Solving Moment of Inertia Homework: Deriving Disk Formula

AI Thread Summary
To derive the moment of inertia of a uniform disk rotating about a central axis, the integral I = ∫ r² dm must be evaluated. The mass of the disk can be expressed as M = πR²dρ, where ρ is the density. To solve the integral, it is suggested to consider concentric rings, allowing for the expression of dm in terms of density and distance from the axis. The confusion arises from the need to evaluate the integral correctly rather than simply taking a derivative. Understanding these steps is crucial for accurately deriving the moment of inertia formula I = 1/2 MR².
revres75
Messages
2
Reaction score
0

Homework Statement



This was an exam question that I got wrong, my teacher tried to explain it but it only left me more confused. I found some websites that also had explanations but they were also confusing.

"Derive the moment of inertia of a uniform disk which rotates along a central axis , radius R , disk thickness d, mass M , density p
I= R^2 dm"


Homework Equations


I= 1/2*M*R^2 ?


The Attempt at a Solution



mass M = Pi*R^2*d*p

My teacher mentioned something about a third integral.
 
Physics news on Phys.org
Presumeably you are to derive the moment of inertia by evaluating \int r^2 dm. So what did you do?
 
If I derive \int r^2 dm I get 2r but I not sure what to do with the dm
 
revres75 said:
If I derive \int r^2 dm I get 2r but I not sure what to do with the dm
By "derive" I didn't mean "take the derivative". You need to evaluate that integral, which is the definition of moment of inertia. Start by expressing "dm" in terms of density and distance from the axis. (Hint: Think in terms of concentric rings.)
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
View full post »

Thread 'Struggling to understand the concept of this question (ice cube in water optics question)'

TL;DR Summary: I came across this question from a Sri Lankan A-level textbook. Question - An ice cube with a length of 10 cm is immersed in water at 0 °C. An observer observes the ice cube from the water, and it seems to be 7.75 cm long. If the refractive index of water is 4/3, find the height of the ice cube immersed in the water. I could not understand how the apparent height of the ice cube in the water depends on the height of the ice cube immersed in the water. Does anyone have an...
View full post »

Similar threads

Moment of inertia of a disk about an axis not passing through its CoM
Replies
11
Views
3K
Parallel Axis Theorem Not Working
Replies
28
Views
1K
What is the "r" in the moment of inertia?
Replies
13
Views
2K
Correct Use of the Parallel Axis Theorem for Moment of Inertia
Replies
2
Views
2K
Analyzing torque of an automobile engine
Replies
6
Views
923
Moment of inertia of two coaxial disks
Replies
8
Views
2K
Calculating Moment of Inertia for Hollow Cylinder w/ Water
Replies
40
Views
5K
Solving for 'a' with Torque, Force, and Mass Moment of Inertia
Replies
3
Views
2K
Moment of inertia of a cuboid parallel to one of its faces (repost with diagram)
Replies
25
Views
2K
Moment of inertia of a disc using rods as differential elements
Replies
8
Views
2K

Hot Threads

Recent Insights

Back
Top