What is a sphere in spherical coordinates?
Spherical coordinates determine the position of a point in three-dimensional space based on the distance ρ from the origin and two angles θ and ϕ. If one is familiar with polar coordinates, then the angle θ isn’t too difficult to understand as it is essentially the same as the angle θ from polar coordinates.
What is the rotational inertia of the sphere about its axis?
Figure 1: The rotational inertia of a hollow sphere is given by I=23MR2 I = 2 3 M R 2 and the rotational inertia of a solid ball is given by 25MR2 2 5 M R 2 .
How do you draw a sphere in spherical coordinates?
A sphere that has the Cartesian equation x2 + y2 + z2 = c2 has the simple equation r = c in spherical coordinates.
How do you find r in spherical coordinates?
These equations are used to convert from spherical coordinates to cylindrical coordinates.
- r=ρsinφ
- θ=θ
- z=ρcosφ
What is moment of inertia of a sphere?
Moment of inertia of sphere is normally expressed as; I = (⅖ )MR2. Here, R and M are the radius and mass of the sphere respectively.
What is the formula of moment of inertia of a sphere?
The smaller sphere having mass “M/8” and radius “R/2” has a moment of inertia calculated below: Ismall sphere = 2 . M/8 . (R/2)2 /5.
How do you find the spherical coordinate equation?
r = ρ sin φ These equations are used to convert from θ = θ spherical coordinates to cylindrical z = ρ cos φ coordinates. and ρ = r 2 + z 2 These equations are used to convert from θ = θ cylindrical coordinates to spherical φ = arccos ( z r 2 + z 2 ) coordinates.
What is Z in spherical coordinates?
z=ρcosφr=ρsinφ z = ρ cos φ r = ρ sin and these are exactly the formulas that we were looking for. So, given a point in spherical coordinates the cylindrical coordinates of the point will be, r=ρsinφθ=θz=ρcosφ r = ρ sin φ θ = θ z = ρ cos
What is moment of inertia of a sphere about its diameter?
Moment of a inertia of a sphere about its diameter is 2/5 MR2.
What is the moment of inertia of a circle?
Moment of inertia of a circle or the second-moment area of a circle is usually determined using the following expression; I = π R4 / 4. Here, R is the radius and the axis is passing through the centre. This equation is equivalent to I = π D4 / 64 when we express it taking the diameter (D) of the circle.
What is the moment of inertia of a spherical shell?
The moment of inertia of spherical shell about its centroidal axis is 32MR2. Thus using parallel axis theorem we get the moment of inertia about a tangent axis is 32MR2+MR2=35MR2.
What is dA for a sphere?
where dA is an area element taken on the surface of a sphere of radius, r, centered at the origin. We have just shown that the solid angle associated with a sphere is 4π steradians (just as the circle is associated with 2π radians).
Why is the equation of a sphere?
Answer: The equation of a sphere in standard form is x2 + y2 + z2 = r2.
How do you determine the moment of inertia?
Calculation of Moment of Inertia. Consider a uniform rod of mass M and length L and the moment of inertia should be calculated about the bisector AB.
How do I find each moment of inertia?
What is the moment of inertia of the rod with respect to the axis?
What does the “moment” in moment of inertia mean?
a. The moment in moment of inertia means it is the rotational analogy of inertia. If you prefix any linear motion physical variable by moment, it means that you are talking about the rotational equivalent of that physical variable. Example – As force is for linear motion, Moment of force (or Torque) is for angular motion.
How do we feel the moment of inertia?
Well, if the beam is not very deep, it will sag a lot. If the beam is deep, it won’t sag much. Moment of Inertia is the property of the beam that tells you how much the beam is going to sag by. By the way, the sagging we’re talking about is called “deflection”, or in other words, strain.