A 5-kg thin disk with radius 3-m is rolling along a flat surface at constant angular velocity 2 rad/s. Calculate the angular momentum about the lowest point [units: kg-m2] on the disk (i.e. the no-slip point that makes contact with the surface). g
Answers
The angular momentum of the disk about the lowest point is:= 45 kg-\(m^2/s\)
The angular momentum (L) of the disk about the lowest point can be calculated using the formula:
L = Iω
where I is the moment of inertia of the disk and ω is its angular velocity.
For a thin disk rotating about its axis perpendicular to its plane, the moment of inertia is given by:
I = (1/2) * m *\(r^2\)
where m is the mass of the disk and r is its radius.
Plugging in the given values, we get:
I = (1/2) * 5 kg *\((3 m)^2\)
I = 22.5 \(kg-m^2\)
Therefore, the angular momentum of the disk about the lowest point is:
L = Iω = 22.5 kg-m^2 * 2 rad/s = \(45 kg-m^2/s\)
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Related Questions
If a ball is given a push so that it has an initial velocity of 6 m/s down a certain inclined plane, then the distance it has rolled after t seconds is given by the following equation s(t) = 6t + 5t2 (a) Find the velocity after 2 seconds. m/s (b) How long does it take for the velocity to reach 40 m/s? (Round your answer to two decimal places.)
Answers
The velocity after 2 seconds is 26 m/s. It takes 11.50 seconds to reach a velocity of 40m/s.
Given the equation s(t) = 6t + 5t²
Let's find the velocity after 2 seconds.
(a) Velocity = (ds)/(dt)
Differentiating s(t) with respect to t, we get; ds(t) / dt = 6 + 10t
At t = 2,ds(t) / dt = 6 + 10t = 6 + 10(2) = 26m/s
Therefore, the velocity after 2 seconds is 26 m/s.
(b) We are to find how long it takes for the velocity to reach 40m/s. We know that the initial velocity of the ball, u = 6m/s. Acceleration = gsinθ = 9.81 x sin (angle of inclination) = 9.81 x 0.3 = 2.943 m/s²From the first equation of motion, v = u + at
We know v = 40m/s, u = 6m/s, a = 2.943m/s² and t is what we are to find, hence the equation becomes;40 = 6 + 2.943(t)t = (40 - 6) / 2.943t = 11.50s
Therefore, it takes 11.50 seconds to reach a velocity of 40m/s.
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how does dark energy affect the expansion of the universe
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Dark energy is a hypothetical form of energy that makes up about 68% of the total energy density of the universe. It has negative pressure and acts as a repulsive force, counteracting the attractive force of gravity. This leads to the expansion of the universe at an accelerating rate.
Dark energy is a hypothetical form of energy that is believed to exist in the universe and is responsible for the observed accelerated expansion of the universe. It is thought to make up about 68% of the total energy density of the universe. Dark energy is different from other forms of energy, such as matter and radiation, as it has negative pressure, causing it to have a repulsive gravitational effect.
This repulsive force counteracts the attractive force of gravity, leading to the expansion of the universe at an accelerating rate. While gravity tends to pull matter and galaxies together, dark energy acts as a sort of anti-gravity, pushing them apart. This means that the expansion of the universe is not only continuing but also accelerating.
The exact nature of dark energy is still not well understood, and it remains one of the biggest mysteries in modern physics. Scientists continue to study and explore the properties of dark energy in order to gain a deeper understanding of its role in the expansion of the universe.
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Dark energy is an unknown energy that seems to exist in the universe and causes the expansion of the universe. In the 1990s, scientists discovered that the universe was expanding at an accelerated rate. This acceleration was caused by a mysterious force, which they named dark energy.
Scientists still know very little about dark energy. Dark energy affects the expansion of the universe because it counteracts gravity. As a result, the universe is expanding at an ever-increasing rate. This means that galaxies are moving away from one another at a faster and faster rate. The force of dark energy is thought to be constant, which means that as the universe expands, the amount of dark energy in it increases. The result is that the universe is expanding at an accelerated rate and will continue to do so in the future.
In conclusion, dark energy plays a vital role in the expansion of the universe by counteracting gravity and causing the universe to expand at an ever-increasing rate. The precise nature of dark energy is still unknown, and scientists are working hard to understand this mysterious force.
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only a relatively small part of the electromagnetic spectrum is visible. what determines which bands of the electromagnetic spectrum are invisible to humans?
Answers
Answer:
hsnzbssj
Explanation:
skibbbbbbbbbbbbibiWhich statement best describes how greenhouse gases result in higher surface temperatures?
a. Greenhouse gases scatter solar radiation high in the atmosphere, distributing it evenly around the globe.
b. The bonds in greenhouse gases absorb infrared radiation emitted from Earth's surface and then emit this
energy as infrared radiation, some of which is redirected back at Earth's surface.
c. Greenhouse gases decompose under UV, releasing infrared radiation in the direction of Earth's surface.
d. Greenhouse gases absorb high energy solar radiation, increasing their kinetic energy.
Answers
The statement that best describes how greenhouse gases result in higher surface temperatures is greenhouse gases absorb and re-emit infrared radiation, some of which is redirected back at Earth's surface.
Greenhouse gases, such as carbon dioxide, methane, and water vapor, are present in the Earth's atmosphere. They absorb some of the infrared radiation emitted from Earth's surface, causing the gases to become excited. As they return to their lower-energy state, they emit the absorbed energy as infrared radiation in all directions.
Some of this radiation is directed back towards the Earth's surface, effectively trapping heat and raising surface temperatures. This phenomenon is known as the greenhouse effect, which is crucial for maintaining Earth's habitable climate but can cause global warming when an excess of greenhouse gases is present.
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If a moving object is 75% kinetic energy how much potential energy does it have?
Answers
Answer:
the ebergy
Explanation:
energy does it have if a miving object
A well lagged bar of 100cm has its end maintained at 90°C and 50°C respectively what is the temperature at a point 50cm from the hotter end.
Answers
two parakeets sit on a swing with their combined center of mass 19.5 cm below the pivot. at what frequency (in hz) do they swing?
Answers
The parakeets swing with a frequency of approximately 1.14 Hz.
The frequency at which the two parakeets swing on the swing can be determined by considering the relationship between the period of oscillation and the length of the pendulum.
The period of a pendulum is the time it takes for one complete oscillation, while the frequency is the reciprocal of the period, representing the number of oscillations per unit time.
In this case, we are given the distance of the combined center of mass of the parakeets below the pivot, which is 19.5 cm.
To find the frequency, we need to calculate the length of the equivalent pendulum, which is the distance from the pivot point to the combined center of mass of the parakeets.
Assuming the length of the equivalent pendulum is L, we can substitute the given values into the formula for the period of a pendulum, T = 2π√(L/g), where π is a constant (approximately 3.14159) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Converting the distance from centimeters to meters, we have L = 0.195 m. Substituting the values into the formula, we obtain T = 2π√(0.195/9.8).
Simplifying the equation, we have T = 2π√(0.02), which evaluates to approximately 0.879 s.
To calculate the frequency, we take the reciprocal of the period: f = 1/T = 1/0.879 ≈ 1.14 Hz.
Therefore, the parakeets swing with an approximate frequency of 1.14 Hz.
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A ball is thrown straight up into the air, with an initial speed of 28.2 m/s.
How fast is it moving after 1.00 s?
Answers
v = 28.2 -10 x 1
v = 18.2 m/s
what parts are in a tv
Answers
The exterior or housing, the audio reception and speaker system, the picture tube, and a complicated mass of electronics including cable.
What are the principal sets of parts in TV?Antennae input and output devices, a built-in antenna in most sets, a remote control receiver, and computer chips, make up the four main sets of parts that make up a television.
Typically, in addition to coaxial cable, HDMI, and other audio-video connectors, smart TVs also enable Ethernet, WiFi, USB, Bluetooth, and flash memory cards from digital cameras.
Therefore, the TV consists of exterior, audio reception, speaker system, and picture tube.
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A friend wants to know whether her wedding ring is made of pure gold or of a mixture of gold and silver.
a)Describe an investigation to answer her question.
b)Explain what is likely to be the most significant source of error in your answer.
The density of gold is 19. 3g/cm and the density of silver is 10. 5g/cm
Answers
The investigation provides a useful method for determining the composition of a wedding ring, but caution should be taken in interpreting the results due to potential sources of error.
a) To investigate whether the wedding ring is made of pure gold or a mixture of gold and silver, we can use the density of the ring as a clue. Firstly, we need to weigh the ring using a scale with high precision. Then, we can calculate the volume of the ring by measuring its dimensions and using the formula for the volume of a cylinder (V=πr²h). Once we have the weight and volume of the ring, we can calculate its density by dividing the weight by the volume. If the density of the ring is close to the density of pure gold (19.3g/cm³), then the ring is likely to be made of pure gold. However, if the density of the ring is lower than that of pure gold, it may indicate that the ring is made of a mixture of gold and silver.
b) The most significant source of error in our investigation is that the ring may contain other metals or impurities that affect its density. Additionally, the precision of the scale and measurements of the ring's dimensions can also affect the accuracy of our calculations. Therefore, we need to use high-precision instruments and repeat our measurements several times to ensure the accuracy of our results.
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Starburst galaxies within clusters of galaxies are seen to be extremely bright in the visible light of many newborn stars and in the infrared light of warm dust. These galaxies are probably direct evidence of:
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Starburst galaxies within clusters of galaxies provide direct evidence of intense star formation and the presence of warm dust, resulting in their brightness in visible and infrared light.
Starburst galaxies within clusters of galaxies serve as direct evidence of intense star formation and the presence of warm dust. These galaxies exhibit high levels of brightness both in visible light, due to the emission from newborn stars, and in infrared light, due to the thermal radiation from warm dust.
The term "starburst" refers to a burst of intense star formation happening at an accelerated rate compared to typical galaxies. In these galaxies, large quantities of gas and dust are gravitationally disturbed, triggering the formation of new stars. The newborn stars emit copious amounts of visible light, contributing to the brightness of starburst galaxies in the visible spectrum.
Furthermore, the intense star formation also leads to the production of significant amounts of warm dust within these galaxies. This warm dust absorbs the ultraviolet and visible light emitted by the newborn stars and re-emits it in the infrared part of the electromagnetic spectrum. As a result, starburst galaxies exhibit strong infrared emissions, providing direct evidence of the presence of warm dust.
In conclusion, the extreme brightness of starburst galaxies in both visible and infrared light serves as direct evidence of intense star formation and the presence of warm dust within these galaxies.
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1. A car starts from the rest on a circular track with a radius of 300 m. It accelerates with a constant tangential acceleration of a = 0.75 m/s?. Determine the distance traveled and the time elapsed"
Answers
Starting from rest on a circular track with a radius of 300 m and a constant tangential acceleration of 0.75 m/s², the car will travel a distance of approximately 0.2119 meters or 21.19 centimeters in 0.75 seconds.
To determine the distance traveled and the time elapsed by the car starting from rest on a circular track with a radius of 300 m and a constant tangential acceleration of 0.75 m/s², we can use the equations of circular motion.
The tangential acceleration is the rate of change of tangential velocity. Since the car starts from rest, its initial tangential velocity is zero (v₀ = 0).
Using the equation:
v = v₀ + at
where v is the final tangential velocity, v₀ is the initial tangential velocity, a is the tangential acceleration, and t is the time, we can solve for v:
v = 0 + (0.75 m/s²) * t
v = 0.75t m/s
The tangential velocity is related to the angular velocity (ω) and the radius (r) of the circular track:
v = ωr
Substituting the values:
0.75t = ω * 300
Since the car starts from rest, the initial angular velocity (ω₀) is zero. So, we have:
ω = ω₀ + αt
ω = 0 + (0.75 m/s²) * t
ω = 0.75t rad/s
We can now substitute the value of ω into the equation:
0.75t = (0.75t) * 300
Simplifying the equation gives:
0.75t = 225t
t = 0.75 seconds
The time elapsed is 0.75 seconds.
To calculate the distance traveled (s), we can use the equation:
s = v₀t + (1/2)at²
Since the initial velocity (v₀) is zero, the equation becomes:
s = (1/2)at²
s = (1/2)(0.75 m/s²)(0.75 s)²
s = (1/2)(0.75 m/s²)(0.5625 s²)
s = 0.2119 meters or approximately 21.19 centimeters
Therefore, the car travels a distance of approximately 0.2119 meters or 21.19 centimeters.
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why naked wire cannot use in electromagnet
Answers
Answer:
Explanation:
The wire in the windings needs to be insulated. Electricity takes the shortest path from one place to another. If the wire is bare, then adjacent turns are touching. The result is the current flows through the shortest path between the leads of the electromagnet, never flowing through a single turn of the coil.
If you found my answer useful then please mark me brainliest.
Answer: It will short between the windings; the current rather than passing around and around and around the core will just pass across the surface and create no magnetic field (which is proportional to the number of turns it takes around the core).
Explanation: If naked wire is used then the iron rod also acts as a conductor and short circuit takes place instead of showing magnetic effect of electricity to the molecules of the rod. Hence, naked wire is not used while making electromagnet.
How can you calculate the mechanical advantage a simple machine provides?
•divide the output force by the output displacement
•divide the output displacement by the input displacement
•Simple machines do not provide a mechanical advantage.
•divide the output force by the input force
Answers
Answer: divide the output force by the input force
Explanation: your welcome
The mechanical advantage which a simple machine provides can be calculated by dividing the output displacement by the input displacement. Thus, the correct option is B.
What is Mechanical advantage?The mechanical advantage of a simple machine can be defined as the ratio of output force applied on the object to input force. Mechanical advantage is also the ratio of load overcame by a simple machine to the effort which is applied by that machine. The formula for mechanical advantage of a simple machine is given as:
M.A. = output force / input force
M.A. = Load / effort
Thus, the mechanical advantage of a simple machine depends both on the output force of the simple machine and the input force of the machine.
Therefore, the correct option is B.
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according to a simplified model of a mammalian heart, at each pulse approximately 20 g of blood is accelerated from 0.25 m/s to 0.35 m/s during a period of 0.10 s. what is the magnitude of the force exerted by the heart muscle?
Answers
The magnitude of the force exerted by the heart muscle is 0.02N
since
F = m dv/dt
= 0.02(0.35-0.25)/0.10
F =0.02N.
A force is an influence in physics that can change the motion of an object. A force can cause a mass object to change its velocity, or accelerate. Intuitively, force can be described as a push or a pull. A force is a vector quantity because it has both magnitude and direction.
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Elvira throws a meatball at the wall. When the meatball hits the wall, it makes a sound. The kinetic energy of the meatball is transferred into the surroundings through what? (Hint: your answer should be a type of wave. )
Answers
Answer:
sound waves
Explanation:
The kinetic energy of the meatball is transferred into the surroundings through sound waves.
When the meatball hits the wall, the kinetic energy of the meatball causes vibrations in the wall and the surrounding air. These vibrations create sound waves which transfers the kinetic energy to sound energy until it runs out of energy.
When an object / medium vibrates, the particle of the object / medium starts to move. This movement of particles is called as sound waves. It is a type of mechanical wave.
Therefore, the kinetic energy of the meatball is transferred into the surroundings through sound waves.
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Using two boxes, design two solar ovens. Before you build, describe your designs. Discuss elements such as materials, box shape, and color, and explain why you think the oven will work.
Answers
Answer:
Find two boxes just a bit smaller than the other. The long sides of the box should be less than twice as long as the short side of the boxes. The smaller box should fit inside the larger box with about 1 inch in each direction to spare. The boxes can be cut down so that they fit together properly. Leave the flaps on the boxes. Buy a small sheet of Plexiglas (tm) a little bit smaller than the width and length of the top of the box. You will also need four pieces of cardboard to use for reflectors.
Explanation:
Yelling
Question 3
Exposure to abnormal or prolonged amounts of heat and humidity without adequate fluid
intake can cause various types of heat related illnesses.
True
1 pts
False
Answers
True. Various heat-related disorders can be brought on by inadequate hydration combined with exposure to extreme heat and humidity.
What are the effects of prolonged exposure to high heat and humidity?One of three disorders brought on by excessive heat, with heat cramps being the least dangerous and heatstroke being the most serious, is heat exhaustion. High temperatures, particularly when there is also a high humidity level, and intensive physical activity are the main causes of heat illness.
Which four types of heat disease are there?According to T8 CCR Section 3395, "Heat Illness" refers to a dangerous medical illness caused by the body's incapacity to handle a specific amount of heat. Examples of this ailment include heat cramps, heat exhaustion, heat syncope, and heat stroke.
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If a simple circuit connected to a battery had its wires replaced with thinner (finer) wires, what would the ammeter read
Answers
The ammeter would read a lower value if the wires in a simple circuit connected to a battery were replaced with thinner (finer) wires.
In a simple circuit, the ammeter is used to measure the electric current flowing through the circuit. According to Ohm's Law, the current (I) in a circuit is directly proportional to the voltage (V) applied and inversely proportional to the resistance (R) in the circuit, given by the equation I = V/R.
When thinner wires are used in the circuit, they have higher resistance compared to thicker wires of the same material and length. This is because the resistance of a wire is inversely proportional to its cross-sectional area. Thinner wires have a smaller cross-sectional area, resulting in higher resistance.
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10.0 points
A reconnaissance plane flies 446 km away from
its base at 684 m/s, then flies back to its base
at 1026 m/s.
What is its average speed?
Answer in units of m/s.
007
Answers
The average speed in m/s, given the data is 820.80 m/s
What is speed?Speed is the distance travelled per unit. Mathematically, it can be expressed as:
Speed = distance / time
What is average speed?This is the total distance travelled divided by the total time taken to cover the distance.
Average speed = total distance / total time
How to determine the time taken to fly away from the baseDistance = 446 km = 446 × 1000 = 446000 mSpeed = 684 m/sTime to fly away =?Speed = distance / time
684 = 446000 / time
Cross multiply
684 × time = 446000
Divide both side by 684
Time = 446000 / 684
Time to fly away = 652.05 s
How to determine the time taken to return the baseDistance = 446 km = 446 × 1000 = 446000 mSpeed = 1026 m/sTime to return =?Speed = distance / time
1026 = 446000 / time
Cross multiply
1026 × time = 446000
Divide both side by 1026
Time = 446000 / 1026
Time to return = 434.70 s
How to determine the average speedTotal distance = 446000 + 446000 = 892000 mTime to fly away = 652.05 sTime to return = 434.70 sTotal time = 652.05 + 434.70 = 1086.75 sAverage speed =?The average speed can be obtained as follow:
Average speed = total distance / total time
Average speed = 892000 / 1086.75
Average speed = 820.80 m/s
Thus, the average speed is 820.80 m/s
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Which statement describes gravity?
There is no defined unit of measurement for gravity.
O Gravity is the force that pulls objects toward Earth's center.
Objects that have a small mass will have no gravitational pull.
Gravitational pull between two objects decreases as the mass of one increases.
Answers
Gravity is a fundamental, universal force that pulls objects toward Earth's center. It increases with mass and decreases with distance. Measured in Newtons, it affects all objects.
Gravity is the force that pulls objects towards Earth's center. Gravitational pull increases as the mass of one object increases, while it decreases as the distance between two objects increases. These statements describe gravity.Gravity is a fundamental force of nature, which means that it is always present. It holds planets and stars in their orbits around the sun, and it keeps objects on Earth's surface.Gravity is a universal force, meaning that it affects all objects in the universe. The gravitational pull between two objects is proportional to their masses and the distance between them.There is a defined unit of measurement for gravity known as Newtons. Newtons are used to measure the force of gravity acting on an object. Objects that have a small mass still have a gravitational pull, but it is weaker than objects with a larger mass.For more questions on Gravity
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The correct question would be as
Which statement describes gravity? Select three options. There is no defined unit of measurement for gravity.
Gravity is the force that pulls objects toward Earth’s center.
Objects that have a small mass will have no gravitational pull.
Gravitational pull between two objects increases as the mass of one increases.
Gravitational pull decreases when the distance between two objects increases
Which boat experience the most force? *
Answers
Answer:
not sure u dont show anything??
Explanation:
Answer:
no idea
Explanation:
A 63.0 kg skier starts from rest at the top of a ski slope 70.0 m high. Part A If frictional forces do -11.0 kJ of work on her as she descends, how fast is she going at the bottom of the slope? Express your answer in meters per second to three significant figures. ΨΗ ΑΣφ ? m/s Submit Request Answer Part B Now moving horizontally, the skier crosses a patch of soft snow where we = 0.24. If the patch is 82.0 m wide and the average force of air resistance on the skier is 170 N, how fast is she going after crossing the patch? Express your answer in meters per second to three significant figures. ΡΟ ΑΣφ ? Part The skier hits a snowdrift and penetrates 3.0 m into it before coming to a stop. What is the magnitude of the average force exerted on her by the snowdrift as it stops her? Express your answer in newtons to three significant figures. VAX¢ ? F- N
Answers
A 63.0 kg skier starts from rest at the top of a ski slope 70.0 m high. The solution to the given problem is explained below. Part A The potential energy of the skier is converted to kinetic energy when she descends down the slope.
The kinetic energy is given by: K = PE - W f
where, PE = mgh
Wf = -11 kJ
= -11000 J
m = 63 kg
g = 9.8 m/s²h
= 70 m
Substituting the given values in the above formula, we get:
K = 63 × 9.8 × 70 - 11000J
= 42954J
The kinetic energy is converted to kinetic energy of motion of the skier at the bottom of the slope. Therefore,
K = 1/2mv²
wherev is the speed of the skier at the bottom of the slope.
Substituting the given values, we get:
42954
= 1/2 × 63 × v²
v = √(42954 / (1/2 × 63))
= 27.8 m/s (rounded to three significant figures)
Therefore, the skier is going at 27.8 m/s at the bottom of the slope.
Part B We know that the work done by the air resistance is given by:
W = f d
where: f = frictional force acting on the skier
d = distance traveled by the skier
We = 0.24d = 82.0 mf = 170 N
Substituting the given values in the above formula, we get: W = 170 × 82.0 × 0.24J= 3230.4J
The kinetic energy of the skier after crossing the patch of soft snow is the same as the work done against the air resistance. K = 1/2mv²where v is the speed of the skier after crossing the patch.
Substituting the given values, we get:
3230.4 = 1/2 × 63 × v²
v = √(3230.4 / (1/2 × 63))
= 11.1 m/s (rounded to three significant figures)
Therefore, the skier is going at 11.1 m/s after crossing the patch of soft snow.
Part C We know that the work done by the snowdrift is given by:
W = F d c
where : F = force exerted on the skier by the snowdrift
d = distance traveled by the skier into the snowdrift We know that the change in kinetic energy of the skier is equal to the work done by the snowdrift. Therefore, K = W where K = 1/2mv²v = final velocity of the skier
Substituting the given values in the above formula, we get:
1/2 × 63 × v²
= F × 3.0
F = (1/2 × 63 × v²) / 3.0
where
v = 27.8 m/s (obtained from Part A)
Substituting the given value of v in the above formula, we get: F = 6067 N (rounded to three significant figures)
Therefore, the magnitude of the average force exerted on the skier by the snowdrift as it stops her is 6067 N.
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Consider this situation: A book is at rest upon a table. Of the
forces listed, identify which act upon the book.
Answers
Answer:
Explanation:
The book is affected by the force of gravity from the Earth and the reaction of the support from the table
Please please please please please please please please please please please please please please please please please please please
Answers
What are the layers of sedimentary rock called?
Answers
A single layer of sedimentary rock is known as a bed or stratum (plural: strata). Each bed contains rock of a single composition and texture and can clearly be distinguished from the beds or strata above and below. This layering of strata is unique to sedimentary rocks and occurs because of the way that sediments are deposited.
Describe the components of health related fitness edge 2021
Answers
Answer:
There are five components of physical fitness: (1) body composition, (2) flexibility, (3) muscular strength, (4) muscular endurance, and (5) cardiorespiratory endurance. A well-balanced exercise program should include activities that address all of the health-related components of fitness.
Explanation:
Is this right. Please help me ITS SOCIOLOGY
Answers
Answer:
Yes
Explanation:
sorry if im wrong
Equivalent forces derivation problem. Figures see Prelab questions 5.1.1, page 51-52 on lab manual. EXAMPLE: Derive the formula for F 3in terms of the experimentally measured quantities m 1, m 2,θ 1, and θ 2
. [Answer: F 3=m 1gcosθ 1+m2gcosθ 2.] Make sure you understand how this formula was derived. QUESTION: If the mass of both weights is 225gm, the first mass is located 20 degrees north of east, the second mass is located 20 degrees south of east, and the transducer sensitivity is 0.5 volts/Newton, how large a voltage do you expect to measure? Assume the transducer has been properly zeroed so that V=0 when F 3=0. Please express your answers with 1 decimal place. Volts
Answers
The voltage is expected to measure as 1759 volts. The formula for F₃, as per the experimentally measured quantities of m₁, m₂, θ₁, and θ₂, is F₃=m₁gcosθ₁+m₂gcosθ₂
The transducer sensitivity is 0.5 volts/Newton and the mass of both weights is 225 gm. The first mass is located 20 degrees north of east, and the second mass is located 20 degrees south of east.
Given the transducer has been properly zeroed so that V = 0 when F₃ = 0.The formula for F₃, as per the experimentally measured quantities of m₁, m₂, θ₁, and θ₂, is given below:
F₃=m₁gcosθ₁+m₂gcosθ₂
Here, m₁ = m₂= 225 gm, and θ₁ = 20° north of east and θ₂ = 20° south of east. Let's put these values in the above formula:
F₃=225×9.8cos20°+225×9.8cos(20°)
F₃= 879.5 N
= 879.5/0.5 V
= 1759 volts
Therefore, the voltage is expected to measure as 1759 volts.
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Using the formula for the wave function given in Equation (9), ym (x,t) = Am cos(2fmt)cos(mx / L) show that the wave
function for a machine that is unclamped at both ends has antinodes at both ends.
Answers
Since the cosine function is equal to 1 when its argument is equal to an even multiple of π/2, and -1 when its argument is equal to an odd multiple of π/2, we can see that the wave function has antinodes at both ends of the machine .
The boundary conditions for a machine that is unclamped at both ends are that the displacement (y) of the machine at both ends is equal to zero. Mathematically, this can be written as:
y(0, t) = 0 and y(L, t) = 0
Substituting these boundary conditions into the given formula for the wave function, we get:
y(0, t) = Am cos(2πfmt)cos(0) = 0
y(L, t) = Am cos(2πfmt)cos(mπL/L) = 0
Since the cosine function is equal to zero when its argument is equal to an odd multiple of π/2, we can set the argument of the second cosine function equal to (2n - 1)π/2, where n is an integer. This gives us:
mπL/L = (2n - 1)π/2
Solving for m, we get:
m = (2n - 1)(2/L)
Therefore, the possible values of m are odd multiples of (2/L), i.e., m = 1/L, 3/L, 5/L, and so on.
Substituting these values of m back into the wave function, we get:
y(x,t) = Am cos(2πfmt)cos(mπx/L) = Am cos(2πfmt)cos((2n - 1)πx/2L)
Since the cosine function is equal to 1 when its argument is equal to an even multiple of π/2, and -1 when its argument is equal to an odd multiple of π/2, we can see that the wave function has antinodes at both ends of the machine (i.e., at x = 0 and x = L) when n is even, and nodes at both ends when n is odd.
Therefore, we have shown that the wave function for a machine that is unclamped at both ends has antinodes at both ends.
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