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meriva
3 years ago
6

An undamped spring-mass system contains a mass that weighs and a spring with spring constant . It is suddenly set in motion at b

y an external force of . Determine the position of the mass at any time . Use as the acceleration due to gravity. Pay close attention to the units.

Physics
1 answer:
balandron [24]3 years ago
8 0

Answer:

Explanation:

When all other forces acting on the mass in a damped mass-spring system are grouped together into one term denoted by F(t), the differential equation describing

motion is

Mx''+ βx' + kx = F(t).

Note for an undamped system

β=0,

Then, the differential equation becomes

Mx'' + kx = F(t).

The force is in the form

F=Fo•Sinωo•t

Let solved for the homogeneous or complementary solution, I.e f(t) = 0

Using D operator

MD² + k = 0

MD²=-k

D²=-k/M

Then, D= ±√(-k/m)

D=±√(k/m) •i

So we have a complex root

Therefore, the solution is

x= C1•Cos[√(k/m)t] + C2•Sin[√(k/m)]

This is simple harmonic motion that once again we prefer to write in the form

x(t) = A•Sin[ √(k/M)t + φ]

Where A=√(C1²+C2²)

and angle φ is defined by the equations

sin φ = C1/A and cos φ = C2/A.

Quantity √(k/M), often denoted by ω, is called the angular frequency.

This is called the natural frequency (ωn) of the system

ωn=√(k/M)

ωn²= k/M

Now, for particular solution

Xp=DSinωo•t

Xp' = Dωo•Cosωo•t

Xp"=-Dωo²•Sinωo•t

Now substituting this into

Mx'' + kx = F(t).

M(-Dωo²•Sinωo•t) + k(DSinωo•t)=FoSinωo•t

Now, let solve for D

D(-Mωo²•Sinωo•t +kSinωo•t) = FoSinωo•t

D=Fo•Sinωo•t/(-Mωo²•Sinωo•t +kSinωo•t)

D=Fo•Sinωo•t / Sinωo•t(-Mωo²+k)

D=Fo / (-Mωo²+k)

D=Fo / (k-Mωo²)

Divide through by k

D=Fo/k ÷ (1 -Mωo²/k)

Note from above

ωn²= k/M

Therefore,

D=Fo/k ÷ (1-ωo²/ωn²)

D=Fo/k ÷ [1-(ωo/ωn)²]

Then,

Xp=DSinωo•t

Xp=(Fo/k ÷ [1-(ωo/ωn)²]) Sinωo•t

Then the general solution is the sum of the homogeneous solution and particular solution

Xg(t)=(Fo/k ÷ [1-(ωo/ωn)²]) Sinωo•t + A•Sin[ √(k/M)t + φ]

Check attachment for the graph of homogeneous, particular and general solution.

Also, check for better way of writing the equations.

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Explanation:

That is why the cell needs to be small

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3 years ago
4. Which of the following would NOT be a major process in the formation of sedimentary
sladkih [1.3K]

Answer:

b. melting

Explanation:

it is made of sediments and that is not necessary

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2 years ago
The depth of the Pacific Ocean in the Mariana Trench is 36,198 ft. What is the gauge pressure at this depth
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Answer:

the pressure at the depth is 1.08 × 10^{8} Pa

Explanation:

The pressure at the depth is given by,

P = h \rho g

Where, P = pressure at the depth

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4 0
3 years ago
Explain each of the following phenomena and how they extend our understanding of genetics from the work done by Gregor Mendel an
Korolek [52]

Answer:

Incomplete Dominance: In this case, the 3rd different phenotype is created by the heterozygous(CwCr) (for example, pink)

Co-Dominance: the heterozygous is observed to show both the homozygous conditions (for example the blood type iAiB shows both A and B carbohydrates)

Multiple alleles: This is when a gene has over 2 alleles (i.e. blood has ia ib i)

Polygenic inheritance: This is when over two different gene leads to the expression of a trait.

Pleiotropy: This is the when a gene shows several effects on an organism.

Epistasis: This is when more than one gene have strong control on the  expression of a trait (for example, E/e allele creates pigments)

Sex-linkage: This refers to the genes found in sex chromosomes

Gene Linkage: These are the genes situated in a similar chromosome.

Explanation:

Incomplete Dominance: In this case, the 3rd different phenotype is created by the heterozygous(CwCr) (for example, pink)

Co-Dominance: the heterozygous is observed to show both the homozygous conditions (for example the blood type iAiB shows both A and B carbohydrates)

Multiple alleles: This is when a gene has over 2 alleles (i.e. blood has ia ib i)

Polygenic inheritance: This is when over two different gene leads to the expression of a trait.

Pleiotropy: This is the when a gene shows several effects on an organism.

Epistasis: This is when more than one gene have strong control on the  expression of a trait (for example, E/e allele creates pigments)

Sex-linkage: This refers to the genes found in sex chromosomes

Gene Linkage: These are the genes situated in a similar chromosome.

7 0
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