Calculus II question really challenging and fun.? - old fashioned record player
Consider the following questions:
kinetic energy of an object of mass m moves at speed V given by 0.5 mV ^ 2 If a system is composed of multiple objects in motion, the kinetic energy of the system, the sum of the energies of the various parties.
Assuming a uniform density disk rotates around the center, like the tray in an old record player in fashion. M is the mass of the disk (in kg), R is the radius of the disc (in meters). Va and the angular velocity of the disc (in radians per second).
Find the kinetic energy of the disc. Show your work.
Thank you for your interest ... it came from the integration by slicing
Monday, January 11, 2010
Old Fashioned Record Player Calculus II Question Really Challenging And Fun.?
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3 comments:
The famous answer is 1 / ω 2 * R ^ 2
when the moment of inertia I = 1 / 2 mR ^ 2
Imagine a small ring at a radial distance r from the center with Dr. infinitely thick. Each point on this ring has a linear velocity ωr
The mass of the ring is 2π ρ * * r * dr
where ρ the mass per unit area
Thus, the EC of the ring, DKE = 1 / 2 * (ρ * 2π * r * dr) * (ωr) ^ 2
DKE 2πr = ^ ^ 3 ρω HD
To find the KE of all the rings, the sum of DKE or integration from 0 to R.
KE = ρωπ ∫ r ^ 3 HD
= 1 / 4 * ^ ^ 4 ρω 2πR
= 1 / 2 * (1 / 2 ρ πR ^ 2) * R ^ 2 * ω ^ 2
= 1 / R 2 m ^ 2 ω ^ 2
= 1 / 2 I ω ^ 2
The famous answer is 1 / ω 2 * R ^ 2
when the moment of inertia I = 1 / 2 mR ^ 2
Imagine a small ring at a radial distance r from the center with Dr. infinitely thick. Each point on this ring has a linear velocity ωr
The mass of the ring is 2π ρ * * r * dr
where ρ the mass per unit area
Thus, the EC of the ring, DKE = 1 / 2 * (ρ * 2π * r * dr) * (ωr) ^ 2
DKE 2πr = ^ ^ 3 ρω HD
To find the KE of all the rings, the sum of DKE or integration from 0 to R.
KE = ρωπ ∫ r ^ 3 HD
= 1 / 4 * ^ ^ 4 ρω 2πR
= 1 / 2 * (1 / 2 ρ πR ^ 2) * R ^ 2 * ω ^ 2
= 1 / R 2 m ^ 2 ω ^ 2
= 1 / 2 I ω ^ 2
The famous answer is 1 / ω 2 * R ^ 2
when the moment of inertia I = 1 / 2 mR ^ 2
Imagine a small ring at a radial distance r from the center with Dr. infinitely thick. Each point on this ring has a linear velocity ωr
The mass of the ring is 2π ρ * * r * dr
where ρ the mass per unit area
Thus, the EC of the ring, DKE = 1 / 2 * (ρ * 2π * r * dr) * (ωr) ^ 2
DKE 2πr = ^ ^ 3 ρω HD
To find the KE of all the rings, the sum of DKE or integration from 0 to R.
KE = ρωπ ∫ r ^ 3 HD
= 1 / 4 * ^ ^ 4 ρω 2πR
= 1 / 2 * (1 / 2 ρ πR ^ 2) * R ^ 2 * ω ^ 2
= 1 / R 2 m ^ 2 ω ^ 2
= 1 / 2 I ω ^ 2
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