A force of 2 lb stretches a spring 1 ft. An 8-lb weight is attached to the end of the spring and released 4in. above the equilibrium position from rest. If the medium offers a resistance to the motion of the weight numerically equal to 3/2 times the instantaneous velocity, find the equation of motion. Determine if the system is underdamped, overdamped, or critically damped.
Answers
The solution to the differential equation is then:
x(t) = 0.25e⁽⁻⁶·⁰²⁵t⁾ cos(2.181t)
And the system is underdamped because the roots of the characteristic equation have a non-zero imaginary part.
The force required to stretch a spring is directly proportional to the amount the spring is stretched. This relationship is known as Hooke’s Law. It can be expressed mathematically as:
F = -kx
Where F is the force applied to the spring, x is the displacement of the spring from its equilibrium position, and k is the spring constant.
For a mass-spring system under the influence of a damping force, the differential equation governing the system is:
m(d2x/dt2) + c(dx/dt) + kx = 0
where m is the mass of the object attached to the spring, c is the damping coefficient, and k is the spring constant.
The given force of 2 lb stretches the spring by 1 ft, so the spring constant is given by k = F/x = 2/1 = 2 lb/ft.
The 8-lb weight is released 4 in. above the equilibrium position, which is 0.25 ft. The initial displacement is therefore x(0) = 0.25 ft, and the initial velocity is v(0) = 0. The damping force is given by f_d = -3/2v. Using the values given, the differential equation for the system is:
m(d2x/dt2) + c(dx/dt) + kx = 0 (1)
The values of m, k, and c are given by
m = 8/g = 8/32.2 = 0.248 kg
k = 2/lb/ft * 0.4536 kg/lb * 0.3048 m/ft = 0.294 kg/sm = 0.248 kgc = 3/2
The equation of motion is then:
d2x/dt2 + 12.05dx/dt + 1.186x = 0 (2)
where we have substituted the values of m, c, and k into equation (1).
The characteristic equation is:
r2 + 12.05r + 1.186 = 0 (3)
Solving for the roots of the characteristic equation, we find:
r = (-12.05 ± √(12.052 - 4(1.186)))/2= -6.025 ± 2.181i
The roots are complex conjugates, so the solution to the differential equation can be written as:
x(t) = e⁽⁻⁶·⁰²⁵t⁾(C₁ cos(2.181t) + C₂ sin(2.181t)) (4)
The initial displacement and velocity are given by x(0) = 0.25 and v(0) = 0.
Substituting these values into equation (4) and taking the derivative, we get:
x(0) = C₁ = 0.25dx/dt|t=0 = -6.025
C₂ = 0
Solving for C₁ and C₂, we get:
C₁ = 0.25C2 = 0
The solution to the differential equation is then:
x(t) = 0.25e⁽⁻⁶·⁰²⁵t⁾ cos(2.181t) (5)
The system is underdamped because the roots of the characteristic equation have a non-zero imaginary part.
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Related Questions
Which type or types of motion is particle motion in surface waves? responses longitudinal longitudinal transverse transverse a combination of longitudinal and transverse a combination of longitudinal and transverse neither longitudinal nor transverse
Answers
A combination of longitudinal and transverse motion is particle motion in surface waves.
The displacement of particles in a transverse motion is perpendicular to the direction of wave propagation. A one-dimensional transverse plane wave moving from left to right can be seen in the animation down below. The particles don't follow the wave; instead, as it passes them by, they just oscillate up and down around their particular equilibrium locations. Watch the movement of one particular particle.
Transverse waves in an earthquake include the S waves (Secondary waves). Waves that include longitudinal and transverse motions are known as S waves, and they travel more slowly than P waves, arriving behind them by a few seconds. The particles move in clockwise rings when a wave moves through the waver. As you go deeper into the water, the circles' radius gets smaller.
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Part E Now convert the time from seconds to years. This value is the approximate age of the universe. Write the age in scientific notation. Use the conversion, 1 year = 3.154 × 107 seconds.
Answers
Explanation:
The age of the universe is 13.8 billion years.
We know that,
1 billion years = 10⁹ years
So,
\(13.8\ \text{billion years}=13.8\times 10^9\ \text{years}\)
We need to convert the age of the universe to the scientific notation.
Since, \(1\ \text{year}=3.154\times 10^7\ s\)
So,
\(13.8\times 10^9\ \text{years}=13.8\times 10^9\times 3.154 \times 10^7\\\\=4.35\times 10^{17}\ s\)
So, the age of the universe is \(4.35\times 10^{17}\ s\).
Answer:
So the universe is approximately 1.34 x 10^10 years old
Explanation:
Edmentum
A 15- cm -tall closed container holds a sample of polluted air containing many spherical particles with a diameter of 2. 5 μm and a mass of 1. 9 × 10 −14 kg. How long does it take for all of the particles to settle to the bottom of the container
Answers
To calculate the time it takes for all the particles to settle to the bottom of the container, we can use Stokes' law, which relates the settling velocity of a particle in a fluid to its size and density.
Stokes' law states that the settling velocity (v) of a spherical particle in a fluid is given by the equation:v = (2/9) * (g * r^2 * (ρp - ρf)) / η, where g is the acceleration due to gravity, r is the radius of the particle, ρp is the density of the particle, ρf is the density of the fluid, and η is the dynamic viscosity of the fluid.
Given:
Height of the container, h = 15 cm = 0.15 m,
Diameter of the particles, d = 2.5 μm = 2.5 × 10^(-6) m,
Mass of the particles, m = 1.9 × 10^(-14) kg.
First, we need to find the radius of the particles:
r = d/2 = 2.5 × 10^(-6) / 2 = 1.25 × 10^(-6) m.
Assuming air as the fluid, we can use the following values:
Density of air, ρf = 1.2 kg/m³
Dynamic viscosity of air, η = 1.8 × 10^(-5) Pa·s
calculate the settling velocity of a single particle using Stokes' law:
v = (2/9) * (g * r^2 * (ρp - ρf)) / η
= (2/9) * (9.8 * (1.25 × 10^(-6))^2 * (1.9 × 10^(-14) - 1.2)) / (1.8 × 10^(-5))
≈ 7.59 × 10^(-6) m/s.
To calculate the time it takes for a particle to settle from the top to the bottom of the container, we can use the equation:
t = h / v
= 0.15 / (7.59 × 10^(-6))
≈ 19751 seconds.
Therefore, it would take approximately 19751 seconds for all the particles to settle to the bottom of the container.
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Suppose the Sun suddenly shrinks in size, but the mass remains the same. According to the law of conservation of angular momentum, what would happen? Group of answer choices The Sun would rotate slower than it does now The rotation of the Sun will stay the same The force of gravity between the Sun and the planets will decrease The Sun would rotate faster than it does now The Sun's angular size in the sky will stay the same
Answers
According to the law of conservation of angular momentum, if the Sun suddenly shrinks in size, but the mass remains the same, then the Sun would rotate faster than it does now. This is because the angular momentum of the Sun is constant, so if the size decreases, then the rotation must increase to maintain the same level of angular momentum.
The law of conservation of angular momentum states that the total angular momentum of a system remains constant if no external torque is acting on it.
This means that if the Sun suddenly shrinks in size, then the total angular momentum of the system (Sun + planets) must remain constant.
Since the mass of the Sun remains the same, its moment of inertia must decrease when it shrinks in size, which means that its angular velocity must increase in order to maintain the same level of angular momentum.
Therefore, the Sun would rotate faster than it does now if it suddenly shrinks in size while its mass remains the same.
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The image produced by a concave lens is always virtual. Is this true or false?
Answers
  do batteries produce or make energy
Answers
Answer:
Batteries are intended to store chemical energy and then it converts into electricidal energy. So overall, batteries produce energy.
Explanation:
a rotating fan completes 1200 revolutions every minute. consider the tip of a blade, at a radius of 0.15 m. (a) through what distance does the tip move in one revolution? what are (b) the
Answers
The angular velocity of the tip of the blade is angular velocity = 2 * π * 1200 rotations / minute = 7539.82 rad/min.
(a) To find the distance the tip of the blade moves in one revolution, we can use the circumference of a circle:
distance = 2 * π * radius = 2 * π * 0.15 m = 0.94 m
So, the tip of the blade moves a distance of 0.94 meters in one revolution.
(b) To find the linear velocity of the tip of the blade, we can use the formula for linear velocity:
linear velocity = distance / time = 2 * π * radius * rotations / minute
Since the fan completes 1200 revolutions per minute, the linear velocity is:
linear velocity = 2 * π * 0.15 m * 1200 rotations / minute = 94.25 m/s
To find the angular velocity of the tip of the blade, we can use the formula for angular velocity:
angular velocity = 2 * π * rotations / minute
So, the angular velocity of the tip of the blade is:
angular velocity = 2 * π * 1200 rotations / minute = 7539.82 rad/min.
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Which statement best describes the direction of heat flow by conduction between two samples of the same material? heat flows from faster molecules to slower molecules when they are near. Heat flows from faster molecules to slower molecules when they collide. Heat flows from slower molecules to faster molecules when they are near. Heat flows from slower molecules to faster molecules when they collide.
Answers
Answer:
B
Explanation:
The statement that best describes the direction of heat flow by conduction between two samples of the same material is that heat flows from faster molecules to slower molecules when they collide.
How does heat flow by conduction?Heat refers to the difference in temperature between two substances. Heat flows in the direction of an hot substance to cold.
However, in a hot substance, the particles move faster than that of a cold substance.
Therefore, the statement that best describes the direction of heat flow by conduction between two samples of the same material is that heat flows from faster molecules to slower molecules when they collide.
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how does adding batteries to a system affect its energy?
Answers
Sonar is a device that uses reflected sound waves to measure underwater depths. If a sonar signal has a frequency of 288 Hz and the speed of sound in water is 1450 m/s, what is the wavelength of the sonar signal?
Answers
Which of these objects converts light to chemical energy?
A. ethanol plant
B. light bulb
C. corn
D. solar cell
Answers
d)A solar cell is an object that converts light into chemical energy.
It works by using sunlight to convert photons into electrons, which are then used to generate electricity. Solar cells can be used to power a variety of objects, such as calculators, phones, and other electronic devices. They can also be used to generate electricity for homes and businesses. Solar cells are made of special materials that allow them to capture light and convert it into energy. When sunlight hits the solar cell, electrons in the material become excited and move freely, creating an electric current. This current is then used to produce electricity.
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is anyone good at physics who can help me??
A dog trots from -11 m to 4 m in 14 s . whats is its velocity ?
Answers
Answer:
1.07 m/s
Explanation:
This is a simple speed distance time question.
first, we need to determine the distance between -11 and 4.
(think of a number line).
We add 4 to 11 using absolute value (i.e taking away negatives), and we get 15 m, our distance. We are attempting to find speed, and we have our time.
Speed is distance/time.
Now we have our equation, we plug in our variables.
speed (aka velocity)= 15/14.
speed= 1.07142857143
If we round to 2 place values, our speed is 1.07 m/s.
The velocity of the dog's trots is 1.07 m/s after using the speed-distance-time relationship.
What is the distance?Distance is a numerical representation of the distance between two items or locations. Distance refers to a physical length or an approximation based on other physics or common usage considerations.
It is given that:
A dog trots from -11 m to 4 m in 14 s.
As we know, the speed-distance-time relationship:
Speed = distance/time
The distance = 4 - (-11)
The distance = 4 + 11
The distance = 15 m
Time = 14 seconds
Speed = 15/14
The velocity of the dog's trots is:
Speed = 1.07 m/second or 1.07 m/s
Thus, the velocity of the dog's trots is 1.07 m/s after using the speed-distance-time relationship if the dog trots from -11 m to 4 m in 14 s.
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State two units bigger than kilogram and how are they related to kilogram
Answers
Answer:
quintal and metric tone.
Explanation:
1 quintal= 100kg
1 metric =1000kg
quintal measures mass of cars, trucks etc, but metric measures bigger things like cargo ships
hope it helps
because relative humidity is dependent on air temperature, it is ____________ in the cooler mornings and ____________ in the warmer afternoons.
Answers
Because relative humidity is dependent on air temperature, it is generally higher in the cooler mornings and lower in the warmer afternoons.
This is because warm air can hold more water vapor than cool air, so as the temperature rises in the afternoon, the air has the capacity to hold more moisture. This means that the same amount of water vapor in the air will result in a lower relative humidity when the temperature is higher. Conversely, in the cooler mornings, the air has less capacity to hold moisture, so the same amount of water vapor in the air will result in a higher relative humidity.
It's important to note, however, that this pattern is not always true and can be affected by other factors such as wind, precipitation, and geography. Additionally, relative humidity is not the same as absolute humidity, which refers to the actual amount of water vapor in the air, regardless of temperature.
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Calculate the velocity of an apple that falls freely from rest and drops for 3.5 seconds.
(brainliest and 100 points)
Answers
Answer:
initial velocity(u)=0
time(t)=3.5sec
acceleration (a)=9.8m/s²
final velocity (v)=?
Explanation:
we have
v=u+at=0+9.8×3.5=34.3m/s
Superman is flying high above the earth on a straight horizontal path. His wallet falls
out of his tights and falls towards Earth, at some point the wallet's velocity is 45.0
m/s [E 53 S].
a) At the point in question, what were the horizontal and vertical components
of the wallet's velocity?
b)
At the point in question, for how long had the wallet been falling?
c) At the point in question, how far horizontally and vertical had the wallet
traveled?
Answers
(a) The horizontal and vertical components of the wallet's velocity is 45 m/s and 53 m/s respectively.
(b) The wallet had been falling for 0 seconds at the point in question.
(c) At the point in question, the wallet had traveled 0 meters horizontally and 0 meters vertically.
What are the component of the velocity?At the point in question, the horizontal component of the wallet's velocity is 45.0 m/s towards the east (or "to the right" from Superman's perspective) and the vertical component of the velocity is 53.0 m/s towards the south (or "downwards" from Superman's perspective).
Assuming that the wallet was initially at rest relative to Superman, and neglecting air resistance, we can use the fact that the vertical motion of the wallet is equivalent to that of an object in free fall.
Using the formula y = 1/2(g)t^2,
where;
y is the vertical displacement g is the acceleration due to gravity and t is the time elapsed,we can solve for t to get:
t = √(2y/g)
t = √(2 x 0/9.8)
t = 0 seconds
c) Again neglecting air resistance, we can use the equations of motion for constant acceleration to determine the horizontal and vertical displacements of the wallet during the time it falls. Since the horizontal velocity of the wallet remains constant at 45.0 m/s, and the vertical acceleration is constant at -9.8 m/s^2, we can use the following formulas:
y = v0yt + 1/2(a) t^2
x = v0xt
where;
v0x is the initial horizontal velocity, = 45.0 m/sSubstituting in the known values, we get:
y = 0 + 1/2(-9.8)(0)^2
y = 0 meters
x = 45.0 (0) = 0 meters
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A car is driving down the highway at a 50 m/s. Suddenly, the driver sees an accident and slams on his breaks, coming to a complete stop in 10 seconds. How far did the car travel before coming to a stop?
Answers
Given parameters:
Speed of the car = 50m/s
Time = 10s
Unknown:
Distance traveled before the car stopped = ?
We need to establish speed - time - distance relationship.
Speed is the rate of change of distance with time. It is a scalar quantity .
Mathematically;
Speed = \(\frac{distance}{time}\)
Distance = speed x time
Input the parameters and solve;
Distance = 50 x 10 = 500m
The car traveled 500m before it stopped.
as you stand on the floor, does the floor exert an upward force against your feet? how much force does it exert?
Answers
The upward force exerted by the floor is equal to the downward force that you apply to the floor (Your weight).
The net force acting on an object is equal to the rate at which its momentum varies over time, according to Newton's second law in its original formulation. This law suggests that an object's acceleration is directly proportional to the net force acting on it, is in the direction of the net force, and is inversely proportional to the mass of the object if the object's mass is constant.
Thrust, which increases an object's velocity; drag, which decreases an object's velocity; and torque, which causes changes in an object's rotational speed, are all concepts related to force.
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A 5.0 Newton force is applied to the right of a 6.0 kg object to move across a smooth surface. The coefficient of friction is zero. a. Draw and label all of the forces acting on the object b. Calculate the acceleration of the object
Answers
Answer:
\(0.833\ \text{m/s}^2\)
Explanation:
m = Mass of block = 6 kg
F = Force on block = 5 N
\(\mu\) = Coefficient of friction = 0
a = Acceleration of the block
From Newton's second law we have
\(F=ma\\\Rightarrow a=\dfrac{F}{m}\\\Rightarrow a=\dfrac{5}{6}\\\Rightarrow a=0.833\ \text{m/s}^2\)
The acceleration of the block is \(0.833\ \text{m/s}^2\)
Over 4 seconds, a car's momentum decreases by 1000 kg m/s how much force did it take to make this happen?
Answers
Answer:
250N
Explanation:
Given parameters:
Time = 4s
Momentum = 1000kgm/s
Unknown:
Force = ?
Solution:
To solve this problem, we use Newton's second law of motion;
Ft = Momentum
F is the force
t is the time
So;
F x 4 = 1000kgm/s
F = 250N
Jenny is going to California. Her trip will be a total distance of 4,698.5 Km. How many meters is that in total?
Answers
Answer:
ytytyt
Explanation:ytytytyt
There are 1000m in 1km. In order to turn meters in kilometers, you need to multiply km by 1000.
What percentage of the starting matter in our solar system went into the formation of our sun?
Answers
Answer:
Eventually the pressure in the core was so great that hydrogen atoms began to combine and form helium, releasing a tremendous amount of energy. With that, our Sun was born, and it eventually amassed more than 99 percent of the available matter
The density of pure diamond is 3.5 g/cm3. what is the volume of a diamond with a mass of 0.25 g?
Answers
The volume is 0.21 cm³.
The density of the pure diamond is: ρdiamond = 3.5g/cm³
The mass of the diamond is: m = 0.75g
Write the expression for the density of the diamond.
ρdiamond = m/V
Here, the volume is V.
Substitute the known values.
(3.5g/cm³) = 0.75gV
V = 0.75g/3.5g/cm³
V = 0.2142857143cm³
V ≃ 0.21cm³
Thus, the volume is 0.21 cm³.
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Part A
Choose the correct SI form for kN/ms.
a. MN/s
b. mN/s
c. GN/s
d. N/s
Answers
mN/s is the correct SI form for kN/ms.
Hence, Option B is correct.
Système Internationale, or SI, is the abbreviation for our metric system of measurements. It is a system that has been universally standardized, providing a common language between countries and among the various branches of science and technology. The International System of Units (SI), also known as the French Système International d'Unités, is a weights-and-measures system used internationally that extends the metric system of units. The metric system, also known as the International System of Units (SI), is the accepted unit of measurement on a global scale. The International Treaty of the Meter was signed on May 20, 1875, in Paris by seventeen nations, including the United States, and is now recognized as World Metrology Day worldwide.
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Water has a specific heat capacity of 4200 J/kg/°C. Calculate the temperature change produced when 0.35 kg of water is heated using 9600 J.
Answers
The temperature change produced when 0.35 kg of water is heated using 9600 J is 6.34 °C.
What is Specific Heat Capacity?
Specific heat capacity is the amount of heat energy required to raise the temperature of one unit of mass of a substance by one degree Celsius (or one Kelvin). It is a measure of the substance's ability to store heat energy. The specific heat capacity is a physical property of the substance and is usually denoted by the symbol c.
The amount of heat required to raise the temperature of a substance is given by the formula:
Q = mcΔT
where Q is the amount of heat energy transferred, m is the mass of the substance, c is the specific heat capacity of the substance, and ΔT is the change in temperature.
In this problem, we are given:
m = 0.35 kg (mass of water)
c = 4200 J/kg/°C (specific heat capacity of water)
Q = 9600 J (amount of heat energy transferred)
We can rearrange the formula to solve for ΔT:
ΔT = Q/(mc)
Substituting the given values, we get:
ΔT = (9600 J) / (0.35 kg x 4200 J/kg/°C)
ΔT = 6.34 °C
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does the force change if the density change in a object
Answers
Answer:
No
Explanation:
It doesn't depend on the density
Answer:No. If you take all your masses and and squeeze them into a tiny volume,
Miles and Joselin are riding a two person long board down an extremely dangerous hill while Chloe watches in horror and Talia in delight. At the end of the hill Alex and Jason are holing a crash pad to stop the dynamic duo from falling off the cliff. If the riders have a combined mass of 200 kg and an acceleration of 30 m/s^2, what Force will they crash the mattress with? Use the correct Force for full credit.
Answers
Answer:
They will crash with the force of 6,000N
Explanation:
Force is equal to mass time acceleration. When we multiply 200 kg, the mass, and 30 m/s2, the acceleration, the answer is 6,000.
what relationship between electron orbits and light emission did bohr postulate? what relationship between electron orbits and light emission did bohr postulate? the energy difference between two electron orbits would equal the energy of an emitted photon. the frequency of electrons circling a nucleus was equal to the frequency of the emitted light. the energy of an electron orbit was equal to the energy of the emitted light. the sum of the energies of two electron orbits would equal the energy of an emitted photon.
Answers
Bohr's theory postulated that there was a relationship between electron orbits and light emission. According to his theory, the energy difference between two electron orbits would equal the energy of an emitted photon.
This means that when an electron jumps from a higher orbit to a lower one, it releases energy in the form of a photon. Furthermore, Bohr proposed that the frequency of electrons circling a nucleus was equal to the frequency of the emitted light. In other words, the energy of the photon is related to the frequency of the light.
Finally, Bohr suggested that the energy of an electron orbit was equal to the energy of the emitted light. This means that the energy of the photon corresponds to the difference in energy between the two electron orbits.
Overall, Bohr's theory provided a framework for understanding the relationship between electron orbits and light emission, and paved the way for further advances in the field of atomic physics.
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why all large astronomical telescopes are reflectors because?
Answers
Large astronomical telescopes are reflectors because they use mirrors to collect and focus light from distant astronomical objects.
The advantage of reflector telescopes is that they can capture more light than refractor telescopes due to their larger aperture, resulting in clearer images and more accurate measurements of celestial objects. The larger apertures also allow for a much wider field of view, which allows for larger portions of the night sky to be viewed at once.
Additionally, reflector telescopes have fewer optical elements than refractors, meaning there is less light loss and therefore the image resolution is higher. Reflector telescopes are also often much more affordable than refractors due to their simpler construction.
Reflector telescopes are the best choice for large astronomical telescopes due to their increased light gathering power, wide field of view, and cost effectiveness. All large astronomical telescopes are reflectors because of their high resolution and larger aperture. The larger aperture provides more light-gathering power, which increases the resolution of the telescope.
In addition, the reflecting surface can be shaped to a high degree of accuracy, which allows for a higher level of precision in focusing the telescope. As a result, the reflecting telescope can see objects that are much farther away than a refracting telescope.
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if two firecrackers produce a sound level of 86 db when fired simultaneously at a certain place, what will be the sound level if only one is exploded? [hint: add intensities, not db 's.] express your answer to two significant figures and include the appropriate units.
Answers
The sound level of 86dB when fired simultaneously will have sound level of 82.9898dB.
Theory-
According to the Beta factor model, the common cause will have an equal impact on every member of a component group that shares a common cause.
The beta factor model is simple to use, models common cause failures, and only requires one parameter to be determined.
A drawback of the beta factor model is that it is impossible to quantify the failure of k–m components within a common cause component group.
As a result of multiplying the partial beta factors, the beta factor is calculated.
Mathematics-
\(\beta =10log\frac{2I}{i} \\\\\beta 1=10log\frac{I}{i} \\\\\beta =10log2 +\beta 1\\\\\beta 1=\beta -10log2\)
Calculation-
\(\beta 1=86-10log2\\\\\beta 1=86-3.0102\\\\\beta 1= 82.9898dB\)
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