A certain vibrating system satisfies the equation u'' + γu' + u = 0. Find the value of the damping coefficient γ for which the quasi period of the damped motion is 90% greater than the period of the corresponding undamped motion.
Answers
Final answer:
In damped harmonic motion, we calculate damping coefficient γ by comparing the periods of damped and undamped motion. For the given situation where the quasi-period is 90% greater than the undamped period, the damping coefficient is approximately 0.7416.
Explanation:
The subject of this question involves Damped Harmonic Motion, a concept in Physics, related to vibrations and waves. The equation given, u'' + γu' + u = 0, describes the motion where γ denotes the damping coefficient. Here, we have to calculate this damping coefficient when the quasi period of the damped motion is 90% greater than the period of the corresponding undamped motion.
To solve this, we must use the relationship between damped and undamped periods. The quasi-period T' of a damped harmonic motion relates to the undamped period T as: T' = T/(sqrt(1 - (γ/2)^2)). Now, given that T' = 1.9T, we can but these two equations together:
1.9 = 1/(sqrt(1 - (γ/2)^2))
Solving this for γ, we get γ ≈ 0.7416. Hence, the damping coefficient γ for which the quasi period of the damped motion is 90% greater than the period of the corresponding undamped motion is approximately 0.7416.
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Final answer:
The value of the damping coefficient γ for which the quasi period of the damped motion is 90% greater than the period of the undamped motion is the one that satisfies γ=2*ω*0.9, where ω is the natural frequency of oscillation.
Explanation:
The given equation is for a damped harmonic oscillator, a physical system that oscillates under both a restoring force and a damping force proportional to the velocity of the system. The damping coefficient γ determines the behavior of the system and in this case, we need to find the value of γ such that the quasi period of the damped motion is 90% greater than the period of the undamped motion.
The period of the undamped motion, T₀, is calculated by the formula T₀=2π/sqrt(ω), where ω is the natural frequency of oscillation. The quasi period of the damped motion, Td, is increased by a factor of 1+η (in this case, 1.9 as the increase is 90%) and calculated by the formula Td=T₀(1+η) = T₀*1.9.
The damping ratio η is determined by the damping coefficient γ as η=γ/2ω. Therefore, by combining these expressions and rearranging the terms, we extract γ from these formulas as γ=2ω*η => γ=2*ω*(0.9). Thus, the value of the damping coefficient γ for which the quasi period of the damped motion is 90% greater than the period of the corresponding undamped motion is the one which satisfies γ=2*ω*0.9.
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Related Questions
The perimeter of a rectangle is 80 cm. The width is 10 cm more than twice the length. Whatis the length and width of this rectangle? P=2W+2L
Simplify one or both sides by combing like terms X=8-3
Select all the values that make the inequality true n<-15A. -25B. -1C. -30D. -19
NEED HELP PLEASEEEEEEEEEEEPoints A′, B′, and C′ are the images of 180-degree rotations of A, B, and C, respectively, around point O.
f(x)=-2x^2-7x+6
Find f(−3)
Answers
Answer:
f(-3) = 9
Step-by-step explanation:
Step 1: Define
f(x) = -2x² - 7x + 6
f(-3) = x = -3
Step 2: Substitute and evaluate
f(-3) = -2(-3)² - 7(-3) + 6
f(-3) = -2(9) + 21 + 6
f(-3) = -18 + 21 + 6
f(-3) = 3 + 6
f(-3) = 9
Substituting x = -3 into the function f(x) = -2x^2 - 7x + 6, we get f(-3) = -33. Therefore, the value of the function at x = -3 is -33.
To evaluate the function f(x)=−2x ^2 −7x+6 at x=−3, substitute −3 for x in the function: f(−3)=−2{(−3) }^2 −7(−3)+6
Now, calculate each part of the expression:
(−3) ^2 is 9 because the square of −3 is 9.
−2 times 9 is −18 because −2⋅9 =−18.
−7 times −3 is 21 because −7⋅−3=21.
Now, plug these values back into the expression:
f(−3)=−18−21+6
Finally, add and subtract:
f(−3)=(−18−21)+6=−39+6=−33
So, f(−3)=−33.
The value of the function
f(x)=−2x^ 2 −7x+6 at x=−3 is −33.
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Answers
Answer:
a = 4, b = 5
Step-by-step explanation:
Given the exponential function
f(x) = a
Use the given points to find a and b
Using (0, 4 ) , then
4 = a (
= 1 ) , thus
a = 4 , so
f(x) = 4
Using (1, 20 ) , then
20 = 4b ( divide both sides by 4 )
b = 5
b. Then find (to 3 decimal places) the number b such that the line y = b divides R into two parts of equal area.
Answers
You have
and
Setting these equal, you get
For part (b), you have
and you want to find
You have
and
Setting them equal gives
As x goes toward -9
Answers
Use the factoration
a^2 - b^2 = (a - b)(a + b)
Then,
x^2 - 81 = x^2 - 9^2
x^2 - 9^2 = ( x - 9).(x + 9)
Then,
Lim (x^2- 81) /(x+9)
= Lim (x -9)(x+9)/(x+9)
Simplity x + 9
Lim (x -9)
Now replace x = -9
Lim ( -9 -9)
Lim -18 = -18
_______________
The second method without using factorization would be to calculate the limit by the hospital rule.
Lim f(x)/g(x) = lim f(x)'/g(x)'
Where,
f(x)' and g(x)' are the derivates.
Let f(x) = x^2 -81
f(x)' = 2x + 0
f(x)' = 2x
Let g(x) = x +9
g(x)' = 1 + 0
g(x)' = 1
Then the Lim stay:
Lim (x^2 -81)/(x+9) = Lim 2x /1
Now replace x = -9
Lim 2×-9 = Lim -18
= -18
Answers
36 teams
Final answer:
To form teams, divide the total number of players by the number of players per team. In this case, 36 teams are needed.
Explanation:
To find out how many teams will be formed, we divide the total number of players by the number of players assigned to a team. In this case, there are 432 players and 12 players per team. So, we divide 432 by 12:
Number of teams = 432 ÷ 12 = 36 teams
Therefore, the league will need to form 36 teams to accommodate all the players.
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