All of the possible answers are correct. Pumping or draining water out of the ground along a slope can reduce the chances of a mass movement by decreasing the amount of water pressure that is pushing against the slope.
What is water pressure?Water pressure is the force that water exerts on the walls of pipes and containers when it is confined and forced to move. Water pressure is generated by the weight of the water above the point of measurement and is measured in pounds per square inch (psi). The higher the water pressure, the harder it is to move the water. High water pressure can cause plumbing problems, such as leaks and bursts. Low water pressure can lead to inadequate water flow and lack of pressure in showers and taps.
Redistributing the mass on a slope by terracing can create a more stable surface and reduce the chances of a mass movement. Planting vegetation on slopes can help retain moisture and reduce erosion, thus decreasing the chances of a mass movement.
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Calculate the power required to move a 2,000-kilogram automobile to the top of a 100-meter hill in 15. 0 seconds. Express the power both in
units of watts and horsepower.
The power required to move the automobile to the top of the hill is 130,666.67 watts or 175.41 horsepower.
The power required to move an object can be calculated using the formula: power = work / time.
First, let's calculate the work done in lifting the automobile to the top of the hill. The work done against gravity is given by the formula: work = force × distance.
The force required to lift the automobile is equal to its weight. The weight of an object is given by the formula: weight = mass × acceleration due to gravity.
Substituting the given values, we have: weight = 2,000 kg × 9.8 m/s^2 (acceleration due to gravity) = 19,600 N.
The distance the automobile is lifted is 100 meters.
Therefore, the work done against gravity is: work = 19,600 N × 100 m = 1,960,000 J (joules).
The time taken to reach the top of the hill is given as 15.0 seconds.
Now, we can calculate the power using the formula: power = work / time.
power = 1,960,000 J / 15.0 s = 130,666.67 W (watts).
To convert watts to horsepower, divide the power in watts by 746 (1 horsepower = 746 watts).
power in horsepower = 130,666.67 W / 746 = 175.41 hp (horsepower).
Rounding to two decimal places, the power required to move the automobile to the top of the hill is approximately 130,666.67 watts or 175.41 horsepower.
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The to problem of calculating the power required to move a 2,000-kilogram automobile to the top of a 100-meter hill in 15.0 seconds is given
Given, Mass of the automobile, m = 2000 knight of the hill, h = 100 time, t = 15.0 the gravitational potential energy of the automobile when at the bottom of the hill is equal to the work done in lifting it up the hill
.W = mgh= (2000 kg) (9.81 m/s²)
(100 m)= 1,962,000 J
Power is defined as the rate at which work is done, or the work per unit time. Therefore,
Power = Work / Time= 1,962,000 J / 15.0 s
= 130,800 WIn horsepower, Power = (130,800 W) / (746 W/hp)
= 175.3 hp
Therefore, the required power to move a 2,000-kilogram automobile to the top of a 100-meter hill in 15.0 seconds is 130,800 W or 175.3 hp.
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Match the items on the left with the best corresponding statement on the right. 1 . chromosomes contains tightly packaged DNA molecules 2 . Carl Correns the person who experimented with four o'clocks 3 . incomplete dominance a change in a gene that results from a reproductive mistake 4 . DNA cell division in which the new cells have the same number chromosomes of as the parent cell 5 . mitosis the person who discovered the principles of genetics 6 . Gregor Mendel the special molecule that is able to store and recover information about traits 7 . R.C. Punnett the person who devised a system of squares used to record genetic problems 8 . mutation a condition in which neither gene of a pair is dominant; instead, they show a blended effect 9 . meiosis cell division in which the new cells have one-half of the original number of chromosomes
Answer:
1 and 4, 2 and 3, 5 and 9, 6 and 8
Explanation:
7 would not have a pair because there is not enough statements
how do we calculate the value of work?
Answer:
Work = Force × Distance
Explanation:
Work = Force × Distance
The SI unit of work is Joule
-TheUnknownScientist
Where is the choke valve in the carburetor always located?
The choke valve is normally found in the upper part of a conventional carburetor, close to the air inlet or throttle plate. In order to assist the engine start and run smoothly in cold temperatures, the choke valve is made to restrict the flow of air into the carburetor.
This raises the fuel-to-air ratio and offers a richer fuel mixture during cold starts. There are various types of carburetors with various designs, so the precise location of the choke valve may vary depending on the design and type of carburetor. However, the choke valve is typically found close to the air intake at the top of the carburetor.
Using a lever or thermostat, it can be controlled manually or automatically, depending on the carburetor design, a bi-metallic mechanism.
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1. What are the two main gases responsible for lowering down the pH.
2. Can we control an acid rain? Why or Why not?
3. When pH changes from 6 to 3, how many times does acidity increase?
Answer:
1. The three main acidic gases responsible for lowering the pH of rainwater are non-metal oxides produced by the burning of fossil fuels like coal, oil and natural gas. Sulfur dioxide is produced when fossil fuels containing sulfur impurities are burned.
2. A great way to reduce acid rain is to produce energy without using fossil fuels. Instead, people can use renewable energy sources, such as solar and wind power. Renewable energy sources help reduce acid rain because they produce much less pollution.
3. The pH scale is logarithmic, meaning that an increase or decrease of an integer value changes the concentration by a tenfold. For example, a pH of 3 is ten times more acidic than a pH of 4.
Starting from rest, you pushed a 10.0 kg lawnmower a distance of 25 meters by applying a force of 75 N at an angle 35 degrees below horizontal, as shown below. Use work and energy principles to calculate how fast the lawn mower is moving (in m/s) at the end of the 25 meters.
The lawn mower is moving with a velocity of 17.53 m/s
Energy is the ability to do work while Work is defined as the product of force and distance moved in the direction of the force. Both work and energy are measured in Joule (J)
Work = Force (F) × distance (d) = FdCosθ
With the above formula, we can obtain the energy used in pushing the lawn mower as illustrated below
From the question given above, the following data were obtained:
•Force (F) = 75 N
•Distance (d) = 25 m
•Angle (θ) = 35°
•Energy (E) =?
E = FdCosθ
E = 75 × 25 × Cos 35
E = 1535.91 J
Definition and Determination of the velocityVelocity is the rate of change of displacement with time.
Velocity and energy are related according to the following equation:
E = ½ × mass (m) × square velocity (v²)
E = ½mv²
With the above formula, we can obtain the velocity of the lawn mower as follow
•Mass (m) = 10 Kg
•Energy (E) = 1535.91 J
•Velocity (v) =?
E = ½mv²
1535.91 = ½ × 10 × v²
1535.91 = 5 × v²
Divide both side by 5
v² = 1535.91 / 5
Take the square root of both side
v = √(1535.91 / 5)
v = 17.53 m/s
Thus, the lawn mower is moving with a velocity of 17.53 m/s
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now, consider model b. according to model b, what is a point estimate for a concert that has attendance of 35,000 people, 5 concession stands, and the song ranked in no. 8 in the billboard ranking?
The point estimate for a concert that has attendance of 35,000 people, 5 concession stands, and the song ranked in no. 8 in the Billboard ranking is $3.147M.
Show Country, INC. is a rock concert promoter with a national reach. The leader of the organization needs to foster a model to gauge the income of a significant show occasion at large venues (for example, Passage Field, Madison Square Gardens) for arranging promoting systems. The company has gathered information about 32 recent major concert events' revenue.
They have also recorded the number of people who attended each concert, the number of concession stands in the venue, and the artist's Billboard chart during the event's week. This data can be found under "Tickets." The revenue could be explained by one of two models they have. These are the two competing models:
A Model: = + + + + 0123
Now, consider model B. According to model B, what is a point estimate for a concert that has attendance of 50000 people, 5 concession stands, and the song ranked in no. 15 in the Billboard ranking is $3.147M.
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Which graph accurately shows the relationship between kinetic energy and speed as speed increases?
Answer:
B
Explanation:
kinetic energy (KE) is the energy possessed by moving bodies. It can be expressed as:
KE = \(\frac{1}{2}\)m\(v^{2}\)
Where: m is the mass of the object, and v its speed.
For example, a stone of mass 2kg was thrown and moves with a speed of 3 m/s. Determine the kinetic energy of the stone.
Thus,
KE = \(\frac{1}{2}\) x 2 x \(3^{2}\)
= 9
KE = 9.0 Joules
Assume that the speed of the stone was 4 m/s, then its KE would be:
KE = \(\frac{1}{2}\) x 2 x \(4^{2}\)
= 16
KE = 16.0 Joules
Therefore, it can be observed that as speed increases, the kinetic energy increases. Thus option B is appropriate.
In 3-5 sentences, discuss one non-psychologist/non-psychiatrist career (it can be one from 5.01c in the lesson or a different career) that you think benefits most from having a degree in Psychology and explain why.
One career that believe benefits greatly from a degree in Psychology is Human Resources (HR). HR professionals are responsible for managing employee relations, recruiting, training, and ensuring compliance with labor laws and company policies.
A degree in Psychology can provide HR professionals with a deeper understanding of human behavior and the ability to better manage workplace dynamics.
Firstly, a degree in Psychology can help HR professionals understand and navigate the complexities of human behavior. HR professionals often deal with sensitive issues such as workplace conflict, discrimination, and harassment. A strong foundation in psychology can help HR professionals better understand the motivations and behaviors of employees, allowing them to handle these situations with greater sensitivity and compassion.
Secondly, psychology can help HR professionals develop effective communication and leadership skills. HR professionals are often responsible for training employees, providing feedback, and managing performance. A degree in Psychology can provide HR professionals with a deeper understanding of communication patterns and techniques, allowing them to better convey information to employees and resolve conflicts.
Lastly, a degree in Psychology can help HR professionals develop critical thinking and problem-solving skills. HR professionals often face complex issues that require careful analysis and creative solutions. Psychology provides a strong foundation in research methods, statistics, and analytical thinking, which can help HR professionals make informed decisions and develop effective policies.
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Nuclide X decays more quickly than nuclide Y. Based on this information, which of the following statements must be true?
A. Nuclide Y is less abundant than nuclide X
B. Nuclide Y has a larger half-life than nuclide X
C. Nuclide Y has a shorter half-life than nuclide X
D. The daughter nuclides of X are more abundant than those of Y
Nuclide X decays more quickly than nuclide Y because Nuclide Y has a shorter half-life than nuclide X. Option C.
The decay rate of a radioactive nuclide is determined by its half-life. A shorter half-life means that a nuclide is more unstable and will decay more quickly than a nuclide with a longer half-life.
Therefore, if nuclide X decays more quickly than nuclide Y, it can be concluded that nuclide X has a shorter half-life than nuclide Y. This rules out option B, which suggests the opposite.
There is no direct relationship between the abundance of a nuclide and its decay rate, so it cannot be conclusively stated that nuclide Y is less abundant than nuclide X (option A). This information would require further context.
Similarly, the decay rate of a nuclide does not provide information about the abundances of its daughter nuclides, so option D cannot be confirmed. So Option C is correct.
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Which of the statements regarding potential energy is true?A.Potential energy is the energy that an object has based on its position.B.Potential energy corresponds to the expansion and compression of spaces between molecules.C.Potential energy corresponds to the motion of electrons through a conductor.D.Potential energy cannot be stored and is immediately converted to kinctic energy.E.All of the sbove are true.
Potential energy is the energy that an object has based on its position is true .
What is the name for the potential energy created when an item contracts or expands?
Elastic potential energy is the potential energy resulting from an object's form. When a force stretches or compresses an elastic substance, this energy is produced. The potential energy of an item increases with the amount it is compressed or stretched.
Potential energy is a form of accumulated energy that is dependent on the relationship between different system components. When a spring is compressed or expanded, its potential energy increases. If a steel ball is raised above the earth as opposed to falling to the ground, it has more potential energy.
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what is a positive lachman maneuver/test
A positive Lachman maneuver or test refers to the presence of increased anterior translation of the tibia in relation to the femur when performing a manual examination of the knee joint.
This finding suggests a possible anterior cruciate ligament (ACL) injury, as the ACL provides stability to prevent this type of movement. A positive Lachman test is often used as part of the clinical evaluation of a knee injury, along with other tests and imaging studies, to determine the extent of the damage and guide treatment. Place the patient's knee in about 20-30 degrees flexion. According to Bates' Guide to Physical Examination, the leg should also be externally rotated slightly. The examiner should place one hand behind the tibia and the other on the patient's thigh. It is important that the examiner's thumb be on the tibial tuberosity. On pulling the tibia anteriorly, an intact ACL should prevent forward translational movement of the tibia on the femur ("firm end-feel"). So, a positive Lachman maneuver or test refers to the presence of increased anterior translation of the tibia in relation to the femur when performing a manual examination of the knee joint.
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Two layers of fluid are contained between parallel plates, each of 0. 8 m2 area. The fluid viscosities are η1 = 0. 12 N. S. M-2 and η2 = 0. 18 N. S. M-2. The thickness of each layer of fluid is L1 = 0. 62 mm and L2 = 0. 56 mm. What is the fluid relative velocity at the interface between the two plates, if the upper plate has a speed of 1. 3 m. S-1 at the interface?
According to the given statement , the fluid relative velocity at the interface between the two plates is approximately 0.684 m/s.
To find the fluid relative velocity at the interface between the two plates, we can use the concept of shear rate and the formula for the velocity gradient.
First, let's calculate the shear rate (γ) using the formula:
γ = Δv / Δx
Where:
- Δv is the velocity difference between the two plates, which is given as 1.3 m/s (since the upper plate has a speed of 1.3 m/s at the interface).
- Δx is the distance between the two plates, which is the sum of the thicknesses of the two fluid layers:
Δx = L1 + L2.
Given that L1 = 0.62 mm and L2 = 0.56 mm, we need to convert these values to meters:
L1 = 0.62 mm = 0.62 × 10⁻³ m
L2 = 0.56 mm = 0.56 × 10⁻³ m
Now we can calculate Δx:
Δx = L1 + L2
Substituting the values, we get:
Δx = 0.62 × 10⁻³ m + 0.56 × 10⁻³ m
= 1.18 × 10⁻³ m
Now we can calculate the shear rate:
γ = Δv / Δx
= 1.3 m/s / 1.18 × 10⁻³ m
Performing the division, we find:
γ ≈ 1101.7 s^-1
The shear rate (γ) represents the velocity gradient between the two fluid layers. To find the fluid relative velocity at the interface, we need to multiply the shear rate by the thickness of the first layer (L1).
Relative Velocity = γ * L1
Substituting the values, we get:
Relative Velocity = 1101.7 s⁻¹ * 0.62 × 10⁻³ m
Performing the multiplication, we find:
Relative Velocity ≈ 0.684 m/s
Therefore, the fluid relative velocity at the interface between the two plates is approximately 0.684 m/s.
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average method and it reports the tollowing unit data tor the rorming department. Units completed in the torming department are transferred to the painting department. Production cost information for the forming department follows. . Calculate the equivalent units of production for both direct materials and conversion for the Forming department. o. Calculate the costs per equivalent unit of production for both direct materials and conversion for the Forming department c. Using the weighted average method, assign costs to the forming department's output-specifically, its units transferred to painting and its endina work in brocess inventorv. Calculate the costs per equivalent unit of production for both direct materials and conversion for the For Jsing the weighted average method, assign costs to the forming department's output-specifically, its 4 d its ending work in process inventory. Complete this question by entering your answers in the tabs below. Calculate the equivalent units of production for both direct materials and conversion for the forming department. a. Calculate the equivalent units of production for both direct materials and conversion for the Forming departm b. Calculate the costs per equivalent unit of production for both direct materials and conversion for the Forming c. Using the weighted average method, assign costs to the forming department's output-specifically, its units tra and its ending work in process inventory. Complete this question by entering your answers in the tabs below. Calculate the costs per equivalent unit of production for both direct materials and conversion for the forming department Required information Using the weighted average method, assign costs to the forming department's output-specifically, its units trar painting and its ending work in process inventory.
Given information: The average method reports the following unit data for the forming department. Units completed in the forming department are transferred to the painting department. Production cost information for the forming department follows.
Direct materials:
Units completed during the period = 45,000 units
Ending work in process inventory = 5,000 units
Direct materials cost = $202,500
Conversion costs:
Units completed during the period = 45,000 units
Ending work in process inventory = 5,000 units
Conversion cost = $189,000
a. Calculation of equivalent units of production for both direct materials and conversion for the forming department:
Equivalent units of production = Units completed during the period + (Ending work in process inventory * Degree of completion)
Direct materials:
Equivalent units of production = 45,000 + (5,000 * 50%) = 47,500 units
Conversion costs:
Equivalent units of production = 45,000 + (5,000 * 60%) = 48,000 units
b. Calculation of the cost per equivalent unit of production for both direct materials and conversion for the forming department:
Cost per equivalent unit of production = Total cost for the period / Equivalent units of production
Direct materials:
Cost per equivalent unit of production = $202,500 / 47,500 units = $4.26 per unit
Conversion costs:
Cost per equivalent unit of production = $189,000 / 48,000 units = $3.94 per unit
c. Calculation of the cost assigned to the forming department's output using the weighted average method:
Total cost = Cost of units transferred out + Cost of ending work in process inventory
Cost of units transferred out = Number of units transferred out * Cost per equivalent unit of production
Cost of ending work in process inventory = Number of units in ending work in process inventory * Cost per equivalent unit of production
Direct materials:
Cost of units transferred out = 40,000 * $4.26 per unit = $170,400
Cost of ending work in process inventory = 5,000 * $4.26 per unit = $21,300
Total cost = $170,400 + $21,300 = $191,700
Conversion costs:
Cost of units transferred out = 40,000 * $3.94 per unit = $157,600
Cost of ending work in process inventory = 5,000 * $3.94 per unit = $19,700
Total cost = $157,600 + $19,700 = $177,300
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This chart shows four atoms, labeled W, X, Y, and Z. These atoms can combine with each other to form molecules.
Which combination of atoms will form a molecule, but not a compound?
W and X
X and Y
W and Z
Y and Z
Answer:
Where is the picture?
All molecules that contain carbon (C) and at least hydrogen (H) atoms is one example until I see what that missing diagram says.
hai quả cầu nhỏ mang hai điện tích có độ lớn bằng nhau, đặt cách nhau 10 cm trong chân không thì tác dụng lên nhau một lúc là 9.10^-3 N xác định điện tích của hai quả cầu đó
A solid cylinder of mass M = 1.25 kg and radius R = 13.5 cm pivots on a thin fixed frictionless bearing a string wrapped around the cylinder pulls downward with a force of F = 7.259 N
What is the magnitude of the angular acceleration of the cylinder?
86.03259 rad/s^2
Consider that instead of force F, a block with mass 0.74 kg with force = 7.259 N is attached to the cylinder with a mass less string.
What is now the magnitude of the angular acceleration of the cylinder
39.3943 rad/s^2
How far does the mass M travel downward before T equals 0.49S and T equals 0.69 S.
0.62755 m
The cylinder is changed to one with the same mass and radius but a different moment of inertia starting from mass starting from rest. The mass is now moved. The distance of 0.448 mass in the time interval of 0.47 seconds.
Find the Inertia of the new cylinder
The inertia of the new cylinder is 0.0566 kgm². Other answers provided are correct.
How to find inertia?The moment of inertia of the new cylinder can be calculated using the formula:
I = (M × d²) / (4 × Δθ)
Where:
M = mass of the cylinder
d = distance moved by the mass
Δθ = change in angular displacement (in radians)
Substituting the given values:
I = (1.25 × 0.448²) / (4 × 0.47)
I = 0.0566 kgm²
Therefore, the moment of inertia of the new cylinder is 0.0566 kgm².
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Which of the following is NOT an application of laser light?Select one:a.cutting materialb.precision surgeryc.satellite communication d.calibration of telescopes
Given:
The uses of LASER light
To find:
Which is not an application of LASER light
Explanation:
When cutting stainless steel or aluminium, LASER beams are used.
To remove warts, moles, sunspots, and tattoos in human bodies LASER lights are used.
LASER is used in communication also. Laser communication in space is the use of free-space optical communication in outer space.
One of the most important uses for lasers in astronomy is to reduce distortion on images taken with ground-based telescopes. So directly we cannot say that LASER is used for the calibration of telescopes.
Hence, calibration of telescopes is not an application of LASER light.
How can we make a zero correction in a length of 5. 30cm while the zero error is +0. 05cm?
After making the zero correction, the length is 5.25cm. To make a zero correction in a length, you need to subtract the zero error from the measured length.
1. Start by identifying the given values:
- Measured length = 5.30cm
- Zero error = +0.05cm
2. To correct for the zero error, subtract it from the measured length:
5.30cm - 0.05cm = 5.25cm
So, after making the zero correction, the length is 5.25cm.
In this case, the zero error is positive (+0.05cm), which means the actual length is slightly smaller than the measured length. By subtracting the zero error from the measured length, you are compensating for this discrepancy and obtaining a more accurate value.
Remember, zero correction involves adjusting measurements to account for any systematic error associated with the measuring instrument's zero mark. It ensures that the measurements reflect the true value.
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Question 76
When trying to establish economical hauling, what is most important?
a. hauling time
b. hauling distance
c. means of hauling
d. what is being hauled
When trying to establish economical hauling, hauling distance is usually the most important factor to consider. Hauling distance refers to the point of origin and the destination of the goods being transported.
The farther the distance, the higher the cost of transportation, as it requires more time, fuel, and resources to haul the goods. While other factors such as hauling time, means of hauling, and what is being hauled can also impact the cost and efficiency of transportation, the distance typically has the most significant impact on the overall cost. Therefore, it is crucial to consider the hauling distance when trying to establish an economical hauling plan. To minimize the cost of transportation, various strategies can be employed, such as optimizing the routing to reduce the distance traveled, utilizing more efficient means of transportation, such as rail or water transport, and using technology to improve logistics and reduce waste. Ultimately, the goal of establishing an economical hauling plan is to minimize transportation costs while ensuring that goods are delivered safely, reliably, and on time. By prioritizing hauling distance and utilizing efficient transportation strategies, businesses can achieve a more cost-effective and sustainable transportation plan.
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A charge of 5.67 x 10^-18 C is placed 3.5 x 10^-6 m away from another charge of -
3.79 x 10^-19 C. What is the force of attraction between them?
What is the mass of a truck if it is accelerating at a rate of 7m/s2 and hits a car with a force of 6,500N?
Answer:
Given
Force of car (F) =6500N
acceleration (a) =7m/s2
R.tc maas=x
Form
F =ma (divid by m both sides)
m=F/a
m=6500N/7m/s2
m=928.6kg
mass =928.6kg
PLEASE ASAP
Giving a brainlist
Answer:
Stationary
Explanation:
This is a position-time graph, meaning it shows an object's position over time. If the line is flat, it means that the object has not changed position during that time period, therefore it is stationary between the points g and h.
Hope this helped!
in a meter of 3 how many beats are in a measure also this is for music
Answer:
3 i think
Explanation:
hope this helps also Hiiii
the young’s modulus of aluminum is 69gpa, of nylon is 3gpa, of tungsten is 400gpa, and of copper is 117gpa. if equal-size samples were put under equivalent tensile stresses, how would you rank the materials by the degree to which they would be elongated?
Answer:
Least to most elongated: tungsten, copper, aluminum, nylon.
Explanation:
Materials with high Young's modulus are difficult to stretch. σ = Yε and ε = ΔL/L so an object with a high Young's modulus (Y) subject to a certain tensile stress (σ) will have a smaller strain than an object with a smaller Young's 's modulus subject to the same tensile stress. If strain (ε) is smaller, then ΔL will also be smaller.
the young’s modulus of aluminum is 69gpa, of nylon is 3gpa, of tungsten is 400gpa, and of copper is 117gpa. if equal-size samples were put under equivalent tensile stresses, Least to most elongated materials are from tungsten, copper, aluminum, nylon.
what is tensile stress ?Tensile stress can be defined the tension forces on a material, this force acts perpendicularly away from the surface of the substance.
Due to this tensile force, The pull on the material exerts tensile stress on the material, that tends to pull the material apart.
The tensile stress is a type of normal stress, in which a perpendicular force creates the stress to an object’s surface.
A material under tensile stress has axial forces tending to stretch the material and cause elongation. It is the force per unit area externally applied that results in stretching
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How long would it take a 500 w electric motor to do 150,000 j of work
Explanation:
300 second = 5 minutes
details
500 watts = 500 joules per second
Time = (1.50 x 10^5 joules)/(5 x 10^2 joules per second) = .3 x 10^3 seconds =300 second
a car of mass 1500 kg collides with a wall during a crush test. all of the energytrcnsferred during the collision is absorbed by the crumble zone. If the energy absorbed by the crumble zone is 4.3 x 10 to the power of 5 J, show that the speed of the car before the collision is 24 m/s.
KE = 1/2mv²
v² = 2KE/m
v² = 2 x 4.3 x 10⁵ : 1500
v = 24 m/s
based on your answer,what idea comes in to your mind?
Answer:
Homework · Free food · Loosing and robbing pencil · Forgetting book at home · Waiting for a Play time · My first bit · Lunch sharing · Assembly lines- standing in last just ...
Explanation:
Let h : Z → R be the point mass function of some distribution.
a) Let Ω = Z × Z. Show that if we define each ω = (ω1, ω2) ∈ Ω,
pω = hω1 hω2, then (pω)ω∈Ω is the point mass function of some distribution.
b) Consider the random variable X : Ω → Z, X(ω) = ω1 + ω2. Show that X's
the point mass function of the distribution, i.e. PX, is

Hints: the a) point is largely a repetition of the old one, but the latter point may require some thought. In particular, you should think about why it is enough to calculate
probability P({ω ∈ Ω : X(ω) = x}). For this, you should think about
that what this event has to do with the event
x - n}
and why it can be applied to calculate the probability of this event
definition of probability distribution.
We have demonstrated that (pω)ω∈Ω is the point mass function of some distribution, and that the random variable X has a point mass function PX equal to (pω)ω∈Ω.
In order to show that (pω)ω∈Ω is the point mass function of some distribution, we need to demonstrate that it satisfies the properties of a probability distribution.
a) Let's consider the properties of a probability distribution. Firstly, the values of pω must be non-negative for all ω ∈ Ω. This is true since pω is defined as the product of two non-negative values hω1 and hω2.
Secondly, the sum of probabilities over all possible outcomes must be equal to 1. In this case, we need to show that the sum of (pω)ω∈Ω over all possible ω in Ω is equal to 1. To do this, we can consider the sum:
Σ(pω)ω∈Ω = Σ(hω1 hω2)ω∈Ω
By the properties of the point mass function h, we know that Σhω1 = 1 and Σhω2 = 1. Therefore, the above expression becomes:
Σ(pω)ω∈Ω = Σ(hω1 hω2)ω∈Ω = 1 * 1 = 1
Thus, we have shown that (pω)ω∈Ω satisfies the properties of a probability distribution.
b) Now let's consider the random variable X(ω) = ω1 + ω2 and show that its point mass function PX is equal to (pω)ω∈Ω.
To calculate PX(x) = P({ω ∈ Ω : X(ω) = x}), we need to consider the event where the sum of the components ω1 and ω2 is equal to x. This can be expressed as:
{ω ∈ Ω : X(ω) = x} = {(ω1, ω2) ∈ Ω : ω1 + ω2 = x}
Now, notice that this event is equivalent to the event {ω1 = n, ω2 = x - n} for any fixed n. The probability of this event is given by pω1 pω2 = hω1 hω2, which matches the point mass function (pω)ω∈Ω.
By considering all possible values of n, we can cover all the cases for X(ω) = x, and therefore, we have shown that PX(x) is equal to (pω)ω∈Ω.
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