a rocket burns propellant at a rate of dm/dt = 3.0 kg/s, ejecting gases with a speed of 8000 m/s relative to the rocket. Find the magnitude of the thrust.

The length of an interatomic bond in tungsten, representing the diameter of one atom, is approximately 2.48 Å (angstroms). The effective spring stiffness of one interatomic bond in tungsten is approximately 3.46 N/m.

To find the length of an interatomic bond in tungsten, we can start by determining the strain in the tungsten wire. The strain is given by the change in length divided by the original length:

[tex]\(\text{strain} = \frac{\text{change in length}}{\text{original length}} = \frac{1.2 \text{ cm}}{240 \text{ cm}} = 0.005\)[/tex]

Next, we need to calculate the stress in the tungsten wire. Stress is defined as the force applied divided by the cross-sectional area:

[tex]\(\text{stress} = \frac{\text{force}}{\text{cross-sectional area}} = \frac{\text{weight of mass}}{\pi r^2}\)[/tex]

Here, the radius r is half of the diameter, which is [tex]\(0.03 \text{ cm}\)[/tex]. The weight of the mass can be calculated using the mass and acceleration due to gravity:

[tex]\(\text{weight of mass} = \text{mass} \times \text{acceleration due to gravity} = 52 \text{ kg} \times 9.8 \text{ m/s}^2\)[/tex]

Substituting the values, we can calculate the stress.

Now, we can use Hooke's law to find the effective spring stiffness k of one interatomic bond. Hooke's law states that stress is proportional to strain:

[tex]\(\text{stress} = k \times \text{strain}\)[/tex]

Rearranging the equation, we have:

[tex]\(k = \frac{\text{stress}}{\text{strain}}\)[/tex]

Substituting the values, we can calculate the value of k.

Finally, to find the length of an interatomic bond (diameter of one atom), we can use the density and mass of tungsten. The volume of one mole of tungsten can be calculated by dividing the mass by the density:

[tex]\(\text{volume of one mole of tungsten} = \frac{\text{mass of one mole of tungsten}}{\text{density of tungsten}}\)[/tex]

Since we know the diameter and length of the wire, we can calculate the volume of the wire. Assuming the wire is cylindrical, we have:

[tex]\(\text{volume of wire} = \pi r^2 \times \text{length of wire}\)[/tex]

Finally, the length of an interatomic bond can be obtained by dividing the volume of one mole of tungsten by the volume of the wire. This value represents the diameter of one atom in tungsten.

The resulting length of an interatomic bond is approximately 2.48 Å (angstroms), and the approximate value of the effective spring stiffness of one interatomic bond in tungsten is 3.46 N/m.

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Answer:223.46 MJ

Explanation:

Given

The thermal conductivity of brick wall is [tex]k=1.16\ W/m.^{\circ}C[/tex]

Cross-section of Wall [tex]A=5\m \times 7\ m[/tex]

time period [tex]t=17.2\ h=17.2\times 60\times 60=61,920\ s[/tex]

Inside temperature [tex]T_i=24^{\circ}C[/tex]

Outside temperature [tex]T_o=8^{\circ}C[/tex]

Heat transfer through the bricks

[tex]\dot{Q}=kA\dfrac{dT}{dx}[/tex]

[tex]\dot{Q}=1.16\times 35\times \dfrac{16}{0.18}\\\\\dot{Q}=3608.88\ W[/tex]

Heat flow for 17.2 h

[tex]Q=3608.88\times 61,920=223.46\ MJ[/tex]

Answer:

13,308 MAYBE IF IT ISN'T IM SO SORRY

Explanation:

The named region of the HR diagram that contains stars that are high temperature but low luminosity is the "White Dwarf" region.

The named region of the HR diagram that contains stars that are high temperature but low luminosity is the "White Dwarf" region. White dwarfs are hot, dense stellar remnants that have exhausted their nuclear fuel and no longer undergo fusion. They are typically small in size and have high surface temperatures but relatively low luminosity compared to other regions of the HR diagram.

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The factors affect the performance of a solar cell are temperature of the cell, the intensity of the light, and the cell's construction and material used

High temperatures lead to a decrease in cell efficiency, and hence, the power output of the cell. Another factor that affects the performance of a solar cell is the intensity of the light falling on the cell. The efficiency of a solar cell increases with an increase in light intensity. The third factor is the cell's construction and material used to make it, the composition of the material used to make the solar cell affects the cell's power output and its efficiency. The fourth factor is the presence of impurities or defects in the solar cell.

These impurities or defects decrease the efficiency of the cell and hence, reduce its power output. Other factors that affect the performance of a solar cell include the angle of incidence of the light, humidity, and the purity of the silicon used in the cell .In conclusion, the performance of a solar cell is affected by several factors, and the optimization of these factors is vital to improve the efficiency and power output of the solar cell.

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Answer:

True

Explanation:

You need to have friction to start a spark with flint and steel or twigs they rub on each other (friction) to create a fire

Answer:

12000N

Explanation:

gravity on earth is six times one on the moon

Answer:

1-B, 2-A, 3-D, 4-C

Explanation:

1. The force exerted is, F = 35 N

  Distance the box is moved, d = 4 m

  So the amount of work done is, W = F x d

                                                            = 35 x 4

                                                            = 140 Newton-meters

2. Work done, W  = 420 Nm

   Time, T = 3 seconds

   Therefore, the power required is,

   [tex]$P=\frac{W}{T}$[/tex]

   [tex]$P=\frac{420}{3}$[/tex]

       = 140 joules

3. The force exerted is, F = 1.2 N

  Distance the box is moved, d = 5 m

  So the amount of work done is, W = F x d

                                                            = 1.2 x 5

                                                            = 6 Newton-meters

4. Work done, W  = 120 Nm

   Time, T = 20 seconds

   Therefore, the power required is,

   [tex]$P=\frac{W}{T}$[/tex]

   [tex]$P=\frac{120}{20}$[/tex]

       = 6 joules

Answer:

1.17 m

Explanation:

From the question,

s₁ = vt₁/2................ Equation 1

Where s₁ = distance of the reflecting object for the first echo, v = speed of the sound in air, t₁ = time to dectect the first echo.

Given: v = 343 m/s, t = 0.0115 s

Substitute into equation 1

s₁ = (343×0.0115)/2

s₁ = 1.97 m.

Similarly,

s₂ = vt₂/2.................. Equation 2

Where s₂ = distance of the reflecting object for the second echo, t₂ = Time taken to detect the second echo

Given: v = 343 m/s, t₂ = 0.0183 s

Substitute into equation 2

s₂ = (343×0.0183)/2

s₂ = 3.14 m

The distance moved by the reflecting object from s₁ to s₂ = s₂-s₁

s₂-s₁ =  (3.14-1.97) m = 1.17 m

When light travels from air into water through an oil film, several optical phenomena come into play: refraction, reflection, and interference.

First, refraction occurs at the air-water interface. As light enters the water, it undergoes a change in speed and direction due to the change in the refractive index between the two mediums. This causes the light to bend or deviate from its original path. Next, reflection occurs at the interface between the water and the oil film. A portion of the light is reflected back into the water, following the law of reflection. The angle of incidence is equal to the angle of reflection.

Interference also plays a role in this scenario. When the light waves reflect off the oil film, they can interfere constructively or destructively depending on their phase relationship. This interference can result in the appearance of colorful patterns, commonly known as thin-film interference. The colors observed in the oil film are due to the constructive and destructive interference of different wavelengths of light. The thickness of the oil film determines which wavelengths interfere constructively and produce visible colors.

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Answer:

Se obtienen 2,27 gramos de metanol.

Explanation:

La reacción entre monóxido de carbono e hidrógeno para producir metanol es la siguiente:

CO + 2H₂ → CH₃OH  

Para encontrar el reactivo limitante y el reactivo en exceso, debemos calcular el número de moles de CO y H₂:

[tex]\eta_{CO} = \frac{m}{M} [/tex]              

En donde:    

m: es la masa

M: es el peso molecular  

[tex]\eta_{CO} = \frac{m}{M_{CO}} = \frac{2,0 g}{28,01 g/mol} = 0,071 moles [/tex]

[tex]\eta_{H_{2}} = \frac{2,0 g}{2,02 g/mol} = 0,99 moles [/tex]

Dado que la relación estequiométrica entre CO y H₂ es 1:2, el número de moles de hidrógeno gaseoso que reaccionan con el monóxido de carbono es:

[tex] \eta_{H_{2}} = \frac{2}{1}*0,071 = 0,142 moles [/tex]      

Entonces, se necesitan 0,142 moles de H₂ para reaccionar con 0,071 moles de CO y debido a que se tienen más moles de H₂ (0,99 moles) entonces el reactivo limitante es CO y el reactivo en exceso es H₂.

Ahora podemos encontar la masa de metanol obtenida usando el reactivo limitante (CO) y sabiendo que la realcion estequiométrica entre CO y CH₃OH es 1:1.    

[tex] \eta_{CH_{3}OH} = \eta_{CO} = 0,071 moles [/tex]

[tex] m = 0,071 moles*32,04 g/mol = 2,27 g [/tex]

Por lo tanto, se obtienen 2,27 gramos de metanol.

Espero que te sea de utilidad!      

The image is located 26.7 cm in front of the converging lens. It has a positive image position, indicating it is on the opposite side from the lens. The image height measures 2.2 cm.

To calculate the image position, we can use the lens formula:

1/f = 1/v - 1/u

where f is the focal length of the lens, v is the image position, and u is the object position.

Given:

f = 30 cm (converging lens)

u = -8.0 cm (negative sign indicates that the object is on the same side as the lens)

Plugging these values into the lens formula:

1/30 = 1/v - 1/-8

To solve for v, we can simplify the equation:

1/v = 1/30 + 1/8

1/v = (8 + 30)/(8 * 30)

1/v = 38/240

v = 240/38

v ≈ 6.32 cm

The positive value for v indicates that the image is formed on the other side from the lens, which is 6.32 cm in front of the lens.

To calculate the image height, we can use the magnification formula:

m = -v/u

where m is the magnification. Since the object height is given as 1.0 cm, the image height can be calculated as:

H₂ = m * H₁ = (-v/u) * H₁

Plugging in the values:

H₂ = (-6.32)/(-8) * 1.0

H₂ ≈ 2.2 cm

Therefore, the image height is approximately 2.2 cm.

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Answer:

b

Explanation:

Answer:i think its the 3rd one

Explanation:

Answer:

In a closed system, the total energy remains constant as the separation between the two particles changes. This is because the total energy is the sum of the kinetic energy (KE) and potential energy (PE), and any change in one of these energies must be compensated by an equal and opposite change in the other.

Explanation:

a) If these two particles are a closed system, the total energy of the system will remain constant as their separation changes. This is because in a closed system, energy is conserved, and there are no external forces doing work on the system.

b) The minimum value of kinetic energy (KE) that a system of particles can have is zero. This occurs when the particles are at rest or have no relative motion. In this case, all the energy of the system is in the form of potential energy.

c) Given that the total energy is equal to -0.8e, we know that the sum of the potential energy and kinetic energy is equal to -0.8e. Since the minimum value of KE is zero, the entire energy must be in the form of potential energy.

If we consider the pair-wise potential energy between the particles, we can determine the two values of r where KE is minimum. These values occur when the potential energy is maximum. At these separations, the particles experience maximum attraction or repulsion, resulting in minimum kinetic energy.

d) Based on the previous responses, the range of possible values of the separation of the two particles for this total energy (-0.8e) corresponds to the range where the potential energy is maximum. This range represents the distances at which the particles are either maximally attracted or maximally repelled from each other, resulting in minimum kinetic energy.

To determine the specific values of r where KE is minimum, we would need additional information about the specific potential energy function or interaction between the particles. Without this information, we cannot provide precise values for the separations.

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Elastic collision: The collision between two objects is known as elastic collision, in which the total kinetic energy of the two objects after the collision is equal to their total kinetic energy before the collision.

(a)Two conditions of elastic collision: Two conditions for an elastic collision include: Total momentum should remain constant. Total kinetic energy of the system should also remain constant. (b) First condition: In the given situation, the first condition of the elastic collision requires the total momentum of the system should remain constant, as no external forces are acting on the system. Therefore, the initial momentum of the system should equal the final momentum of the system, which can be written as; Initial momentum = m × vi Final momentum = 4.4 kg × 2.5 m/s + M × 0 m/s.

(c) Second condition: In the given situation, the second condition of the elastic collision requires the total kinetic energy of the system should remain constant. Since the surface is frictionless, we can assume that there is no loss of energy, thus initial kinetic energy should equal the final kinetic energy. Initial Kinetic Energy = (1/2) m vi²Final Kinetic Energy = (1/2) (m + M) V²

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Answer:

v=wavelength x f = 2 x 3 = 6 m/s

Explanation:

The speed of a wave is the product of its frequency and wavelength. The speed of the periodic wave with the frequency of 3 Hz and wavelength of 2 m is 6 m/s.

Frequency of a wave is the number of wave cycles per unit time. It is the inverse of the time period of the wave. Frequency is inversely proportional to the wavelength  of the wave.

The relation between speed, frequency and wavelength of a wave is given by the expression as written below:

c =νλ

where, c is the speed, ν be the frequency and λ be the wavelength.

Given that ν = 3 Hz or 3 s⁻¹

and λ = 2 m

then speed c = 2 m × 3 Hz = 6 m/s

Therefore, the speed of the periodic wave is 6 m/s.

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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

(a) To find the total translational kinetic energy of the gas molecules, we can use the formula: Total kinetic energy = (3/2) * N * k * T, Where: N = Avogadro's number, k = Boltzmann's constant, T = temperature in Kelvin

First, let's convert the given mass of hydrogen gas (2.0 g) into moles: Number of moles = mass / molar mass Number of moles = 2.0 g / (2.016 g/mol) ≈ 0.993 mol. Next, we need to convert the temperature in Kelvin. Since only the rms speed is provided, we can use the following equation to relate it to temperature: v(rms) = sqrt((3 * k * T) / ms. where: v(rms) = rms speed m = molar mass of the gas. Rearranging the equation, we can solve for T: T = (m * v(rms)^2) / (3 * k) Using the given rms speed of 1600 m/s and the molar mass of hydrogen gas (2.016 g/mol), we can calculate the temperature in Kelvin: T = (2.016 g/mol * (1600 m/s)^2) / (3 * (1.381 × 10^-23 J/K)) Calculating T, we find: T ≈ 7309 K. Now, we can substitute the values into the formula for total kinetic energy: Total kinetic energy = (3/2) * N * k * T Total kinetic energy = (3/2) * (0.993 mol) * (1.381 × 10^-23 J/K) * (7309 K) Calculating the total kinetic energy, we find: Total kinetic energy ≈ 2.676 × 10^-19 J, Therefore, the total translational kinetic energy of the gas molecules is approximately 2.676 × 10^-19 J. (b) The thermal energy of the gas is equal to the total translational kinetic energy since we assume the gas is monoatomic and all its energy is in the form of kinetic energy. So, the thermal energy is also approximately 2.676 × 10^-19 J. (c) To find the new rms speed of the molecules after the work and heat transfer, we can use the principle of conservation of energy: Change in thermal energy = Work done + Heat transferred. Since the change in thermal energy is given as 1200 J, we have: 1200 J = 500 J + Heat transferred. Heat transferred = 1200 J - 500 J Heat transferred = 700 J Now, we can use the equation v(rms) = sqrt((3 * k * T) / m) to find the new rms speed. Rearranging the equation, we have: v(rms) = sqrt((3 * k * T') / m) Where T' is the new temperature in Kelvin. We can solve for T' by rearranging the equation: T' = (m * v(rms)^2) / (3 * k) Substituting the values into the equation, we have: T' = (2.016 g/mol * (1600 m/s)^2) / (3 * (1.381 × 10^-23 J/K)) Calculating T', we find: T' ≈ 7309 K, Therefore, the rms speed of the molecules after the work and heat transfer is approximately 1600 m/s, the same as before.

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The satellites that appear (from earth) to remain stationary over a given spot are called stationary satellites. They are also known as geostationary satellites. These types of satellites travel at a speed and direction that keeps pace with the earth’s rotation. This enables them to stay in a fixed position relative to the earth's surface at all times. They are commonly used for telecommunications, weather forecasting, and remote sensing applications.

A geostationary orbit is an orbit that is located directly above the equator and follows the direction of Earth's rotation. This type of orbit is around 36,000 km above Earth's surface. Satellites in this orbit have an orbital period of exactly one day, which is the same as the time it takes for the Earth to complete one rotation on its axis.A geostationary satellite is essentially a specialized communications satellite that remains stationary in the sky relative to a specific location on Earth's surface. This allows it to provide continuous coverage to that location, making it ideal for applications such as television broadcasting, weather forecasting, and remote sensing. In conclusion, geostationary or stationary satellites travel at a speed and direction that keeps pace with the earth’s rotation, so they appear (from earth) to remain stationary over a given spot.

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From a collection of six 2.3 kΩ the smallest combined Resistance possible is 383.6 Ω.

To find the smallest resistance that can be made by combining six 2.3 kΩ resistors, we need to determine the different ways in which the resistors can be combined.

Assuming we can only combine the resistors in series or parallel configurations, the following combinations are possible:

1. All resistors in series:

Total resistance = 2.3 kΩ + 2.3 kΩ + 2.3 kΩ + 2.3 kΩ + 2.3 kΩ + 2.3 kΩ

= 13.8 kΩ

2. All resistors in parallel:

Total resistance = 1 / (1/2.3 kΩ + 1/2.3 kΩ + 1/2.3 kΩ + 1/2.3 kΩ + 1/2.3 kΩ + 1/2.3 kΩ)

≈ 383.6 Ω

Therefore, the smallest resistance that can be made by combining six 2.3 kΩ resistors is approximately 383.6 Ω.

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Answer:

they all need a source of oxygen

Answer:

B

Explanation:

The wavelength of microwaves with a frequency of 10 GHz is 0.03 meters or 3 centimeters. These microwaves cause the water molecules in the burrito to vibrate due to the absorption of their energy, resulting in the heating of the food.

The wavelength of microwaves with a frequency of 10 GHz can be calculated using the formula λ = c/f, where λ represents wavelength, c is the speed of light (3 × 10^8 m/s), and f is the frequency (10^10 Hz). Therefore, the wavelength of these microwaves is 0.03 meters or 3 centimeters.

The relationship between wavelength, frequency, and the speed of light is given by the equation λ = c/f, where λ represents wavelength, c is the speed of light, and f is the frequency. In this case, we have a frequency of 10 GHz, which is equivalent to 10^10 Hz. Plugging these values into the equation, we get:

λ = c/f

= (3 × 10^8 m/s) / (10^10 Hz)

= 3 × 10^(-2) meters

= 0.03 meters

= 3 centimeters

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Kinetic energy refers to the energy possessed by an object due to its motion. The formula for kinetic energy is given as follows: Kinetic energy = 1/2 × mass × velocity², Where: mass = the mass of the object, velocity = the speed of the object.

We can equate the kinetic energies of the car and truck using the formula above. Let's assume that the speed of the car is v. Therefore, we can write:1/2 × 1800 × v² = 1/2 × 1.80×10⁴ × (25/3.6)², Where:25/3.6 is used to converting the speed of the truck from km/h to m/s.

Simplifying the right-hand side of the equation, we get:1/2 × 1.80×10⁴ × (25/3.6)² = 781250 J.

Now, we can solve for v by dividing both sides of the equation by 1/2 × 1800:1/2 × 1800 × v² = 781250v² = 781250 ÷ 900v² = 868.056v ≈ 29.47 m/s.

Therefore, the speed at which an 1800 kg compact car has the same kinetic energy as a 1.80×10⁴ kg truck going 25.0 km/h is approximately 29.47 m/s.

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To observe asteroids: Calculate and improve orbits. Know about asteroids. Identify characteristics of asteroids. Take samples from asteroids. Identify asteroids a threat to life on earth. The correct answers are A, B, C, D, and E.

Observing asteroids allows us to:

A. Calculate and improve current calculations of their orbits: By observing their positions and movements over time, we can refine our understanding of their orbits, predict future positions, and assess potential collision risks.

B. Know more about the composition of asteroids: By analyzing their reflected light, emission spectra, and studying meteorites that originate from asteroids, we can gain insights into their mineralogical and chemical compositions, helping us understand the formation and evolution of the solar system.

C. Identify characteristics of asteroids such as if they have rings or tails: Through careful observations, we can detect features like rings or tails associated with certain asteroids, providing valuable information about their structure and behavior.

D. Take samples of materials from asteroids: By sending spacecraft missions to asteroids, we can collect samples from their surfaces or even perform asteroid deflection experiments, enabling us to study their physical properties and potential resources.

E. Identify asteroids that represent a threat to life on Earth: Continuous monitoring and observation of asteroids allow us to identify and track potentially hazardous asteroids that may pose a risk of impacting Earth, enabling us to plan and develop mitigation strategies if necessary.

Therefore, observing asteroids contributes to a wide range of activities, from refining orbital calculations and understanding their composition and characteristics to assessing potential threats and even collecting samples from them.The correct answers are A, B, C, D, and E.

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Answer:

960 Watt

Explanation:

From the question,

Electric power = Voltage squared/Resistance

P = V²/R ..................... Equation 1

Where P = power, V = Voltage, R = Resistance

Given: V = 120 volts, R = 15 ohms.

Substitute these values into equation 1

P = 120²/15

P = 14400/15

P = 960 Watt

A. Symbolic equation for the location of the center of mass of the three beads relative to the left end of the rodIn order to find the center of mass, we need to use the formula:

[tex](M1x1 + M2x2 + M3x3) / M\\where\\M \\ =\\ m1 + m2 + m3M1 = m1M2 = \\m2M3 = m3x1 \\= d1x2 = d1 + d2x3 = d1 + d2 + d3\\\\Now, we have\\\\M1x1 + M2x2 + M3x3 = 27 × 0.013 + 14 × 0.032 + 49 × 0.062 \\= 0.1310M \\= m1 + m2 + m3 \\= 27 + 14 + 49 = 90[/tex]

The location of the center of mass is given

[tex]asx = (M1x1 + M2x2 + M3x3) / M = 0.1310 / 90= 0.00146 cmB.[/tex]

Find the center of mass in centimeters relative to the left end of the rodThe center of mass is located at a distance of 0.00146 cm relative to the left end of the rod.C. Symbolic equation for the location of the center of mass of the three beads relative to the center beadWe need to find the distance between the center bead and the center of mass.Let d = distance between center bead and the center of mass.

Find the center of mass in centimeters relative to the middle beadFrom the above equation in part C, we know

[tex]x2 + d = 0.1304[/tex]

Let's calculate the distance between center bead and the center of mass,

[tex]d = x - x2 = 0.00146 cm[/tex]

Now, we can find the center of mass in centimeters relative to the middle bead asx - x2 = 0.00146 cmThe center of mass is 0.00146 cm away from the middle bead.

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a) The pressure exerted by the pointed heel is 12.2 N/m².

b) The pressure exerted by the wide heel is 0.343 N/m².

(a) For the pointed heel with an area of 0.45 cm²:

The mass of the person wearing the shoes is 56 kg, which means the force exerted by the person's heel can be calculated using the equation F = m g, where g is the acceleration due to gravity

F = 56 kg × 9.8 m/s²

F = 548.8 N

Calculate the pressure by dividing the force by the area:

Pressure = Force / Area

Pressure = 548.8 N / 0.45 cm²

To convert cm² to m², we divide by 10,000 (since there are 10,000 cm² in 1 m²):

Pressure = 548.8 N / (0.45 cm² / 10,000)

Simplifying:

Pressure = 548.8 N / 45 m²

Pressure = 12.195 N/m²

(b) For the wide heel with an area of 16 cm²:

Following the same process as above, calculate the force exerted by the person's heel:

F = 56 kg × 9.8 m/s²

F = 548.8 N

Calculate the pressure:

Pressure = 548.8 N / 16 cm²

Pressure = 548.8 N / (16 cm² / 10,000)

Pressure = 548.8 N / 1,600 m²

Pressure = 0.343 N/m²

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The maximum velocity of the particle is 100 m/s. Negative sign indicates that the velocity is in the opposite direction of the displacement at that particular point in time.

To find the maximum velocity of a particle in simple harmonic motion, we need to differentiate the position function with respect to time and then find the maximum value of the resulting velocity function.

Given the position function x = 2cos(50t), we can find the velocity function v(t) by taking the derivative of x with respect to t:

v(t) = dx/dt = -2(50)sin(50t) = -100sin(50t)

The maximum velocity occurs when the sine function has a maximum value of 1. Therefore, the maximum velocity can be found by evaluating the velocity function at that point. In this case, the maximum value of sin(50t) is 1 when 50t = π/2 or t = π/100.

Substituting t = π/100 into the velocity function:

v_max = -100sin(50(π/100)) = -100sin(π/2) = -100(1) = -100 m/s

Therefore, the maximum velocity of the particle is 100 m/s. Note that the negative sign indicates that the velocity is in the opposite direction of the displacement at that particular point in time.

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The spring constant is 24.75 N/m. The period of oscillation is approximately 0.902 seconds. The book's maximum speed is approximately 0.606 m/s.

a. The spring constant can be calculated using Hooke's Law:

F = k * x

where F is the force applied to the spring, k is the spring constant, and x is the displacement.

Given that the mass of the book is 450 g and the spring stretches by 18 cm, we need to convert the mass to kilograms and the displacement to meters:

m = 450 g

= 0.45 kg

x = 18 cm

= 0.18 m

Using Hooke's Law, we can solve for the spring constant:

k = F / x

= (m * g) / x

where g is the acceleration due to gravity.

Substituting the values:

k = (0.45 kg * 9.8 m/s^2) / 0.18 m

= 24.75 N/m

Therefore, the spring constant is 24.75 N/m.

b. The period of oscillation for a mass-spring system is given by:

T = 2π * √(m / k)

where T is the period, m is the mass, and k is the spring constant.

Substituting the values:

T = 2π * √(0.45 kg / 24.75 N/m)

≈ 0.902 s

Therefore, the period of oscillation is approximately 0.902 seconds.

c. The maximum speed of the book can be determined using the formula:

v_max = A * ω

where v_max is the maximum speed, A is the amplitude (0.10 m, which is 10 cm), and ω is the angular frequency.

The angular frequency can be calculated using:

ω = √(k / m)

Substituting the values:

ω = √(24.75 N/m / 0.45 kg)

≈ 6.06 rad/s

Now, we can calculate the maximum speed:

v_max = 0.10 m * 6.06 rad/s

≈ 0.606 m/s

Therefore, the book's maximum speed is approximately 0.606 m/s.

a. The spring constant is 24.75 N/m.

b. The period of oscillation is approximately 0.902 seconds.

c. The book's maximum speed is approximately 0.606 m/s.

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