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
Lerok [7]
3 years ago
8

A mass of 3.25 kg is attached to the end of a spring that is stretched 22 cm by a force of 15 N. It is set in motion with an ini

tial position x0=0 and initial velocity (in m/s) v0=−12. Find the amplitude, period, and frequency of the resulting motion. You should assume that the system is frictionless.
Mathematics
1 answer:
mylen [45]3 years ago
5 0

Answer:

Step-by-step explanation:

Given that,

Mass of object=3.25kg

The extension e=22cm=0.22m

Force applied to cause extension F=15N

Initial position Xo=0

Initial velocity Vo=-12m/s

We can get the spring constant from Hooke's law

F=ke

Then, k=F/e

k=15/0.22

k=68.182N/m

Also our natural frequency w is given as

w=√(k/m)

Therefore,

w=√(68.182/3.24)

w=√20.98

w=√21

w=4.58rad/s

w=4.6rad/s

There is no damping in this situation, no outside force acting on the system and the equation that governs the system is

mx''+kx=0

3.25x''+68.182x=0

Divide through by 3.25

x''+20.98x=0

We can approximate 20.98 to 21

x"+21x=0

The solution to this differential equation using D operator

D²+21=0

D²=-21

D= ±√-21

D=±√21 •i

Then the solution is

x(t)=A•Sinwt +B•Coswt

x(t)=A•Sin√21 t +B•Cos√21 t

Note that x'(t)=v(t)

and at t=0 Vo=-12m/s

x(t)=A•Sin√21 t +B•Cos√21 t

x'(t)=v(t)=A√21•Cos√21 t - B√21•Sin√21 t

v(t)=A√21•Cos√21 t - B√21•Sin√21 t

Then, using the two initial conditions

v(0)=-12

And X(0)=0

x(t)=A•Sin√21 t +B•Cos√21 t

X(0)=A•Sin√21•0 +B•Cos√21•0

X(0)=A•Sin0+B•Cos0

0=B

B=0

Also,, V(0)=-12m/s

v(t)=A√21•Cos√21 t - B√21•Sin√21 t

V(0)=A√21•Cos√21•0- B√21•Sin√21•0

V(0)=A√21•Cos0- B√21•Sin0

-12=A√21

Therefore,

A=-12/√21

A=-2.62

Therefore the general equation becomes

x(t)=A•Sin√21 t +B•Cos√21 t

x(t)=-2.62Sin√21 t +0•Cos√21 t

x(t)=-2.62Sin√21 t

a. The amplitude

Comparing x(t) to wave equations

x(t)=-Asin(wt+2λ/t)

Then,

A=2.62m

b. We know the natural frequency already to be

w=√21

w=4.58rad/s

c. Period

Comparing the equation again

wt=√21t

Given that w=2πf

Therefore, 2πft=√21t

Then, f=√21t / 2πt

f=√21/2π

f=0.73Hz

Then, period is the reciprocal of frequency

T=1/f

T=1/0.73

T=1.37seconds

The period is 1.37sec,

You might be interested in
Given the piecewise function shown below, select all of the statements that are true.
ZanzabumX [31]

Answer:

A, B & D are true

C is wrong

7 0
3 years ago
I don't need you to work it out, just theorem(s) i need to reach the answer :)
VikaD [51]
Not sure why such an old question is showing up on my feed...

Anyway, let x=\tan^{-1}\dfrac43 and y=\sin^{-1}\dfrac35. Then we want to find the exact value of \cos(x-y).

Use the angle difference identity:

\cos(x-y)=\cos x\cos y+\sin x\sin y

and right away we find \sin y=\dfrac35. By the Pythagorean theorem, we also find \cos y=\dfrac45. (Actually, this could potentially be negative, but let's assume all angles are in the first quadrant for convenience.)

Meanwhile, if \tan x=\dfrac43, then (by Pythagorean theorem) \sec x=\dfrac53, so \cos x=\dfrac35. And from this, \sin x=\dfrac45.

So,

\cos\left(\tan^{-1}\dfrac43-\sin^{-1}\dfrac35\right)=\dfrac35\cdot\dfrac45+\dfrac45\cdot\dfrac35=\dfrac{24}{25}
7 0
3 years ago
I need to show my work help
Tema [17]

Answer:

it will take 90 minutes

Step-by-step explanation:

10 - 6 = 9

9 x 10 = 90

4 0
2 years ago
Read 2 more answers
A school administrator bought 6 desk sets. Each set consists of one table and one chair. He paid $48.50 for each chalr. If he pa
zlopas [31]
The answer is $62.00

Explanation: If you take $48.50 times 6 you get $291 now subtract $291 from $663 you will have an answer of $372 that is how much the total of tables came to now divide 372 by 6 for the number of tables and you will be left with 62
7 0
3 years ago
What is the average for a half marathon time on a 35 year old man?
nevsk [136]
Average time for a 35 year old man to run half a marathon is 9:50:38
4 0
3 years ago
Other questions:
  • The mean of a set of eight numbers is 8 and the mean of a different set of twelve numbers is m. Given that the mean if the combi
    5·1 answer
  • the relative frequency of a drawing pin falling pin up is 3/8. How many times would you expect it to fall pin up in 400 drops?
    11·1 answer
  • What do you call any number that is expressed as an integer or as a ratio between two integers when 0 does not serve as the deno
    11·1 answer
  • Someone help my cousin out, he wanted help for this question but I’m too lazy to solve it so here you go.
    5·1 answer
  • Solve the expression 9.50h+60=460
    12·2 answers
  • A shipping company charges a $3 flat fee for a package, plus a fee based on the weight of the package. The company charges $0.75
    11·1 answer
  • 5 = 500e^-0.00043(t)
    13·2 answers
  • Work out the sum of the interior angles of this irregular hexagon.<br><br> Does anyone know ?
    12·2 answers
  • Tanya and Cam can each wash a car and vacuum its interior in 2 hours. Pat needs 3 hours to do this same job by himself. If Pat,
    7·2 answers
  • On a certain hot​ summer's day, 573 people used the public swimming pool. The daily prices are $1.50 for children and $2.00 for
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!