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Tasya [4]
3 years ago
13

A 1-kilogram mass is attached to a spring whose constant is 14 N/m, and the entire system is then submerged in a liquid that imp

arts a damping force numerically equal to 9 times the instantaneous velocity. Determine the equations of motion if the following is true. (a) the mass is initially released from rest from a point 1 meter below the equilibrium position. (b) the mass is initially released from a point 1 meter below the equilibrium position with an upward velocity of 13 m/s.
Physics
1 answer:
ch4aika [34]3 years ago
3 0

Answer:

Part(a): The equation of motion is \bf{x(t) = \dfrac{7}{5}~e^{-2t} - \dfrac{2}{5}~e^{-7t}}.

Part(b): The equation of motion is  \bf{x(t) = -e^{-2t} + \dfrac{11}{5}~e^{-7t}}.

Explanation:

If 'm' be the mass of the object, 'k' be the force constant and '\beta' be the damping constant, then the equation of motion of the particle can be written as

\dfrac{d^{2}x}{dt^{2}} + \dfrac{\beta}{m} \dfrac{dx}{dt} + \dfrac{k}{m}x= 0.........................................(I)

Given m = 1 Kg, k = 14 N~m^{-1}, \beta = 9. Substituting these values in equation (I),

\dfrac{d^{2}x}{dt^{2}} + 9~\dfrac{dx}{dt} + 14~x= 0

Taking a trial solution x(t) = e^{mt}, the auxiliary equation can be written as

m^{2} + 9m + 14 = 0............................................................(II)

and its solutions are m_{1} = -2~and~m_{2} = -7, resulting the general solution

x(t) = C_{1}~e^{-2t} + C_{2}~e^{-7t}....................................................................(III)

The velocity at any instant of time of the mass is

v(t) = -2C_{1}~e^{-2t} _7~C_{2}~e^{-7t}..............................................................(IV)

Part(a):

Given x(t=0) = 1 m,~and~v(t=0) = 0~m~s^{-1}. Substituting these values in equation (III) and (IV),

&& 1 = C_{1} + C_{2}......................(V)\\&and,& 0 = -2C_{1} - 7C_{2}.......................(VI)

Solving equations (V) and (VI), we have

C_{1} = \dfrac{7}{5}~and~C_{2} = \dfrac{-2}{5}

So the equation of motion is

x(t) = \dfrac{7}{5}~e^{-2t} - \dfrac{2}{5}~e^{-7t}

Part(b):

Given x(t=0) = 1 m,~and~v(t=0) = - 12~m~s^{-1}. Substituting these values in equation (III) and (IV),

&& 1 = C_{1} + C_{2}......................(VII)\\&and,& -12 = -2C_{1} -7C_{2}.......................(VIII)

Solving equations (V) and (VI), we have

C_{1} = -1~and~C_{2} = \dfrac{11}{5}

So the equation of motion is

x(t) = -e^{-2t} + \dfrac{11}{5}~e^{-7t}

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The diameter is 0.000056 m

Explanation:

Lets explain the relation between the meter and the micrometer

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1 micrometer = \frac{1}{1000000}=\frac{1}{10^{6}}=10^{-6}

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To do that we divide this number by 1,000,000 or multiply it by 10^{-6}

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A car horn emits a frequency of 400 Hz. A car traveling at 20.0 m/s sounds the horn as it approaches a stationary pedestrian. Wh
Temka [501]

Answer:

The observed frequency by the pedestrian is 424 Hz.

Explanation:

Given;

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speed of the car as it approaches the stationary observer, Vs = 20 m/s

Based on Doppler effect, as the car the approaches the stationary observer, the observed frequency will be higher than the transmitted (source) frequency because of decrease in distance between the car and the observer.

The observed frequency is calculated as;

F_s = F_o [\frac{v}{v_s + v} ] \\\\

where;

F₀ is the observed frequency

v is the speed of sound in air = 340 m/s

F_s = F_o [\frac{v}{v_s + v} ] \\\\400 = F_o [\frac{340}{20 + 340} ] \\\\400 = F_o (0.9444) \\\\F_o = \frac{400}{0.9444} \\\\F_o = 423.55 \ Hz \\

F₀ ≅ 424 Hz.

Therefore, the observed frequency by the pedestrian is 424 Hz.

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A piece of glass has a temperature of 72.0 degrees Celsius. The specific heat capacity of the glass is 840 J/kg/deg C. A liquid
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Specific heat capacity of the liquid, c(l) = ?

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Since their mass are the same, then

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