a high-strength steel has a yield strength of 1460 mpa and a kic of 98 mpa.m^1/2. calculate the size of a surface crack that will lead to catastrophic failure at an applied stress of 1/2 yield strength

Answers

Answer 1

To calculate the size of a surface crack that will lead to catastrophic failure, we can use the given yield strength, applied stress, and fracture toughness of the high-strength steel.

How can we determine the size of a surface crack that will result in catastrophic failure based on the given yield strength, applied stress, and fracture toughness?

The size of a surface crack that will lead to catastrophic failure can be calculated by considering the applied stress, yield strength, and fracture toughness of the high-strength steel. First, we need to determine the stress intensity factor (K) using the formula:

K = Y * \((σ * π * a)^0.5\)

Where Y is a geometric factor (usually assumed to be 1), σ is the applied stress, and a is the crack length. To determine the crack length, we rearrange the formula and solve for a:

a = \((K^2 / (Y^2 * π^2 * σ^2))\)

By substituting the given values of yield strength, applied stress (1/2 yield strength), and fracture toughness (Kic), we can calculate the size of the surface crack that will result in catastrophic failure for the high-strength steel.

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Related Questions

Which of the following is an example of a passive fall protection system

Answers

Answer:

Guardrail

netting and

safety gates

The one that is an example of passive fall protection system are Safety Nets, and Guardrails. The correct options are A and C.

What is passive fall protection system?

Any safety precautions that are essentially stationary, fixed, or unchanging are included in a passive fall protection system.

After installation, the system is completely automated, and no personal safety equipment is required.

Passive systems are buildings that make the best use of solar heat or light by their siting, design, or construction materials.

Active systems feature components that transform solar energy into a more useful form, such electricity or hot water.

Guardrails and Safety Nets are two examples of passive fall protection systems.

Thus, it can be concluded that the correct options are A and C.

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Your question seems incomplete, the probable complete question is:

Which of the following is an example of a passive fall protection system

A. Safety Nets

B. Personal Fall Arrest Systems

C. Guardrails

D. Travel Restraint Systems

A pool of contaminated water is lined with a 40 cm thick containment barrier. The contaminant in the pit has a concentration of 1.5 mol/L, while the groundwater circulating around the pit flows fast enough that the contaminate concentration remains 0. There is initially no contaminant in the barrier material at the time of installation. The governing second order, partial differential equation for diffusion of the contaminant through the barrier is: dC

Answers

This question is incomplete, the complete question is;

A pool of contaminated water is lined with a 40 cm thick containment barrier. The contaminant in the pit has a concentration of 1.5 mol/L, while the groundwater circulating around the pit flows fast enough that the contaminate concentration remains 0. There is initially no contaminant in the barrier material at the time of installation. The governing second order, partial differential equation for diffusion of the contaminant through the barrier is:

dC/dt = D( d²C / dz²)

where c(z,t) represent the concentration of containment of any depth into the barrier at anytime and D is the diffusion coefficient (a constant) for the containment in the barrier material.

a) write all boundary and initial conditions needed to solve this equation for C(z, t)

b) Find the steady  state solution (infinite time) for C(z)

Answer:

a) At t = 0, z= 0, c = 1.5 mol/L

at t =0, z = 0.4m, c = 0 mol/L

b) C(z) = z² - 4.15z + 1.5

Explanation:

a)

The boundary and initial conditions are as follows

At t = 0, z= 0, c = 1.5 mol/L

at t =0, z = 0.4m, c = 0 mol/L

b)

The governing second order, partial differential equation for diffusion of the contaminant through the barrier is :

(dC/dt) = D*(d²C/dz²) ..............equ(1)

For steady state, above equation becomes,

(d²C/dz²) =0

Integrating above equation,

(dC/dz) = Z + C1  { where C1 is integration constant) }

again integrating above equation,

C = z² + C1*z + C2    ...................equ(2)

applying boundary condition : at t =0, z= 0, c = 1.5 mol/L, to above equation

 C = z² + C1*z + C2

1.5 = 0 + 0*0 + c2

C2 = 1.5

applying boundary condition : at t =0, z= 0.4m, c = 0 mol/L, to equation (2) ,

0 = 0.4² + C1*0.4 +  1.5

0 = 0.16 + 0.4C1 + 1.5

0.4C1 = - 1.66

C1 = -1.66/0.4

C1 = -4.15

So, the steady state solution for C(z) is:

C(z) = z² - 4.15z + 1.5

A steam power plant operates on the reheat Rankine cycle. Steam enters the highpressure turbine at 12.5 MPa and 550°C at a rate of 7.7 kg/s and leaves at 2 MPa. Steam is then reheated at constant pressure to 450 °C before it expands in the low-pressure turbine. The isentropic efficiencies of the turbine and the pump are 85 percent and 90 percent, respectively. Steam leaves the condenser as a saturated liquid. If the moisture content of the steam at the exit of the turbine is not to exceed 5 percent, determine (a) the condenser pressure, (b) the net power output, and (c) the thermal efficiency

Answers

Answer:

A) condenser pressure = 9.73 kPa,

B) 10242 kw

C) 36.9%

Explanation:

given data

entrance pressure of steam = 12.5 MPa

temperature of steam = 550⁰c

flow rate of steam = 7.7 kg/s

outer pressure = 2 MPa

reheated steam temperature = 450⁰c

isentropic efficiency of turbine( nt ) = 85% = 0.85

isentropic efficiency of pump = 90% = 0.90

From steam tables

at 12.5 MPa and 550⁰c ; h3 = 3476.5 kJ/kg,  S3 = 6.6317 kJ/kgK

also for an Isentropic expansion

S4s = S3 .

therefore when S4s = 6.6317 kJ/kg and P4 = 2 MPa

h4s = 2948.1 kJ/kg

nt = 0.85

nt (0.85) = \(\frac{h3-h4}{h3-h4s}\) = \(\frac{3476.5 - h4}{3476.5 - 2948.1}\)

making h4 subject of the equation

h4 = 3476.5 - 0.85 (3476.5 - 2948.1)

h4 = 3027.3 kj/kg

at P5 = 2 MPa and T5 = 450⁰c

h5 = 3358.2 kj/kg,  s5 = 7.2815 kj/kgk

at P6 , x6 = 0.95  and s5 = s6

using nt = 0.85 we can calculate for h6 and h6s

from the chart attached below we can see that

p6 = 9.73 kPa, h6 = 2463.3 kj/kg

B) the net power output

solution is attached below

c) thermal efficiency

thermal efficiency = 1 - \(\frac{qout}{qin}\) = 1 - ( 2273.7/ 3603.8) = 36.9% ≈ 37%

A steam power plant operates on the reheat Rankine cycle. Steam enters the highpressure turbine at 12.5
A steam power plant operates on the reheat Rankine cycle. Steam enters the highpressure turbine at 12.5

A chemical process stream enters a shell-and-tube exchanger at a temperature of 200.0°Fand does two passes on the shell side, exiting the exchange at 170.0°F. The Heat exchanger Spring 2021 has 200 stainless steel tubes that are 2-in.ODand 10.0ft long. Indicate whether the temperature of the process stream will increase, decrease, or remain the sameunder the following scenarios. You must justify your answerto receive full credit.a)The flow rate of the cooling fluid is increased.b)There are 200 tubes that are 1.0-in. OD and 20.0ft long.c)The number of shell passes is doubled.d)The tube material is changed to copper.

Answers

Answer:

a) Decrease

b) Decrease

c) Decrease

d) Decrease

Explanation:

Ti= 200°F ,

Te =  170°F

Area of heat exchanger = \(\pi *(2 )* 10\)  = 20π

A) when The flow rate of the cooling fluid is increased

Temperature of process stream will decrease  and this is because the tube side heat transfer coefficient will increase and this will increase the rate of heat transfer thereby decreasing the temperature of the process stream.

B) when There are 200 tubes that are 1.0-in. OD and 20.0ft long

The temperature of the process stream will decrease and this is because the heat transfer coefficient will increase likewise the heat transfer rate

C) When The number of shell passes is doubled

This will cause an increase in the overall length of the shell, an increase in velocity of constant volumetric flowrate, hence the Temperature of the process steam will decrease as well

D) When The tube material is changed to copper.

Due to the high thermal conductivity of copper when compared to steel , switching to copper will cause a decrease in the temperature of the process steam

Where you attend college could costmore to borrow and refinance loans

Answers

Ans: Researchers did the analysis on the average of the rates offered by Wells Fargo to determine monthly and total payments over the life of the loans. A community college borrower would pay $1,134.31 more on a $10,000 loan than a student with the same credit history at a four-year school.

A germanium diode carries a current of 1 mA at room temperature when a forward bias of 0.15v is applied. Estimate the reverse saturation current at room temperature.

Answers

The reverse saturation current, denoted as I0, can be estimated using the diode equation:

I = I0 * (exp(qV/kT) - 1)

where I is the current through the diode, q is the charge of an electron, V is the voltage across the diode, k is Boltzmann's constant, and T is the temperature in Kelvin.

At room temperature, T = 298 K. We are given that the diode carries a current of 1 mA when a forward bias of 0.15 V is applied. Let's assume that the diode is ideal, meaning that it has no series resistance, so the voltage across the diode equals the forward bias voltage.

Plugging in these values, we get:

1E-3 A = I0 * (exp((1.602E-19 C)(0.15 V) / (1.381E-23 J/K)(298 K)) - 1)

Simplifying, we get:

1E-3 A = I0 * (exp(0.01275) - 1)

1E-3 A / (exp(0.01275) - 1) = I0

I0 = 2.34E-12 A, or approximately 0.23 nA. Therefore, the estimated reverse saturation current at room temperature is 0.23 nA.

given a list of integers, write python code to separate only the squares of the odd numbers from the list my list. my list

Answers

python code is:

my_list = [1, 2, 3, 4, 5, 6, 7, 8, 9]squares_of_odd_numbers = [x*x for x in my_list if x % 2 != 0]print(squares_of_odd_numbers)

How to write python code?

Python code that separates only the squares of odd numbers from a given list of integers my_list:

my_list = [1, 2, 3, 4, 5, 6, 7, 8, 9]squares_of_odd_numbers = [x*x for x in my_list if x % 2 != 0]print(squares_of_odd_numbers)

Output:

[1, 9, 25, 49, 81]

In the code above, we use a list comprehension to filter only the odd numbers from the list my_list using the condition x % 2 != 0.

Then, we use the map() function to map the square of each odd number using the expression x*x.

Finally, we assign the result to a new list called squares_of_odd_numbers.

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What’s cat plus cat







?

Answers

Uhmmmmm, a kitten...? Lol
Cat + Cat = kitten/baby cat

Is reinforcement needed in a retaining wall

Answers

reinforcement means to rebuild. so yeah i guess you need to reinforce a walk
Yea, bc you’ll need help reestablishing it

A low-altitude meteorological research balloon, temperature sensor, and radio transmitter together weigh 2.5 lb. When inflated with helium, the balloon is spherical with a diameter of 4 ft. The volume of the transmitter can be neglected when compared to the balloon's size. The balloon is released from ground level and quickly reaches its terminal ascent velocity. Neglecting variations in the atmosphere's density, how long does it take the balloon to reach an altitude of 1000 ft?

Answers

Answer:

12 mins

Explanation:

The summation of the forces in vertical direction

= Fb - Fd - w = 0 ∴ Fd = Fb - w  ----- ( 1 )

Fb ( buoyant force ) = Pair * g * Vballoon ------- ( 2 )

Pair = air density , Vballoon = volume of balloon

Vballoon = \(\frac{\pi D^3}{6}\) ,  where D = 4  ∴  Vballoon = 33.51 ft^3

g = 32.2 ft/s^2

From property tables

Pair = 2.33 * 10^-3 slug/ft^3

μ ( dynamic viscosity ) = 3.8 * 10^-7 slug/ft.s

Insert values into equation 2

Fb = ( 2.33 * 10^-3 ) * ( 32.2 ) *( 33.51 )  = 2.514 Ib

∴ Fd = 2.514 - 2.5 = 0.014 Ib ( equation 1 )

Assuming that flow is Laminar  and  RE < 1

Re = (Pair * vd) / μair   -------- ( 3 )

where:  Pair = 2.33 * 10^-3 slug/ft^3 ,  vd = ( 987 * 4 ) ft^2/s ,   μair = 3.8 * 10^-7 slug/ft.s

Insert values into equation 3

Re = 2.4 * 10^7 (  this means that the assumption above is wrong )

Hence we will use drag force law

Assume Cd = 0.5

Express  Fd using the relation below

Fd = 1/2* Cd * Pair * AV^2

therefore V = 1.39 ft/s

Recalculate  Reynold's number using  v = 1.39 ft/s

Re = 34091

from the figure Cd ≈ 0.5 at Re = 34091  

Finally calculate the rise time ( time taken to reach an altitude of 1000 ft )

t = h/v

  = 1000 / 1.39 = 719 seconds ≈ 12 mins

A low-altitude meteorological research balloon, temperature sensor, and radio transmitter together weigh

True or false?Inventory-risk costs are a form of carrying costs.

Answers

or...................

Answer:

i believe it is true

Explanation:

A subsystem of the emergency system consisting of feeders and branch circuits is a _____________.
517.2

Answers

A subsystem of the emergency system consisting of feeders and branch circuits is defined as an "Emergency Lighting and Power System" in accordance with the National Electrical Code (NEC) section 517.2.

In an automatic emergency lighting and power system, the emergency power source takes over when the ship's service power and lighting plant's potential drops, supplying electricity to the emergency loads.

Emergency lighting is used to provide light in case of a power outage or other disruption to the main power source. A fire or a power outage may cause the loss of mains electricity.

What is the power source for the emergency lights?

Emergency lighting is connected to the building's electrical system so that internal batteries can be continuously recharged to provide backup power for the lighting. This makes sure that during a power loss, the lights stay on, enabling building residents to find the exits with ease in the event that they need to escape.

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A standard wheel consists of the ___ and the ___.

Answers

Answer:

metal hub and the wire tension spokes?

Answer: metal hub and the wire tension spokes is correct

pplying Performance-Based Engineering: Describe shortly two different example projects (they do not need to be real ones) where performance-based engineering could add value to the project. Why performance-based design could be better option in these projects than prescriptive fire design? o What are the potential benefits available in these projects? o What are the key considerations to take into account?

Answers

Performance-Based Engineering (PBE) is an approach used to design building structures that consider how the structure responds in case of fire. By using this approach, it can be established the effectiveness of fire safety measures and if these measures meet the safety objectives of the building owner and local fire authorities.

There are two different example projects where performance-based engineering could add value to the project. These examples are not real but show how performance-based design could be a better option than prescriptive fire design.

Example 1: High-rise office building The first project is a 20-story high-rise office building that is still in the planning phase. This building has been designed to meet the minimum fire safety requirements of the building codes. The design is based on traditional methods where the objective is to protect the occupants from fire until the fire department arrives.

Example 2: Healthcare facility The second project is a new hospital that is under construction. This hospital will have several departments, including an emergency room, surgery rooms, and patient wards. Due to the complexity of the hospital, the building codes have been difficult to implement in the design of the hospital. Performance-based engineering could be used to demonstrate how the hospital can meet the safety objectives of the building owner and local fire authorities.

The key considerations to take into account include the cost and time required to conduct a performance-based analysis, determining the level of safety objectives, and taking into account the needs of the occupants and staff during an emergency.

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Suppose that a wing component on an aircraft is fabricated from an aluminum alloy that has a plane strain fracture toughness of 28 MPa (25.48 ksi). It has been determined that fracture results at a stress of 106 MPa (15370 psi) when the maximum internal crack length is 7.9 mm (0.3110 in.). For this same component and alloy, compute the stress level at which fracture will occur for a critical internal crack length of 5.5 mm (0.2165 in.).

Answers

The stress level at which fracture will occur for a critical internal crack length of 5.5 mm (0.2165 in.) in the wing component made of the aluminum alloy can be calculated.

To determine the stress level, we can use the formula for fracture toughness:

K_IC = σ √(πa)

Where K_IC is the fracture toughness, σ is the stress level, and a is the crack length.

Given that K_IC is 28 MPa√m (25.48 ksi√in.), and the original crack length is 7.9 mm (0.3110 in.), we can rearrange the formula to solve for the stress level:

σ = K_IC / √(πa)

Plugging in the values, we have:

σ = 28 MPa√m / √(π * 5.5 mm) ≈ 96.83 MPa

Therefore, the stress level at which fracture will occur for a critical internal crack length of 5.5 mm is approximately 96.83 MPa.

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how are sedimentary - gravity flows different from fluid - gravity flows

Answers

Answer:

Sediment gravity flows are mixtures of water and sediment particles where the gravity acting on the sediment particles moves the fluid, in contrast to rivers, where the fluid moves the particles.

Explanation:

Calculate the areas under the stress-strain curve (toughness) for the materials shown in Fig. below, (a) plot them as a
function of temperature, and describe your observations. (b) As expected, the elastic modulus of the polymer decreases as
temperature increases. Using the stress-strain curves in the figure, make a plot of the modulus of elasticity versus the
temperature. Comment on the shape of the curve.

Answers

Answer:

Explanation:

Wow

an incorrect turning radius angle is being discussed. technician a says replacement shims can be used in many vehicles to correct the condition. technician b says steering or suspension parts are damaged. who is correct?

Answers

As long as the rear toe is movable, this is often done during a 4-wheel alignment. If the rear is fixed, the front toe must be set to account for the thrust angle in order to center the steering.

The front wheels' direction of travel can be predicted by the thrust angle. So, failing to consider this angle can compromise even the front suspension that has been precisely aligned. As the front wheels steer to align themselves with the intended direction of the vehicle, it may cause the steering wheel to become misaligned. Positive camber causes the outside suspension to raise on the wheel as the vehicle turns. The weight of the car bears down on the steering axis and aids in straightening the wheel as soon as it turns back to face front.

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Once you have chosen a topic, what should you do before beginning the research process? a. Find as many possible facts and details on your topic c. Discuss your idea with others b. Choose a position d. None of these Please select the best answer from the choices provided A B C D

Answers

Answer:

The answer is C

Explanation:

Once you have chosen a topic, the next thing you should do before beginning the research process is: C. discuss your idea with others.

What is a research topic?

A research topic refers to an event, issue, or subject that a researcher is keenly and deeply motivated or interested in, especially when conducting a study or research.

Based on scientific information and records, it is very important you discuss your idea with others once you have chosen a topic, before beginning the research process.

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Choose an equation for the energy-releasing reaction of PEP. A. PEP + 14.8 kcal/mole rightarrow pyruvate + P_i B. PEP rightarrow pyruvate + Pi + ADP + 14.8 kcal/moleC. PEP rightarrow pyruvate + Pi + 14.8 kcal/mole D. PEP + 14.8kcal/mole + ATP rightarrow pyruvate + Pi

Answers

The equation for the energy-releasing reaction of PEP is option A: PEP + 14.8 kcal/mole → pyruvate + P_i.

Which equation represents the energy-releasing reaction of PEP?

The correct equation for the energy-releasing reaction of PEP is option C: PEP rightarrow pyruvate + Pi + 14.8 kcal/mole.

In this reaction, PEP (phosphoenolpyruvate) is converted into pyruvate and inorganic phosphate (Pi) with the release of 14.8 kcal/mole of energy.

Option A is incorrect as it does not include the release of energy. Option B is incorrect as it includes the addition of ADP, which is not part of the reaction.

Option D is incorrect as it includes the addition of ATP, which is not involved in the energy-releasing reaction of PEP.

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1.Shortcut operators are faster than the conventional arithmetic operators.
2.You can declare more than one variable in a single line.
3.You must use else after every if statement.
what is answer?

Answers

It's important to note that this speed difference is only noticeable for large programs. For small programs, the difference is negligible.

1. Shortcut operators are faster than the conventional arithmetic operators: This statement is true. Shortcut operators are faster because they combine arithmetic operations with variable assignments in a single statement. For example, instead of writing "a = a + 2", you can write "a += 2". This saves time and reduces the amount of code you need to write. However, it's important to note that this speed difference is only noticeable for large programs or when dealing with complex calculations. For small programs, the difference is negligible.
2. You can declare more than one variable in a single line: This statement is also true. In many programming languages, you can declare and initialize multiple variables on the same line. For example, instead of writing "int a; int b; int c;", you can write "int a, b, c;". This saves space and makes your code more concise. However, it's important to note that you should only do this if the variables are related and have the same data type.
3. You must use else after every if statement: This statement is false. It's not necessary to use else after every if statement. You can use if statements on their own if you don't need to execute any code if the condition is not true. However, if you need to execute code in both cases (true and false), then you should use else. It's also important to note that you can use else if to test for additional conditions if the first if statement is not true.

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; How do engineers make a difference in the world and with proof ?​

Answers

Engineers are the reason healthcare has improved so dramatically throughout the years. The advancements in medical technology are down to the work conducted by engineers as well as the creation of devices that help to save lives and improve the quality of life for others. New developments are being created by engineers constantly, for example, the Crossrail project in London is creating a new transport system throughout the South East to create shorter journey times and easier connections. The existing transport system wouldn’t be where it is today without engineers.
Well Engineers make a difference in the world because well for an example when u buy a car or something like that someone has made it because if u had no car were would u go if u had to go somewhere like New York u can’t walk from there and back u would be exhausted and die so with things like car, trucks, and bus’s u wouldn’t have something to get u to point A to B and maybe C that if u have a point c well hoped it helped

Why are most products the result of an innovation instead of an invention?

Answers

Answer:

they were updated rather than being created

Answer:

Invention is about creating something new, while innovation introduces the concept of “use” of an idea or method.

Suppose that x is an int variable, ch is a char variable, and the input is:
276.
Choose the values after the following statement executes:
cin >> ch >> x;
a. ch='2',x=276
c. ch='',x=276
b. ch='276',x='.'
d. ch='b',x=76

Answers

The values for the datatypes variable after the following statement executes are: ch = '2', x = 76

What is datatype?

Data types describe the kinds of values that a variable can store. They establish the kind of operations that can be carried out on the data, the amount of memory needed to hold the data, and the possible range of values. Integer, float, double, char and boolean are a few examples of common data types. A boolean datatype stores either true or false values, while a float or double data type stores floating point numbers, a char data type stores single characters, a string data type stores a series of characters, and an integer data type stores full integers. The type of data to be saved and the unique program requirements both influence the datatype to be used.

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features of a dry leclanche cell​

Answers

Answer:

Primary cell with a nominal open circuit voltage of 1.5 Volts produced in very high volumes. Chemistry based on a zinc anode and a cathode/depolariser of manganese dioxide which absorbs the liberated hydrogen bubbles which would otherwise insulate the electrode from the electrolyte.

Answer:

Explanation:

(1) The electrolyte is a paste of ammonium chloride.

(2) The positive terminal (anode) is a carbon rod surrounded bymangAnese Dioxide as a depolarised .

(3) The negative terminal is zinc container.

Match the tasks with the professionals who would complete them.

Answers

Answer: what are the tasks

Explanation:

no professionals to match with the tasks.

4. A cylinder in the lab contains nitrogen at 2200psia. If the cylinder falls and the valve is sheared off, estimate the initial mass flow rate of nitrogen from the tank. Assume a hole diameter of 0.5in. What is the force created by the jet of nitrogen

Answers

When the cylinder is dropped and the valve is removed, the pressure inside the cylinder drops to atmospheric pressure (about 14.7 psia) quickly. It is considered that the initial mass flow rate is equal to the mass.

flow rate through a hole at atmospheric pressure. This flow rate is calculated using the orifice equation, assuming the flow is choked. Flow rate through the hole can be determined by using the mass conservation equation.

If it is assumed that the gas is at the same temperature inside and outside the cylinder, the mass flow rate through the hole is calculated as follows: m1 = ρ A √(2*ΔP/γ)where = density of gas A = cross-sectional area of the holeΔP = pressure difference across the holeγ = ratio of specific heats for the gas Nitrogen has a ratio of specific heats γ = 1.4.

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Consider two houses that are identical, except that the walls are built using bricks in one house, and wood in the other. The walls of the brick house are twice as thick. Which house do you think will be more energy efficient?

Answers

Answer:

Walls Built Using Bricks and Wood

The brick house is more energy-efficient than the one built with wood.

Explanation:

Because of their high thermal mass, which gives bricks the ability to absorb heat and release it over time, bricks remain more energy-efficient than other building materials, including wood.  In summer, bricks leave your home cool.  In winter, they make it warm.  With these two advantages provided by bricks over other building materials, bricks are the most energy-efficient building material.

What do we need to build a car?

Answers

It’s called www. Popularmechanica.com

A particle of mass m=4 kg is moving along a guide wire with shape given by y(x)=−4sin(2. 5x) m, where x is in meters. The particle's horizontal velocity component is a constant vx=−4 m/s and gravity →g=−9. 8^ȷ m/s^2 acts vertically. What is the force →Fw of the wire on the particle when x=12 m?A particle of mass m


4 kg is moving along a guide wire with shape given by y(r)- -4sin(2. 5x) m, where


is in meters. The parti

Answers

To determine the force of the wire on the particle when x = 12 m, we need to consider both the gravitational force acting on the particle and the normal force exerted by the wire.

1. Gravitational Force:

The gravitational force acting on the particle can be calculated using the formula:

Fg = m * g

Where:

Fg is the gravitational force,

m is the mass of the particle, and

g is the acceleration due to gravity.

Given:

m = 4 kg (mass of the particle)

g = -9.8^ȷ m/s^2 (acceleration due to gravity, negative because it acts vertically)

Substituting the values, we have:

Fg = 4 kg * (-9.8^ȷ m/s^2) = -39.2^ȷ N

2. Normal Force:

The normal force is the force exerted by the wire perpendicular to the wire's surface. It counteracts the vertical component of the gravitational force to keep the particle on the wire.

To calculate the normal force, we need to consider the net force acting in the vertical direction. Since the particle moves along the guide wire, it stays in contact with the wire, which means the net force in the vertical direction is zero. Therefore, the normal force is equal in magnitude but opposite in direction to the vertical component of the gravitational force.

Since the guide wire is given by the equation y(x) = -4sin(2.5x), we can determine the vertical displacement of the particle at x = 12 m:

y(12) = -4sin(2.5 * 12) = -4sin(30) = -4 * 0.5 = -2 m

The vertical component of the gravitational force is equal to the weight of the particle, which is given by:

Fg_vert = m * g_vert

Where:

Fg_vert is the vertical component of the gravitational force,

m is the mass of the particle, and

g_vert is the vertical component of the acceleration due to gravity.

Since g_vert = -g (opposite direction), we have:

Fg_vert = m * (-g) = -m * g

Substituting the values, we have:

Fg_vert = 4 kg * (-9.8 m/s^2) = -39.2 N

Therefore, the normal force exerted by the wire is:

Fn = -Fg_vert = 39.2 N

3. Force of the Wire:

The force of the wire on the particle is the vector sum of the gravitational force and the normal force:

Fw = Fg + Fn

Substituting the values, we have:

Fw = -39.2^ȷ N + 39.2^ȷ N = 0 N

Thus, when x = 12 m, the force of the wire on the particle is zero. This means that the wire is not exerting any horizontal force on the particle at that particular position along the guide wire.

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