Moment of Inertia / Mass Element proof

In summary, the conversation discusses the calculation of mass and moment of inertia for a thin square plate with a given density function. The first part proves that the mass is equal to (1/4)a^4, while the second part involves finding the moment of inertia by integrating r^2*rho over the surface of the plate.
  • #1
raintrek
75
0

Homework Statement



A thin square plate of side a has one corner at the origin and two sides along the positive x and y axes. If the density of the plate is given by p(x,y) = xy show that its mass is M=(1/4)a^4
If the distance of the mass element dM = pdS from the origin is r the moment of inertia of the plate is I= integral [r^2 (pdS)] where S is the surface of the plate. Prove that I=Ma^2

Homework Equations



The Attempt at a Solution



I've proved the first half by showing

M = pdV and taking a small mass element, then integrating

(0->a)int(dx) (0->a)int(dy) xy = (a^2)/2 * (a^2)/2 = (1/4)a^4

However I can't seem to get started with the second part...
 
Physics news on Phys.org
  • #2
Just use the given definition:

[tex]I=\int_0^a \int_0^a r^2\rho dxdy[/tex].

What is r^2 in terms of x and y?
 

What is the concept of moment of inertia?

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It is dependent on an object's mass distribution and the distance of the mass from the axis of rotation.

How is moment of inertia calculated?

Moment of inertia is calculated by multiplying the mass of each element of an object by the square of its distance from the axis of rotation and then adding all these values together. This can be expressed mathematically as I = Σmr², where I is the moment of inertia, m is the mass of the element, and r is the distance from the axis of rotation.

What is the relationship between moment of inertia and rotational motion?

According to Newton's Second Law of Motion, an object will resist changes in its rotational motion unless acted upon by an external torque. The moment of inertia determines how much torque is required to produce a given rotational acceleration.

How is the moment of inertia affected by the shape of an object?

The moment of inertia is greatly affected by the shape of an object. Objects with most of their mass concentrated near the axis of rotation have a lower moment of inertia compared to objects with their mass distributed further away from the axis. This is why a long rod will have a lower moment of inertia than a disk of the same mass.

What is the difference between moment of inertia and mass?

Mass is a measure of the amount of matter in an object, while moment of inertia is a measure of an object's resistance to changes in its rotational motion. Two objects with the same mass can have different moments of inertia depending on their mass distribution and distance from the axis of rotation.

Similar threads

Computing the polar moment of inertia (calculus)
    • Calculus and Beyond Homework Help
Replies
1
Views
939
Three principal axis moments of inertia for hexagonal prism?
    • Calculus and Beyond Homework Help
Replies
3
Views
2K
Finding the Centre of Mass of a Hemisphere
    • Calculus and Beyond Homework Help
Replies
9
Views
960
Moment of inertia of a disc using rods as differential elements
    • Introductory Physics Homework Help
Replies
8
Views
1K
Proof related to the center of mass
    • Calculus and Beyond Homework Help
Replies
1
Views
882
Center of mass of spherical shell inside of cone (Apostol Problem).
    • Calculus and Beyond Homework Help
Replies
3
Views
547
Moment of Inertia of a hollow cone about its base
    • Calculus and Beyond Homework Help
Replies
7
Views
5K
What is the "r" in the moment of inertia?
    • Introductory Physics Homework Help
Replies
13
Views
1K
Solving for 'a' with Torque, Force, and Mass Moment of Inertia
    • Introductory Physics Homework Help
Replies
3
Views
848
I Moment of inertia of human body with limbs at an angle
    • Mechanics
Replies
12
Views
343

Hot Threads

Recent Insights

Back
Top