g two horizontal 5-kip forces are applied to pin b of the assembly shown. knowing that a pin of 0.8-in. diameter is used at each connection, determine the maximum value of the average nor- mal stress (a) in link ab, (b) in link bc.

The femur of the runner compresses by 0.0006 m when the normal force of the ground on the foot reaches three times the runner's body weight.

Given:

Length of the femur, L = 52 cm = 0.52 m

Mass of the runner, m = 79 kg

Cross-sectional area of the bone of the femur, A = 5.2 x 10⁻⁴m²

Young's modulus of the bone of the femur, Y = 1.6 x 10¹⁰ N/m²

The force exerted on the foot when it strikes the pavement, F = 3mg

The stress on the bone can be calculated as:

Stress = Force / Area

Stress = 3mg / A

The strain on the bone can be calculated as:

Strain = Stress / Young's modulus

Strain = (3mg / A) / Y

The change in length of the femur, ΔL, can be calculated as:

ΔL = Strain x Length

ΔL = [(3mg / A) / Y] x L

Substituting the given values, we get:

ΔL = [(3 x 79 x 9.81) / (5.2 x 10⁻⁴ x 1.6 x 10¹⁰)] x 0.52

ΔL ≈ 0.0006 m

Therefore, the femur of the runner compresses by 0.0006 m when the normal force of the ground on the foot reaches three times the runner's body weight.

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Explanation:

force or gravitational force

1.01 × 106 K would the root-mean-square speed of hydrogen, H2, molecules equal 11.2 km/s.

Molecules are the smallest units of matter that have the characteristics of a particular chemical element or compound. They are made up of two or more atoms held together by covalent, ionic, or metallic bonds.

The root-mean-square speed of a gas is given by the equation vrms = (3kT/m)^1/2, where k is the Boltzmann constant, T is the temperature in Kelvin, and m is the mass of the molecule.

Plugging in the given values, we get: vrms = (3*1.38e-23*T/1.67e-27)^1/2

vrms = 11.2 km/s

Solving for T, we get: T = (1.67e-27×(11.2e3)²)/(3×1.38e-23)

T = 1.01e6 K

Therefore, the temperature at which the root-mean-square speed of hydrogen, H2, molecules equals 11.2 km/s is 1.01 × 106 K.

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The wavelength of the absorbed photon when the oscillator undergoes a transition from the n = 2 to the n = 3 energy level is 4.93 μm.

The energy of a photon with a particular wavelength is given by the equation:

E = hc/λ

where E is the energy of the photon, h is Planck's constant (6.626 x 10⁻³⁴ J s), c is the speed of light (2.998 x 10⁸ m/s), and λ is the wavelength of the photon.

When a harmonic oscillator undergoes a transition from the n = 1 to the n = 2 energy level, the energy absorbed is equal to the difference in energy between the two levels:

ΔE = E₂ - E₁ = hν

where ΔE is the energy absorbed, E₁ is the energy of the n = 1 level, E₂ is the energy of the n = 2 level, and ν is the frequency of the absorbed photon. Since the oscillator is a harmonic oscillator, the energy levels are given by the equation:

En = (n + 1/2)hν0

where En is the energy of the nth level, ν0 is the frequency of the oscillator, and n is an integer.

We can rearrange this equation to solve for the frequency of the oscillator:

ν₀ = En / ((n + 1/2)h)

For the n = 1 to n = 2 transition, we can set n = 1 and n = 2 to find the frequencies of the two levels:

ν₁ = E1 / (3/2 h)

ν₂ = E2 / (5/2 h)

Substituting the energy difference ΔE = hν and the wavelength λ = c/ν into these equations, we get:

ν₂ = (E₁ + ΔE) / (5/2 h) = (hc/λ + ΔE) / (5/2 h)

λ = hc / (ν₂ (5/2 h) - ΔE)

λ = (6.626 x 10⁻³⁴ J s x 2.998 x 10⁸ m/s) / ((1.5 x 6.90 x 10⁻⁶ m⁻¹) x (5/2 x 6.626 x 10⁻³⁴J s) - (4.64 x 10⁻¹⁹J))

λ = 4.93 μm

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The star will lose approximately 0.075 m (or 7.5% of its initial mass) over its 3-million-year lifetime.the rate of mass loss can vary depending on a star's age, size, and other factors

To answer this question, we can use the formula for mass loss rate over time, which is:
Mass loss = Mass loss rate x Lifetime
Since the main-sequence star in question has a mass of 25 m, we can substitute that into the formula and solve for the mass loss:
Mass loss = 10^(-6) m/year x 3 x 10^6 years x 25 m
Mass loss = 0.075 m
Therefore, the star will lose approximately 0.075 m (or 7.5% of its initial mass) over its 3-million-year lifetime.
It's important to note that the rate of mass loss can vary depending on a star's age, size, and other factors. However, this calculation gives us an estimate of the amount of mass that could be lost based on the given information.

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The temperature of the colder reservoir is 140 K. so the answer will be none of the above (140 K).

The efficiency of an ideal Carnot engine is given by the formula:
efficiency = 1 - Tc/Th
where Tc is the temperature of the colder reservoir and Th is the temperature of the warmer reservoir. We are given that the engine extracts Qh = 300 J of heat from the warmer reservoir and does W = 200 J of work during each cycle. Using the first law of thermodynamics, we know that Qh = W + Qc, where Qc is the heat released to the colder reservoir. Therefore, Qc = Qh - W = 300 J - 200 J = 100 J.
Using the efficiency formula, we can solve for Tc:
efficiency = 1 - Tc/Th
0.4 = 1 - Tc/233 K
Tc/233 K = 0.6
Tc = 0.6 x 233 K = 140 K

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To determine the location of the image in this scenario, we can use the thin lens equation, which is 1/f = 1/do + 1/di, where f is the focal length of the lens, do is the distance of the object from the lens,

and di is the distance of the image from the lens. We are given that the object is 10.0 cm to the left of the lens (i.e. do = -10.0 cm) and that the focal length of the lens is 16.0 cm.

Plugging these values into the thin lens equation, we get:

1/16.0 = 1/-10.0 + 1/di

Solving for di, we get:

di = (-1/1.6 + 1/-10.0)^-1

di = -6.67 cm

Therefore, the location of the image is 6.67 cm to the left of the lens (i.e. the image is virtual and located on the same side of the lens as the object).

It is important to note that the negative sign indicates that the image is virtual, which means that it cannot be projected onto a screen. Instead, it appears to be behind the lens when viewed from the same side as the object.

In summary, the location of the image is -6.67 cm.

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The specific heat capacity of the water (4.184 J/g°C), and ΔT is the change in temperature (1.05°C). is 13.2 J.

Specific heat is the amount of heat required to raise the temperature of one gram of a material by one degree Celsius. It is usually measured in units of joules per gram per degree Celsius (J/g°C). The concept of specific heat is important in many areas of science, including thermodynamics and chemistry. It is used to calculate the amount of energy required to change the temperature of a given mass of a substance, and it is also used to calculate the thermal conductivity of a material.

The heat involved in the dissolution of KCl can be calculated using the following equation:
q = m x c x ΔT
where q is the heat, m is the mass of the KCl (3.00 g), c is the specific heat capacity of the water (4.184 J/g°C), and ΔT is the change in temperature (1.05°C).
Therefore, q = (3.00 g) x (4.184 J/g°C) x (1.05°C)
= 13.2 J.
The assumption made here is that the coffee cup calorimeter is perfectly insulated, so that all the heat gained or lost by the KCl is equal to the heat gained or lost by the water.

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The tangential acceleration of a point on the rim is twice that of a point halfway between the center and the rim.

The tangential acceleration of a point on the rim of a wheel spinning on an axis through its center is given by:

a_rim = r * α

where r is the radius of the wheel and α is its angular acceleration.

The tangential acceleration of a point halfway between the center and the rim is given by:

a_midpoint = (1/2) * r * α

since the midpoint is only half the distance from the center as the rim.

Therefore, the ratio of the tangential acceleration of a point on the rim to the tangential acceleration of a point halfway between the center and the rim is:

a_rim / a_midpoint = (r * α) / [(1/2) * r * α] = 2

So, the tangential acceleration of a point on the rim is twice that of a point halfway between the center and the rim.

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The columns containing the most useful information for estimating the density of object C are the columns indicating whether it floats in liquids 1 and 3.

What is Density?

The density of a substance can be calculated by dividing its mass by its volume. The resulting value represents the amount of mass in a given volume of the substance. For example, a substance with a density of 1 g/cm³ would have a mass of 1 gram per cubic centimeter of volume.

Comparing the buoyancy of objects A, B, and C in liquids 1 and 3, we can see that object C floats in liquid 1 while objects A and B sink. This suggests that the density of object C is lower than that of liquids 2, 3, and 4. Similarly, in liquid 3, object C floats while objects A and B sink, indicating that the density of object C is lower than that of liquids 2, 4, and also lower than that of liquid 1.

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The speed of the skier on reaching the bottom of the slope. The answer is B) 31 m/s.

To solve this problem, we can use the conservation of energy principle. Initially, the skier has only potential energy, which is given by mgh, where m is the mass of the skier, g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the slope (50 m). When the skier reaches the bottom of the slope, all of this potential energy is converted to kinetic energy, which is given by (1/2)mv^2, where v is the velocity of the skier.

Setting these two equations equal to each other, we have:

mgh = (1/2)mv^2

Simplifying, we get:

v = sqrt(2gh)

Substituting the given values, we have:

v = sqrt(2 x 9.8 m/s^2 x 50 m) = 31 m/s

Therefore, the answer is B) 31 m/s.

It's important to note that neglecting friction is an idealized assumption, as frictional forces are present in most real-world situations. However, this assumption is often made in physics problems to simplify the calculations and focus on the main concepts.

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According to the National Highway Traffic Safety Administration (NHTSA), children must ride in a federally approved safety or booster seat in the rear of the vehicle until they are at least 8 years old or until they reach a height of 4 feet 9 inches (145 cm), whichever comes first.

The National Highway Traffic Safety Administration (NHTSA) is the agency within the U.S. Department of Transportation that works to reduce deaths and injuries and economic costs due to motor vehicle crashes. NHTSA works to deliver safer roads by encouraging Americans to make safer choices when they drive, ride, and walk; advancing lifesaving vehicle safety technologies; and supporting state and local police in their efforts to enforce the rules of the road that protect us all. By researching new vehicle safety technologies, mandating their inclusion on new vehicles, and rooting out defects in vehicles and equipment, NHTSA helps protect Americans when they’re on the road. NHTSA conducts research on how vehicle improvements and other technological advances can better protect people in a crash (crashworthiness) and reduce the likelihood of crashes (crash avoidance).

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Your apparent weight is equal to your weight in an elevator not accelerating.

Apparent weight refers to the force a person feels due to gravity and any other forces acting on them, such as the normal force (the force exerted by a surface that supports an object).

1. In an elevator accelerating upwards: Your apparent weight is greater than your actual weight. This is because the upward acceleration adds to the gravitational force, making you feel heavier.

2. In an elevator in free fall: Your apparent weight is zero. This is because there is no normal force acting on you since the elevator is in free fall, which means you experience weightlessness.

3. In an elevator accelerating downwards: Your apparent weight is less than your actual weight. This is because the downward acceleration is working against the force of gravity, making you feel lighter.

4. In an elevator not accelerating Your apparent weight is equal to your weight because the normal force acting on you is equal to the force of gravity. When the elevator is not accelerating, there is no additional force affecting your apparent weight, so it remains equal to your actual weight.

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Barred spirals are a type of galaxy that have a bar-shaped structure running through their centers, while ordinary spirals have a central bulge and arms that spiral outwards. Based on observations, it's estimated that roughly 30-35% of spiral galaxies are barred, with the remaining 65-70% being ordinary. However, it's important to note that these percentages may not be completely accurate, as different studies may define and identify barred spirals differently. Additionally, the overall number of galaxies in the universe is still uncertain and difficult to estimate, but current estimates suggest that there may be around 2 trillion galaxies in the observable universe alone.
Hi! Barred spiral galaxies and ordinary spiral galaxies are two distinct types of galaxies found in the universe. Barred spirals have a central bar-shaped structure, while ordinary spirals do not.

In general, it is estimated that about two-thirds of spiral galaxies in the universe are barred spirals, while the remaining one-third are ordinary spirals. This indicates that barred spirals are more common than ordinary spirals in the universe.

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The distance along an arc on the surface of the earth, with a central angle of 2 minutes (or 1/30 of a degree), is equal to 264,800 miles

Earth is the third planet from the Sun and the fifth-largest planet in the Solar System. It is the only planet known to have an atmosphere containing free oxygen, oceans of liquid water on its surface, and, of course, life. Earth is the only planet in our Solar System not named after a Greek or Roman deity. Its solid outer surface is called the crust, and its molten interior is the mantle. It has many natural satellites, including the Moon, and its rotation and orbit around the Sun give way to the seasons, day and night, and the length of the year.

The distance along an arc is equal to the circumference of the circle times the central angle (in radians). The circumference of a circle is equal to 2πr, where r is the radius of the circle.
Therefore, the distance along an arc on the surface of the earth, with a central angle of 2 minutes (or 1/30 of a degree), is equal to:
(2π)(3960 miles)(1/30 degree) = 264,800 miles.

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Complete Question:
Find the distance along an arc on the surface of the earth that subtends a central angle of 2 minutes (1 minute = 1/60 degree). The radius of the earth is 3960 miles

Between Mars and Jupiter's orbits is where the asteroid belt is located. Because of orbital resonances with Mars, it exists. Here option B is the correct answer.

The asteroid belt is a region in our solar system that contains numerous small celestial bodies known as asteroids, with some as large as 590 miles (940 kilometers) in diameter. It is located between the orbits of Mars and Jupiter, with an average distance of about 280 million miles (450 million kilometers) from the Sun.

The asteroid belt exists because of gravitational interactions with Jupiter, which prevented the formation of a planet between Mars and Jupiter. However, the asteroids in the belt are not in stable orbits and can be perturbed by the gravitational influence of other planets, leading to collisions and ejections from the belt.

In particular, the asteroid belt's location and shape are strongly influenced by orbital resonances with Mars. Orbital resonance occurs when two objects orbiting the Sun exert a regular gravitational influence on each other, causing their orbits to be synchronized. As Mars orbits the Sun, it exerts a regular gravitational pull on the asteroids in the belt, causing them to follow specific orbital patterns.

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Complete question:

Choose the correct explanation, where is the asteroid belt located, and why.

A - The asteroid belt lies between the orbits of Jupiter and Saturn. It exists because of orbital resonances with Jupiter.

B - The asteroid belt lies between the orbits of Mars and Jupiter. It exists because of orbital resonances with Mars.

C - The asteroid belt lies between the orbits of Jupiter and Saturn. It exists because of orbital resonances with Saturn.

D - The asteroid belt lies between the orbits of Mars and Jupiter. It exists because of orbital resonances with Jupiter.

The wavelength of an electron with energy 25 eV is approximately 5.35 x [tex]10^{-10}[/tex] meters.

To find the wavelength of an electron with energy 25 eV, we can use the de Broglie wavelength formula

λ = h/p

Where λ is the wavelength, h is the Planck constant (6.626 x [tex]10^{-34}[/tex] J*s), and p is the momentum of the electron.

The momentum of an electron with energy E can be found using the formula

p = √(2mE)

Where m is the mass of the electron.

Substituting the given values, we get

p = √(2(9.109 x [tex]10^{-31}[/tex] kg)(25 eV)(1.602 x [tex]10^{-19}[/tex] J/eV)) = 1.24 x [tex]10^{-24}[/tex]kg m/s

Now, we can calculate the wavelength

λ = h/p = (6.626 x [tex]10^{-34}[/tex] J*s)/(1.24 x [tex]10^{-24}[/tex]kg m/s) = 5.35 x [tex]10^{-10}[/tex] m

Therefore, the wavelength of an electron with energy 25 eV is approximately 5.35 x [tex]10^{-10}[/tex] meters.

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The critical angle (the smallest angle that a light ray inside the diamond can make with the normal and reflect back inside the diamond) is θ¹ = arcsin(1/2.42) = 24.4°.

Diamond is the hardest naturally occurring mineral on Earth, composed of pure carbon atoms in a lattice arrangement. Its hardness, brilliance, and sparkle make it ideal for use in jewelry, and it is the traditional choice for engagement rings and wedding bands. It is also used in industry for industrial applications, such as cutting and polishing tools.

The critical angle is determined by the equation n¹sinθ¹ = n²sinθ², where n¹ and θ¹ are the index of refraction and angle of the first medium, respectively, and n² and θ² are the index of refraction and angle of the second medium, respectively.

In this case, the first medium is diamond (n¹ = 2.42) and the second medium is air (n² = 1.00). Solving for the angle of the second medium (θ²) gives θ² = arcsin(2.42 sinθ¹).

Therefore, the critical angle (the smallest angle that a light ray inside the diamond can make with the normal and reflect back inside the diamond) is θ¹ = arcsin(1/2.42) = 24.4°.

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To determine the corresponding values of the mean velocity (um) and mean temperature (tm), we first need to measure the velocity and temperature at different points in the fluid. We can then calculate the mean velocity and temperature using these measurements.



The mean velocity (um) is the average velocity of the fluid over a given area. It is calculated by dividing the total volume flow rate by the cross-sectional area of the flow. The mean temperature (tm) is the average temperature of the fluid over a given area. It is calculated by taking the average of the temperature measurements at different points in the fluid.

To plot the velocity and temperature distributions, we need to measure the velocity and temperature at different points in the fluid and plot them on a graph. We can then connect the data points to get a visual representation of the distribution.

Whether or not the values of um and tm appear reasonable will depend on the specific system being studied. We would need to compare our results to previous studies or theoretical models to determine if our values are reasonable.

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An electron travels due north through a vacuum in a region of uniform magnetic field 4 that is also directed due north. It will: be unaffected by the field.

Electron is an open source library developed by GitHub that allows for the development of cross-platform desktop applications using web technologies such as HTML, JavaScript, and CSS. Electron applications are built with web technologies but have access to native operating system functionalities such as file system access and native notifications. Electron applications can be distributed through the Mac App Store, Windows Store, and Linux distributions such as Ubuntu and Fedora. Electron applications are designed to work on multiple platforms and provide a single codebase for developers looking to develop applications for multiple platforms. Electron applications are fast, secure, and user friendly, providing users with an intuitive interface and a seamless experience. Electron is a powerful tool for creating native applications that can be used on multiple platforms.

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If our star (the Sun) shrinks into a black hole, the orbit of the planets of the solar system would remain same.

What is the planets' gravitational law?

A more precise formulation of the law is "Whenever one body exerts force upon a second body, the second body exerts an equal and opposite force upon the first body." That is, when a planet pulls on the sun with gravity, the planet pulls on the sun with a force that is comparable in strength.

A star the size of our Sun cannot become a black hole. However, if a black hole of the same mass appeared overnight to replace the Sun, the Solar System would remain intact. The gravitational attraction of a solar-mass black hole would be equal to that of the Sun. The planets would actually keep moving forward in their orbits.

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When a large parallel-plate capacitor is being charged, the electric field between the plates increases at a rate of (dE/dt).

According to Maxwell's equations, particularly the Ampere-Maxwell law, a changing electric field generates a magnetic field (B) in the region between the plates.

For the charging capacitor, the magnetic field (B) will form closed loops around the edges of the plates.

The direction of the magnetic field can be determined using the right-hand rule, with your thumb pointing in the direction of the increasing electric field (from the positive plate to the negative plate), and your curled fingers indicating the direction of the magnetic field.

In summary, during the charging process of a parallel-plate capacitor, the increasing electric field (dE/dt) will induce a magnetic field (B) between the plates.

The magnetic field will form closed loops around the edges of the plates, with the direction determined by the right-hand rule.

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The new force of attraction between the two objects would be 1.78 units.

Force is an influence that causes an object to change its velocity, shape or direction. Forces can be categorized into contact forces and non-contact forces. Contact forces are those that require physical contact between two objects, such as a person pushing a box, while non-contact forces are those that act without physical contact, such as gravity or magnetism. Forces can also be described as either balanced or unbalanced.

The force of attraction between two objects is inversely proportional to the square of the distance between them. This means that if the distance is tripled, then the force of attraction will be reduced to one ninth of its original value. Therefore, the new force of attraction between the two objects is 16/9 = 1.78 units.

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The skier would have zero kinetic energy at the bottom of the hill, according to the conservation of energy, since the total energy (potential + kinetic) remains constant without friction. Therefore, the answer is none of the options provided.

In accordance with the principle of energy conservation, the skier's total energy (potential energy plus kinetic energy) stays constant, disregarding any non-conservative factors like friction. So, the potential energy (PE) at the top of the hill can be subtracted from the total energy (TE) at the bottom of the hill to determine the skier's kinetic energy (KE) at the bottom of the hill:

TE equals PE plus KE.

Since there is no energy loss from friction and the skier starts with 10,000 J of potential energy at the top of the hill, the total energy at the bottom of the hill will also be 10,000 J. As a result, the skier's kinetic energy at the bottom of the hill may be determined as follows:

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The frequencies of light absorbed by an apple include those in the visible range, specifically in the blue and green wavelengths.

This is because apples predominantly reflect red light, which gives them their characteristic red color. Apples absorb blue and green light frequencies. To explain further, when light hits an object, some wavelengths are absorbed while others are reflected. In the case of a red apple, it absorbs shorter wavelengths (blue and green) and reflects the longer wavelength (red), resulting in the red color we perceive.

However, the pigments in the apple, such as chlorophyll and carotenoids, absorb specific wavelengths of light and reflect others.

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The distance separating the objects is doubled, then the new force is 0.02 N.

Distance is a numerical measurement of how far apart two objects or locations are. It is a scalar quantity, meaning it can be expressed in terms of a single numerical value without any associated direction.

The force between two charged objects is given by Coulomb's law:

F = k×(q¹×q²)/r²

where k is the Coulomb's constant, q¹ and q² are the charges of the two objects, and r is the distance between them.

In the given problem, the force between the two objects is 0.080 N. This means that: 0.080 N = k×(q¹×q²)/r²

Now, if the charge of one of the objects is increased by a factor of four, and the distance separating the objects is doubled, then the new force will be given by: F' = k×(4×q²)/(2×r)²

Substituting the original values of k, q², and r in the above equation, we get: F' = 0.080×(4)/(2×r)²

F' = 0.080×(4)/(4×r²)

F' = 0.080/r²

F' = 0.080/4×r²

F' = 0.02 N

Therefore, the new force between the two charged objects is 0.02 N.

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397 Hz is the frequency of the sound wave in the pipe with length  216 cm and the air column in the pipe is vibrating and has five nodes

Define sound wave

The pattern of disruption brought about by the movement of energy through a medium (such as air, water, or any other liquid or solid matter) as it spreads away from the source of the sound is known as a sound wave. Pressure waves are generated by item vibrations that cause sound waves, such as those from a ringing phone.

The frequency of an occurrence is the number of times it repeats itself in a given amount of time. The frequency of wave-like patterns, such as sound, light, electrical signals, and other waves, expresses the number of cycles of the repeated waveform per second.

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The amount of heat required to raise the temperature of a 225-g lead ball from 15.0°C to 25.0°C is 41,472 J (128 J/kg ∙ K x 225 g x 10 K).

This can be calculated using the specific heat capacity formula, which relates the amount of heat required to the mass, specific heat, and change in temperature of a substance. In this case, the specific heat of lead is 128 J/kg ∙ K, the mass of the lead ball is 225 g, and the change in temperature is 10 K. Multiplying these values together gives the total amount of heat required, which is 41,472 J.

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When using a fume hood with a vertically opening sash, the best statement to describe the protection afforded when the sash is fully open is that the fume hood still offers protection in the case of an explosion but almost no protection from harmful gases.

It's important to remember that a fume hood is designed to protect the user from harmful chemicals and vapors, but it's not intended to provide protection in the event of an explosion. If an explosion were to occur, the fume hood would provide some level of protection by containing the blast and directing it away from the user.

However, harmful gases may still be released and could potentially pose a danger to the user. Therefore, it's important to always use proper safety procedures and follow the instructions for using a fume hood to minimize the risk of accidents and incidents.

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The power series expansion of the principal branch of the log function about the point z = i is .:

Log(z) = iπ/2 + (z-i)/i - (z-i)²/2i² + (z-i)³/3i³

The logarithm function is described as  the inverse function to exponentiation which means the logarithm of a number x to the base b is the exponent to which b must be raised to produce x.

We will then apply  the formula for the derivative of the principal branch of the log function:

d/dz Log(z) = 1/z

Log(z) = Log(i) + (z-i)/i - (z-i)²/2i² + (z-i)³/3i³

where Log(i) =  the value of the principal branch of the log function at z = i.

So therefore Since Log(i) = ln(1) + iπ/2 = iπ/2,

Next step is to simplify the power series expansion :

Log(z) = iπ/2 + (z-i)/i - (z-i)²/2i² + (z-i)³/3i³

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