1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
lbvjy [14]
3 years ago
13

A spring-mass system has a spring constant of 3 Nm. A mass of 2 kg is attached to the spring, and the motion takes place in a vi

scous fluid that offers a resistance numerically equal to the magnitude of the instantaneous velocity. If the system is driven by an external force of 27cos(3t)−18sin(3t) N,determine the steady-state response in the form Rcos(ωt−δ).
Physics
1 answer:
frosja888 [35]3 years ago
8 0

Answer:

The answer to the question

The steady state response is u₂(t) = -\frac{3\sqrt{2} }{2}cos(3t + π/4)

of the form R·cos(ωt−δ) with R = -\frac{3\sqrt{2} }{2}, ω = 3 and δ = -π/4

Explanation:

To solve the question we note that the equation of motion is given by

m·u'' + γ·u' + k·u = F(t) where

m = mass = 2.00 kg

γ = Damping coefficient = 1

k = Spring constant = 3 N·m

F(t) = externally applied force = 27·cos(3·t)−18·sin(3·t)

Therefore we have 2·u'' + u' + 3·u = 27·cos(3·t)−18·sin(3·t)

The homogeneous equation 2·u'' + u' + 3·u is first solved as follows

2·u'' + u' + 3·u = 0 where putting the characteristic equation as

2·X² + X + 3 = 0 we have the solution given by \frac{-1+/-\sqrt{23} }{4} \sqrt{-1} =\frac{-1+/-\sqrt{23} }{4} i

This gives the general solution of the homogeneous equation as

u₁(t) = e^{(-1/4t)} (C_1cos(\frac{\sqrt{23} }{4}t) + C_2sin(\frac{\sqrt{23} }{4}t)

For a particular equation of the form 2·u''+u'+3·u = 27·cos(3·t)−18·sin(3·t) which is in the form u₂(t) = A·cos(3·t) + B·sin(3·t)

Then u₂'(t) = -3·A·sin(3·t) + 3·B·cos(3·t) also u₂''(t) = -9·A·cos(3·t) - 9·B·sin(3·t) from which  2·u₂''(t)+u₂'(t)+3·u₂(t) = (3·B-15·A)·cos(3·t) + (-3·A-15·B)·sin(3·t). Comparing with the equation 27·cos(3·t)−18·sin(3·t)  we have

3·B-15·A = 27

3·A +15·B = 18

Solving the above linear system of equations we have

A = -1.5, B = 1.5 and  u₂(t) = A·cos(3·t) + B·sin(3·t) becomes 1.5·sin(3·t) - 1.5·cos(3·t)

u₂(t) = 1.5·(sin(3·t) - cos(3·t) = -\frac{3\sqrt{2} }{2}·cos(3·t + π/4)

The general solution is then  u(t) = u₁(t) + u₂(t)

however since u₁(t) = e^{(-1/4t)} (C_1cos(\frac{\sqrt{23} }{4}t) + C_2sin(\frac{\sqrt{23} }{4}t) ⇒ 0 as t → ∞ the steady state response = u₂(t) = -\frac{3\sqrt{2} }{2}·cos(3·t + π/4) which is of the form R·cos(ωt−δ) where

R = -\frac{3\sqrt{2} }{2}

ω = 3 and

δ = -π/4

You might be interested in
A ______ is a push or pull that one object exerts on another.
fiasKO [112]
The answer is A. force. 
8 0
3 years ago
Read 2 more answers
How does the end point differ from the equivalence <br>point of a titration?​
Gwar [14]

<u>Answer:</u>

<em>Equivalence point and end point are terminologies in pH titrations and they are not the same. </em>

<u>Explanation:</u>

In a <em>titration the substance</em> added slowly to a solution usually through a pippette is called titrante and the solution to which it is added is called titrand. In acid-base titrations acid is added to base or base is added to acid.the strengths of the <em>acid and base titrated</em> determines the nature of the final solution.

At equivalence point the <em>number of moles of the acid</em> will be equal to the number of moles of the base as given in the equation.  The nature of the final solution determines the <em>pH at equivalence point. </em>

<em>A pH less than 7 will be the result if the resultant is acidic and if it is basic the pH will be greater than 7. </em>In a strong base-strong acid and weak base-weak acid titration the pH at the equivalence point will be 7 indicating <em>neutral nature of the solution. </em>

3 0
3 years ago
Is this statement true or false? The next generation of nuclear power plants being built in California and South Africa are even
alexira [117]

Answer:

True

Explanation:

Modern safer and cheaper nuclear reactors can not only meet the range of our long term energy demands, they can also fight global warming.

Modern techniques provide ways to reduce radioactive waste amount. "A closed fuel cycle may be switched on for new kinds of nuclear plants. Alternatively, the waste is chemically dissuaded to transform the reusable element into fuel. This implies that nuclear waste would not be buried.

8 0
3 years ago
What's a sentence using work in a nonscientific sense
kvv77 [185]

My Physics teacher keeps giving us way too much work to do at home.

7 0
4 years ago
Read 2 more answers
What distance will a car traveling at a speed of 15 m/s cover in 300 s?
AnnZ [28]

Distance formula: d = st

d = 15 * 300

d = 4,500m

Best of Luck!

3 0
3 years ago
Other questions:
  • A point charge q1 = -2.3 μC is located at the origin of a co-ordinate system. Another point charge q2 = 6.2 μC is located along
    6·1 answer
  • Small rock substances that orbit the sun in groups is called
    7·1 answer
  • A 128.0-N carton is pulled up a frictionless baggage ramp inclined at 30.0∘above the horizontal by a rope exerting a 72.0-N pull
    5·1 answer
  • How would life on earth be different if earth’s axis were not tilted with respect to its orbit?
    5·2 answers
  • Explain how fossil fuels are used to produce electricity?
    14·1 answer
  • Why are not all microorganism harmful?
    5·1 answer
  • Plz answer these questions u don't have to do 4,5,6,or 7 just 1,2, and 3 plz plz plz
    5·1 answer
  • What is the main source of groundwater? A. precipitation falling into man-made wells B. precipitation seeping through pores and
    9·1 answer
  • Identify the parts of the wave below. Please help! I need it quite soon!
    14·1 answer
  • The type of energy possessed by a fluid above its base level is
    11·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!