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Answer:

Equation of motion is x(t) = [tex]-te^{-4t}[/tex] + [tex]\frac{1}{4}[/tex] sin(4t)

Explanation:

P.S - The exact question is -

Given - A mass of 1 slug, when attached to a spring, stretches it 2 feet and then comes to rest in the equilibrium position. Starting at t = 0, an external force equal to  [tex]f(t) = 8 cos(4t)[/tex] is applied to the system.

To find - Find the equation of motion if the surrounding medium offers a damping force that is numerically equal to 8 times the instantaneous velocity.

Proof -

Given that,

Mass = 1 slug

We know that, 1 slug = 32 lb

Now,

Force, f = kx

⇒32 = k(2)

⇒k = 16

Now,

Given that, C = 8 ( 8 times the instantaneous velocity)

Now,

The differential equation of motion is equals to

mx'' + Cx' + kx = 8 cos(4t)

⇒x'' + 8x' + 16x = 8 cos(4t)               ...........(1)

Let the General solution of equation (1) be

x(t) = x(c) + x(p)

Now,

The auxiliary equation is

m² + 8m + 16 = 0

m² + 4m + 4m + 16 = 0

m (m+4) + 4 (m+4) = 0

⇒(m+4)(m+4) = 0

⇒m = -4, -4

So,

The Complimentary equation becomes

x(c) = [tex]Ae^{-4t} + Bte^{-4t}[/tex]                  ...........(2)

Now,

Let the particular solution be

x(p) = C cos(4t) + D sin(4t)

x'(p) = -4C sin(4t) + 4D cos(4t)

x''(p) = -16C cos(4t) - 16D sin(4t)

It also satisfy equation (1)

Equation (1) becomes

-16C cos(4t) - 16D sin(4t) + 8 [ -4C sin(4t) + 4D cos(4t) ] + 16 [ C cos(4t) + D sin(4t) ] = 8 cos(4t)

⇒-16C cos(4t) - 16D sin(4t) - 32C sin(4t) + 32D cos(4t) ] + 16C cos(4t) + 16D sin(4t) ] = 8 cos(4t)

⇒-4C sin(4t) + 4B cos(4t) = cos(4t)

By comparing, we get

4B = 1 , A = 0

⇒ B = [tex]\frac{1}{4}[/tex] , A = 0

So, The particular solution becomes

x(p) =  [tex]\frac{1}{4}[/tex] sin(4t)

Now,

The General solution becomes

x(t) = [tex]Ae^{-4t} + Bte^{-4t}[/tex] + [tex]\frac{1}{4}[/tex] sin(4t)       .......(3)

Now,

Given that, At t = 0, initial velocity is zero and the system starts equilibrium

⇒x(0) = 0, x'(0) = 0

By putting t = 0 in equation (3) , we get

A = 0

Now,

Differentiate equation (3), we get

x'(t) = [tex]-4Ae^{-4t} + Be^{-4t} - 4Bte^{-4t}[/tex] + [tex]\frac{1}{4}[/tex] *4 cos(4t)

Put t = 0, we get

0 = -4A + B + 1

⇒B = -1

∴ we get

The general solution becomes

x(t) = [tex]-te^{-4t}[/tex] + [tex]\frac{1}{4}[/tex] sin(4t)

Equation of motion is x(t) = [tex]-te^{-4t}[/tex] + [tex]\frac{1}{4}[/tex] sin(4t)

Answer:

D. Forces between molecules

Explanation:

Specific heat capacity of water can be defined as the amount of heat a gram of water must lose or absorb in order to change its temperature by a degree Celsius. It is measured in Joules per kilogram per degree Celsius (J/kg°C). Generally, the specific heat capacity of water is 4.182J/kg°C and is the highest among liquids.

Mathematically, the specific heat capacity of a substance is given by the formula;

[tex] c = \frac {Q}{mdt} [/tex]

Where;

Q represents the heat capacity or quantity of heat.

m represents the mass of an object.

c represents the specific heat capacity of water.

dt represents the change in temperature.

Cohesion is a property of water and it typically refers to the attraction between molecules of water which holds them together.

In Science, the property which helps to explain differences in the specific heat capacities of two substances is the forces between molecules.

This ultimately implies that, the more closely bonded the atoms of a substance are, the higher or greater would be the substance's specific heat capacity. Thus, it varies for the various states of matter i.e solid, liquid and gas.

Answer:

m = 0.01 kg

Explanation:

Given that,

Momentum of the marble, p = 0.15 kg-m/s

Speed of the marble, v = 15 m/s

We need to find its mass. We know that,

Momentum, p = mv

Where

m is the mass

[tex]m=\dfrac{p}{v}\\\\m=\dfrac{0.15}{15}\\\\m=0.01\ kg[/tex]

So, the mass of the marble is equal to 0.01 kg.

Answer:

10

Explanation:

lyezurzup?y?ur if fk kgs7rsjzkgzitz

Answer:

b

Explanation:

Answer:

Because the forces acting on the student are creating a net force.

Explanation:

I took the test and that was the answer

Answer:

Because the forces acting on the student are creating a net force

Explanation:

Answer:

ummm i dk what your asking

Explanation:

Answer:

The answer is D.250N.

Explanation:

Answer:

0.5x35=17.5

Explanation:

You throw 0.5 kg the ball leaves your hand with

A speed of 35

Answer:

In business and economics, fundamentals represent the primary characteristics and financial data necessary to determine the stability and health of an asset. This data can include macroeconomic, or large-scale factors, and microeconomic, or small-scale factors to set a value on securities or businesses.

Explanation:

Answer found from:

https://www.investopedia.com/terms/f/fundamentals.asp#:~:text=In%20business%20and%20economics%2C%20fundamentals,value%20on%20securities%20or%20businesses.

Answer:

I’m not okay . Me and my bf are bickering once again

Answer:

It is because of air pressure

Explanation:

Filling a fountain pen with ink involves various methods, but the most common approach uses a piston or converter mechanism.

To begin, the pen is prepared by removing the cap and separating the barrel from the grip section. The type of filling system is checked, whether it's a built-in piston or a removable converter.

For a piston-filling method, the knob is twisted counterclockwise to retract the piston. The nib is submerged in the ink bottle, and by turning the knob clockwise, the piston extends, creating a vacuum that draws ink into the reservoir through capillary action.

The piston is then retracted, sealing the ink inside. In the case of a converter, it is removed from the grip section and dipped into the ink bottle, followed by operating the plunger or squeezing to expel air.

Releasing the plunger allows ink to enter the converter through atmospheric pressure. After cleaning any excess ink, the pen is reassembled.

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Answer:

hi

Explanation:

answer is C

have a nice day

Answer:

Turned up side down

Explanation:

Answer: The Strong Nuclear Force is an attractive force between protons and neutrons that keep the nucleus together and the Weak Nuclear Force is responsible for the radioactive decay of certain nuclei.

Explanation:

Answer:

option (C) is right answer

Explanation:

I think it's helps you

Answer:

Option C is correct

Explanation:

hope it helps you

Answer:

v = 6.195 m / s

Explanation:

For this exercise we can use the conservation of energy, for the system formed by the ball and the pulley

starting point. Higher before releasing the system

        Em₀ = U = M g h

final point. When the ball has lowered h = 2

        Em_f = K = ½ M v² + ½ I w²

the energy is preserved

        Em₀ = Em_f

        M g h = ½ M v² + ½ I w²

angular and linear velocity are related

         v = w r

        w = v / r

indicate that the moment of inertia is

         I = ½ m r²

we substitute

          M g h = ½ M v² + ½ (½ m r²)  (v/r) ²

          ½ v² (M + [tex]\frac{1}{2}[/tex] m) = M g h

          v² = [tex]2gh \ \frac{M}{M + \frac{m}{2} }[/tex]

let's calculate

          v = [tex]\sqrt{ 2 \ 9.8 \ 2 \ \frac{70}{70 + 1.5} }[/tex]

          v = 6.195 m / s

Answer:

i am guessing for reflection but not so sure

Answer:

In a healthy eye, the cornea bends or refracts light rays so they focus precisely on the retina in the back of the eye.  The back of the eye is lined with a thin layer of tissue containing millions of photoreceptor (light-sensitive) cells. This is the retina, where light rays focus into an upside-down image.

Explanation:

Answer:

Kinetic.

Explanation:

Temperature can be defined as a measure of the degree of coldness or hotness of a physical object. It is measured with a thermometer and its units are Celsius (°C), Kelvin (K) and Fahrenheit (°F).

Kinetic energy can be defined as an energy possessed by an object or body due to its motion.

Generally, temperature measures the average kinetic energy of particles in a particular substance. Also, when the temperature of a particular substance is greater than absolute zero, it is said to possess kinetic energy.

Additionally, conduction involves the transfer of electric charge or thermal energy due to the movement of particles.

In the process of heat conduction, thermal energy is usually transferred from fast moving particles to slow moving particles during the collision of these particles. Also, thermal energy is typically transferred between objects that has different degrees of temperature and materials (particles) that are directly in contact with each other but differ in their ability to accept or give up electrons.

The term "temperature" refers to a measurement of how cold or hot an actual physical object is. It is measured with a thermometer, which gives readings in Celsius, Kelvin, and Fahrenheit (°C, K, and °F).

Thus, An energy that an object or body possesses as a result of its motion is known as kinetic energy.

The average kinetic energy of the particles in a given substance is often measured by temperature. Additionally, a substance is said to have kinetic energy when its temperature is higher than absolute zero.

Conduction also entails the transmission of heat or electric charge as a result of the motion of particles. Thermal energy is typically transmitted from quickly moving particles to slowly moving particles during a collision in the process of heat conduction.

Thus, The term "temperature" refers to a measurement of how cold or hot an actual physical object is. It is measured with a thermometer, which gives readings in Celsius, Kelvin, and Fahrenheit (°C, K, and °F).

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Answer:

C) Kinetic energy

Explanation:

Since the skateboarder is ALREADY going down the hill, he is using kinetic energy.

Kinetic energy is when you're moving and in motion

Answer:

a sample of water boils and releases gas.

Answer:

The answer to the question is 7200

Answer:

I think its d

Explanation:

I'm not sure I'm sorry if I'm wrong

Answer:

4.78 degrees east of north

Explanation:

The computation is shown below:

The Speed of the plane is 120 m/s

The Speed of the wind is 10 m/s

Sin [tex]\theta[/tex] = 10 ÷ 120

= 1 ÷ 12

= 0.0833

Now

\theta = 4.78 degrees

It should be 4.78 degrees east of north

Since the wind blows due west plane and it should fly east of north to  reach at the final destination