential V to be zero at infinite separation. Calculate...
  34. KiNGGeexD

    Components of torque in a solid sphere

    A uniform sphere of mass M and radius R has a point on its surface fixed at the origin. Its centre lies along a line in the direction of the position vector r = i + 2k + 3k at length R. Find the components of the torque acting on it due to gravity if the z-direction is upwards and gravity acts...
  35. S

    Taking small element for integration purpose in SOLID sphere?

    Homework Statement The Question originally is to find the m of a solid uniformly charged solid sphere which is rotating uniformly with ω Now Homework Equations Now my question to you is how to take the small element? The Attempt at a Solution i take a small disc with...
  36. M

    Angular and CoM Velocities of a Solid Sphere

    Homework Statement A solid sphere of mass M and radius R is rolling,without slipping, down a curved rail. The sphere is initially at rest at a height of h1. Find the angular velocity ω2 and the center of mass velocity of the sphere vcm at the end of the rail of height h2. You may assume that...
  37. M

    Work, energy stored in solid sphere

    Homework Statement Find the energy stored in a uniformly charged sphere of charge q, radius R Homework Equations The Attempt at a Solution Ein=\frac{qr}{4\pi\epsilon o R^3}, Eout=\frac{q}{4\pi\epsilon o r^2}... W=\int_{0}^ {R}\int_{0}^{2\pi}\int_{0}^{\pi}[\frac{qr}{4\pi\epsilon...
  38. andyrk

    Charged Spherical Shell and Solid Sphere

    A spherical shell and a conducting sphere each of radius R are charged to maximum potential. Which of the two has more charge? My attempt: Since in a conductor, no charge can reside inside the conductor so all charge is on the surface of the conductor just like the spherical shell. Now ...
  39. M

    Effects on water level when a sphere is replaced by a new solid sphere

    A small solid sphere of mass M0, of radius R0, and of uniform density ρ0 is placed in a large bowl containing water. It floats and the level of the water in the dish is L. Given the information below, determine the possible effects on the water level L, (R-Rises, F-Falls, U-Unchanged), when that...
  40. M

    Drag - Terminal Velocity of a solid sphere.

    Homework Statement A solid sphere - 20mm in diameter, σ (specific gravity) = 1.3 dropped in water μ=1*10^-3 and ρ=1000 determine the terminal velocity for the sphere. (hint- guess the value for the drag coefficient then iterate) Homework Equations Fd=(1/2)*Cd*ρ*(U^2)*A...
  41. C

    Proofing Moment of Inertia Solid Sphere

    Homework Statement Can i use "pyramid" method to derive the equation of Moment Inertia solid sphere? The pyramid is such we slice a watermelon. Sorry for my bad english. Regards. Homework Equations 2/5 MR^2 The Attempt at a Solution
  42. S

    Complex vector potential of solid sphere and Heaviside function

    Homework Statement I have done this problem for the case of a spherical shell, however, I am not understanding how to go about this for a solid sphere. Homework Equations \vec{A} = \frac{1}{4 \pi} \int_{\phi' = 0}^{2 \pi} \int_{-1}^1 \int_0^R \rho_o \Theta(R-r) \sum_{l=0}^\infty...
  43. G

    Electric field within a solid sphere

    Homework Statement A nonconducting, solid sphere of radius a is placed at the center of a spherical conducting shell of inner radius b (> a) and outer radius c, as shown in the figure below. A charge +Q is distributed uniformly through the sphere, which thus carries a charge density ρ...
  44. C

    Free electrons in solid sphere of copper

    Homework Statement Use approximations to find the number of free electrons in a 4mm diameter solid sphere of copper. What fraction of its electrons have to be removed to leave a sphere with a charge of +50μC? Note that density of 29_Cu is 8.96 g/cm^3 and molar mass 63.54g/mol Hint: Atomic...
  45. D

    Finding gravitational potential inside solid sphere

    So I am given that the gravitational potential of a mass m a distance r away from the center of a spherical shell with mass m' is -Cm'/r for m outside the shell and constant for m inside ths shell. I am to find the potentials inside and outside a solid sphere (the earth) of radius R as well...
  46. R

    Electric field inside a solid sphere

    Homework Statement We have a uniformly charged solid sphere whose radius is R and whose total charge is q. I'm trying to find the electric field inside a (r<R). The correct answer must be: E=\frac{1}{4 \pi \epsilon_0} \frac{q}{R^3} r \hat{r} How did they get that answer? The Attempt at a...
  47. L

    Finding the potential of a charged, solid sphere using the charge density

    Homework Statement Solid ball of charge with radius R and volume charge density ρ(r) = ρ0r2, centred at the origin. I have already found the electric field for r<R and r>R and also the potential at the origin by using the formula: V = -∫E.dl Now i want to find the potential at the...
  48. S

    Find the moment of inertia of a solid sphere.

    Homework Statement Beginning with Icm = Integral of r^2 dm from r1 to r2, find the moment of inertia of a solid sphere about any tangential axis. Homework Equations Icm = Integral of r^2 dm The Attempt at a Solution I set up the infinitesimally mass of an infinitesimally...
  49. K

    Use brutal force to find E field inside a uniformly charged solid sphere

    Homework Statement Use direct integration to find electric field inside a uniformly charged non-conducting solid sphere. The radius of the sphere is R, observing point is at a way from center of the sphere while a<R. Homework Equations Use Coulomb's law only. No Gauss law is allowed. You may...
  50. D

    Moment of inertia of a uniform solid sphere

    For calculating I of a uniform solid sphere, why can't we use thin spherical shells? When I try to use spherical shells I get (3/5)MR^2. Every single derivation uses thin cylindrical shells and end up with the correct expression((2/5)MR^2) but they never explain why it is correct to use...
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