oldHWK31 - oldhomewk 31 – VALENCIA, DANIEL – Due: Apr...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: oldhomewk 31 – VALENCIA, DANIEL – Due: Apr 28 2008, 4:00 am 1 Question 1, chap 15, sect 1. part 1 of 2 10 points Two massless springs with spring constants 1044 N / m and 5684 N / m are hung from a horizontal support. A block of mass 2 kg is suspended from the pair of springs, as shown. The acceleration of gravity is 9 . 8 m / s . 1044 N / m 5684 N / m 2 kg When the block is in equilibrium, each spring is stretched an additional Δ x . Then the block oscillates with an amplitude of 21 m, and it passes through its equilibrium point with a speed of 441 m / s. What is the angular velocity of this system? 1. ω = 15 rad / s 2. ω = 11 rad / s 3. ω = 2 rad / s 4. ω = 18 rad / s 5. ω = 6 rad / s 6. ω = 10 rad / s 7. ω = 12 rad / s 8. ω = 21 rad / s correct 9. ω = 17 rad / s 10. ω = 13 rad / s Explanation: Let m = 2 kg , A = 21 m , v = 441 m / s , k 1 = 1044 N / m , and k 2 = 5684 N / m . Basic Concepts: Hooke’s law F =- k x = ma = d 2 x dt 2 d 2 x dt 2 + k m x = 0 , (1) whose integral form has a sine function x ( t ) = A sin( ω t + δ ) , (2) v ( t ) ≡ dx dt v ( t ) = ω A cos( ω t + δ ) , and (3) a ( t ) ≡ dv dt a ( t ) =- ω 2 A sin( ω t + δ ) , where (4) ω = radicalbigg k m . (5) The angular velocity ω is the square root of the coefficient of x in Eq. 1. The frequency of oscillation f versus angu- lar frequency ω is f ≡ ω 2 π . (6) k 1 k 2 m Consider the forces from a spring’s point of view. The oscillating mass exerts the same force, F (at some instant in time) on the springs k 1 and k 2 , F = k 1 x 1 ⇒ x 1 = F k 1 F = k 2 x 2 ⇒ x 2 = F k 2 . oldhomewk 31 – VALENCIA, DANIEL – Due: Apr 28 2008, 4:00 am 2 Now consider the effective spring constant, k series , where x = x 1 + x 2 , therefore k series = F x = F x 1 + x 2 = F F k 1 + F k 2 = 1 1 k 1 + 1 k 2 = k 1 k 2 k 1 + k 2 . (7) = (1044 N / m) (5684 N / m) (1044 N / m) + (5684 N / m) = 882 N / m . Solution: The question presents the springs in series, Eq. 6 and Eq. 7, therefore ω series = radicaltp radicalvertex radicalvertex radicalbt k 1 k 2 m bracketleftBig k 1 + k 2 bracketrightBig (8) = radicaltp radicalvertex radicalvertex radicalbt (1044 N / m) (5684 N / m) (2 kg) bracketleftBig (1044 N / m) + (5684 N / m) bracketrightBig = radicalBigg (882 N / m) (2 kg) = 21 rad / s , and f = ω series 2 π (9) = (21 rad / s) 2 π = 3 . 34225 cycles / s , and T = 1 f = 1 (3 . 34225 cycles / s) = 0 . 299199 s...
View Full Document

This note was uploaded on 03/26/2009 for the course PHY 303K taught by Professor Turner during the Spring '08 term at University of Texas.

Page1 / 6

oldHWK31 - oldhomewk 31 – VALENCIA, DANIEL – Due: Apr...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online