Math Problem Statement
An ultracentrifuge accelerates from rest to 100,000 rpm in 2.00 min. (a) What is its angular acceleration in rad/s2? rad/s2 (b) What is the tangential acceleration (in m/s2) of a point 8.50 cm from the axis of rotation? m/s2 (c) What is the radial acceleration in m/s2 and multiples of g of this point at full rpm? a in m/s m/s2 a as a multiple of g g
Solution
Let's solve each part step-by-step.
Given dataLet's solve each part step-by-step.
Given
- Final angular velocity () = 100,000 rpm
- Time () = 2.00 min
- Radius () = 8.50 cm = 0.085 m
First, let's convert angular velocity from rpm to rad/s.
(a) Angular Acceleration
Angular acceleration is calculated using the formula: where (since it starts from rest) and in seconds is .
Answer (a): Angular acceleration, .
(b) Tangential Acceleration
Tangential acceleration at a point on the edge of the disc is given by: where and .
Answer (b): Tangential acceleration, .
(c) Radial Acceleration
Radial (or centripetal) acceleration is calculated using: where and .
To express in terms of , where :
Answer (c):
- Radial acceleration
- As a multiple of :
Would you like further details on these calculations or have any questions?
- What are the differences between tangential and radial accelerations in rotational motion?
- How does converting rpm to rad/s impact calculations in rotational physics?
- How would increasing the radius impact both tangential and radial accelerations?
- What practical considerations limit the maximum rpm achievable in an ultracentrifuge?
- Why are radial accelerations often much larger than tangential accelerations at high angular speeds?
Tip: Always convert units to SI (m, s, rad) before starting calculations to maintain consistency and avoid errors.
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Math Problem Analysis
Mathematical Concepts
Kinematics
Angular Motion
Acceleration
Formulas
Angular acceleration formula: α = (ω_f - ω_i) / t
Tangential acceleration formula: a_t = α * r
Radial acceleration formula: a_r = ω^2 * r
Theorems
Conversion of rpm to rad/s
Centripetal acceleration
Suitable Grade Level
College level - Introductory Physics
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