Find the slit separation (in m) of a double-slit arrangement that will produce interference fringes 0.0218 rad apart on a distant screen when the light has wavelength 531 nm.
Answers
The slit separation required to produce interference fringes 0.0218 rad apart on a distant screen with light of wavelength 531 nm is approximately 0.625 mm.
In a double-slit interference setup, the fringe separation is determined by the wavelength of the light and the slit separation. The formula relating these quantities is given by:
λ = (m * λ) / d
where λ is the wavelength of light, m is the order of the fringe, and d is the slit separation.
In this case, we are given the wavelength of light (531 nm) and the fringe separation (0.0218 rad). Since the fringe separation corresponds to the first-order fringe (m = 1), we can rearrange the formula to solve for the slit separation:
d = (m * λ) / λ
Substituting the given values, we get:
d = (1 * 531 nm) / 0.0218 rad
Converting the wavelength to meters (1 nm = 1 × 10^(-9) m), we have:
d = (1 * 531 × 10^(-9) m) / 0.0218 rad
Calculating this expression gives us approximately 0.625 mm for the slit separation required to produce interference fringes 0.0218 rad apart on the distant screen with light of wavelength 531 nm.
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Related Questions
use a double integral in polar coordinates to find the volume of the solid in the first octant enclosed by the sphere x^2 y^2 z^2 =4 and the cylinder r=2 cos(theta)
Answers
The volume of the solid is approximately 2.094 cubic units.
To find the volume of the solid in the first octant enclosed by the sphere and cylinder, we can use a double integral in polar coordinates.
First, let's graph the two surfaces:
The sphere \(x^{2}\) + \(y^{2}\)+ \(z^{2}\) = 4 can be rewritten in terms of polar coordinates as:
\(r^{2}\) + \(z^{2}\) = 4
This is a sphere with radius 2 centered at the origin.
The cylinder r = 2 cos(θ) can be rewritten as:
x = r cos(θ) = 2 \(cos^{2}\)(θ)
y = r sin(θ) = 2 cos(θ) sin(θ)
z = 0
This is a cylinder with radius 1 centered at (1,0,0).
Now, let's set up the integral. We want to integrate over the first octant, which means:
0 ≤ θ ≤ π/2
0 ≤ r ≤ 2 cos(theta)
0 ≤ z ≤ sqrt(4 - \(r^{2}\))
The volume of the solid is given by:
V = ∫∫∫ dV
where dV = r dz dr dθ.
Substituting in the limits of integration, we get:
V = ∫[0,π/2] ∫[0,2cos(θ)] ∫[0,\(\sqrt{(4-r^{2} )}\)] r dz dr dθ
Evaluating the innermost integral first:
∫[0,\(\sqrt{(4-r^{2} )}\)] r dz = rz |[0,\(\sqrt{(4-r^{2} )}\)] = r \(\sqrt{(4-r^{2} )}\)
Substituting this into the double integral:
V = ∫[0,π/2] ∫[0,2cos(θ)] r \(\sqrt{(4-r^{2} )}\) dr dθ
To evaluate this integral, we can use the substitution u = 4 - \(r^{2}\), du = -2r dr:
V = -1/2 ∫[0,π/2] ∫[4,0] \(\sqrt{u}\) du dθ
= -1/2 ∫[0,π/2] (2/3) \(u^{3/2}\) |[4,0] dθ
= -1/2 ∫[0,π/2] (2/3) (\(4^{3/2}\) - 0) dθ
= -1/2 (2/3) (\(4^{3/2}\))) ∫[0,π/2] dθ
= (4/3) π/2
= 2.094 cubic units
Therefore, the volume of the solid is approximately 2.094 cubic units.
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which one of the following is true, according to newton’s laws of motion? ignore friction.
Answers
The following which is true, according to newton’s laws of motion with friction being ignored is that none of the above is true, according to the third law and is therefore denoted as option E.
What is Newton's third law of motion?This law was coined by Isaac Newton and it states that for every action (force) in nature there is an equal and opposite reaction.
A semitrailer truck crashing all the way through a wall and since the wall collapses, the wall sustains a greater force than the truck does which doesn't depict the third law of motion as the force are meant to be equal.
They pushing off against each other and fly apart and Sam flying off with the greater acceleration also disobeys the third law due to the action and reaction not being equal.
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The options are:
A. A sports utility vehicle (SUV) hits a stationary motorcycle. Since it is stationary, the motorcycle sustains a greater force than the SUV does.
B. A semitrailer truck crashes all the way through a wall. Since the wall collapses, the wall sustains a greater force than the truck does.
C. Sam (18 years old) and his sister (9 years old) go ice skating. They push off against each other and fly apart. Sam flies off with the greater acceleration.
D. Two astronauts on a space walk are throwing a ball back and forth between each other. In this game of catch the distance between them remains constant.
E. None of the above is true, according to the third law.
The retinal molecule has 12 electrons that are free to move about the chain. For reasons that you may learn later, these 12 electons fill the first 6 states of the box (with 2 electrons in each state). (Figure 1) Thus, the lowest energy photon that can be absorbed by this molecule would be the one that moves an electron from the 6th state to the 7th. Use the equation that you found in Part A to determine the wavelength lambda (?) of this photon. Use the length of the retinal molecule given in the introduction as the length of the box and use m=9.11�10?31kg for the mass of the electron.
\lambda = 4mcL^2/(2n+1)\pi (h/2\pi ))
L = 1.5 * 10-9
Answers
The wavelength of the photon with mass m=9.11×10⁻³¹kg and the transition taking place between n=6 and the n=7 state is λ=570.4nm.
The photon or light has dual nature of light(i.e. the light has matter and wave nature). The wavelength of the wave is defined as the distance between two crests and two waves.
From the given, The wavelength of the light makes the electron pumps form the nth state to (n+1)th state.
mass of the electron, m=9.11×10⁻³¹kg
nth state = 6
n+1 state =7
speed of light(c) = 3×10⁸ m/s
L = 1.5×10⁻⁹
Wavelength(λ) = 4mcL²/(2n+1)π(h/2π), where h is the Planck's constant and is equal to 6.626×10⁻³⁴.
λ = 4mcL²/(2n+1)π(h/2π)
= 4×9.11×10⁻³¹×3×10⁸ ×(1.5×10⁻⁹)² / (2×6+1)π( 6.626×10⁻³⁴/2π)
= 4×9.11×10⁻³¹×3×10⁸ ×(1.5×10⁻⁹)² / (2×6+1)×3.14( 6.626×10⁻³⁴/6.28)
= 5.704×10⁻⁷m
=570.4 ×10⁻⁹m
Thus, the wavelength of the photon is 570.4nm.
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Select ALL of the places you might find DNA evidence.
saliva on gum
blood spatter on the wall
inside a femur bone found in the woods
cells on root of hair found at the scene
Answers
Answer:
Inside a femur bone found in the woods,
saliva of gum
10 joules of work (energy) is required to transfer 2 coulombs of charge from X to Y. What is the potential difference between these two points?
Answers
Answer:
5v
Explanation:
W = 10J ;Q = 2C ;V = ?
So,potential between x and y is 5v
Hope it will help you if not so please sorry.
thirty 6kg boxes lifted on to ashelf 1.5m height what is the total work
Answers
We must first determine the amount of force needed to lift the boxes against gravity. The weight of the boxes is calculated as follows: 30 boxes x 6 kg/box = 180 kg
Work = Force x Distance Work = 1765.8 N x 1.5 m Work = 2648.7 Joules Force = Weight x Gravitational Acceleration Force = 180 kg x 9.81 m/s2 Force = 1765.8 N
The total work required to raise thirty 6 kilogram boxes onto a 1.5 m high shelf is therefore 2648.7 Joules.
How can the gravitational pull of a planet be strengthened?Hence, the gravitational pull between two objects grows as their respective masses do as well. The force of gravity between two objects is equal to their respective masses multiplied by two. To put it another way, the gravitational potential energy directly relates to how high an item is above the earth. Consequently, an item needs be elevated higher in order to enhance its gravitational potential energy. The gravitational potential energy increases with height.
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in The unit of power is a
derived unite
The unit of power is
a derived unit be con
CAY
Answers
Answer:
Watts
Explanation:
The SI unit for power is the watt (W), where 1 watt equals 1 joule/second (1 W=1 J/s).
You push on a door and it opens. Explain what happens in terms of action-reaction forces
30 PTS! ASAP
Answers
Given what we know about Newton's third law of motion, we can confirm that when you push a door, you are exerting a force on the door which is enough to push it, while at the same time, the door is exerting the same force back onto you.
What is Newton's third law of motion?The third law of motion states that for every action, there is an equal and opposite reaction. What this means is that when a force is exerted onto something, the object will exert the same force, in an opposite direction. In the case of the door, when you push the door, the door pushes you back with the same force, in the opposite direction that you are pushing. This is why if we try to push a solid wall, we will move instead of the wall.Therefore, we can confirm that since Newton's third law of motion states that each force has an equal reaction of force in an opposite direction, then when you push a door with your hand to open it, the door is actually exerting the same force back onto you.
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A jogger runs 5.0 km on a straight trail at an angle of 60° south of west. What is the southern component of the run rounded to the nearest tenth of a kilometer?
Answers
Answer:
4.33 Km
Explanation:
Given :
\(\theta\ = \ 60\)
\(T=5.0\ km\)
According to the question the southern component of the run can be written as
\(T_{y} \ =T Sin\theta\)
\(T_{y} \ = 5 *sin60 \ \degree\)
\(T_{y} \ = 4.33\ km\)
and the direction is towards the south.
Answer:
D 4.3 km
Explanation:
A vector points -43.0 units along the x axis, and 11.1 units along the y axis. Find the Direction of the vector.
Answers
Answer:
Explanation:
The direction of the vector implies only its angle, not its magnitude. The direction of the vector is found in
\(tan^{-1}(\frac{y}{x})\) so
\(tan^{-1}(\frac{11.1}{-43.0})=-14.5\) but since we are in QII (where x is negative and y is positive) we have to add 180 to this number to get a direction of 165.5 degrees
a pilot flew a jet from city a to city b, a distance of 2100 mi. on the return trip, the average speed was 20% faster than the outbound speed. the round-trip took 3 h 30 min. what was the speed from city a to city b
Answers
The cruising velocity was 20% higher on the trip home than it was on the outward trip. The distance of 506 miles between cities a and b was traveled in 3 hours, 30 minutes.
Outbound data: What is it?The process of extracting data from such a message to produce data objects is described in the outgoing Data section. Each message type has its own definition. For HL7, static, or delimited messages, the Data part is necessary.
Is outbound more advantageous than inbound?The cost savings of inbound marketing over outbound marketing is one of its main advantages. 80% of company decision makers indicate they would prefer to receive information about an organization into a series of blogs rather than through outbound calls, while inbound leads were demonstrated to cost roughly 60% less than massive machine - type communications.
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what's electron affinity? and what is the formula of electric current?
Answers
Answer:
Electron affinity is the energy change that results from adding an electron to a gaseous atom. For example, when a fluorine atom in the gaseous state gains an electron to form F⁻(g), the associated energy change is -328 kJ/mol.
Give brainliest please.
Answer:
Electron affinity is the energy change that results from adding an electron to a gaseous atom.
Sorry I don't know about the second one.
A 33-N force is applied to a 7-kg object to move it with a constant velocity of 6.3 m/s across a level surface. The coefficient of friction between the object and the surface is approximately ____. Round your answer to the hundredths place.(Use the approximation g ≈ 10 m/s2.)
Answers
Given:
The applied force is,
\(F=33\text{ N}\)The mass of the object is,
\(m=7\text{ kg}\)The constant velocity of the object is,
\(v=6.3\text{ m/s}\)To find:
The coefficient of friction between the object and the surface
Explanation:
The object is moving with a constant velocity which means the object is under equilibrium. So, the applied force is equal to the frictional force on the object.
If the coefficient of friction between the object and the surface is
\(\mu\)we can write the frictional force as,
\(f=\mu mg\)For equilibrium condition,
\(\begin{gathered} \mu mg=F \\ \mu=\frac{F}{mg} \\ \mu=\frac{33}{7\times10} \\ \mu=0.47 \end{gathered}\)Hence, the coefficient of friction between the object and the surface is 0.47.
2. CHARLES MANSON WAS DIAGNOSED WITH
PERSONALITY DISORDER.
10 points
(A)NARCISSISTIC
(B) BIPOLAR
(C)ANTI-SOCIAL
(D)BORDERLINE
O Other
Answers
Answer:
The answer is C. anti social
Explanation:
Hope that helps!
Which of the following statements are true about whether atoms tend to gain
or lose electrons?
A. Atoms with four valence electrons will always lose electrons.
B. Atoms with eight valence electrons easily lose electrons.
C. Atoms with five, six, or seven valence electrons will gain
electrons.
D. Atoms with one, two, or three valence electrons will lose
electrons.
Answers
Answer:
The true statements are;
C and D
Explanation:
The tendency of an atom to gain or lose electrons in a chemical reaction are given by its electronegativity and electropositivity respectively
Electronegativity is a measure of an atom's capacity to attract electrons from neighboring particles, with atoms that are more likely to gain an electron, being described as having a higher electronegativity
Electronegativity increases across the period with increasing number of valence electrons, therefore, elements with five, six, or seven electrons, which are located on the far right of the periodic table are said to have higher electronegativity and therefore more likely to gain electrons in a chemical reaction
Electropositivity is the quantification or measure of an element to donate or give electrons and become positively charged ions in a chemical reaction
The fewer the number of valence electrons an element has, the more likely the element will undergo a reaction by donating the valence electrons to a more electronegative atom and become positively charged
Electropositivity decreases across the period and therefore, elements with one, two, or three valence electrons which are located in groups I, II, and III, are more likely to lose electrons in a chemical reaction
Answer:
C+D
Explanation:
any object free to rotate about a pivot will come to rest with its center of gravity the pivot.
Answers
Objects free to rotate around a pivot will eventually rest with their center of gravity directly below the pivot.
When an object is free to rotate around a pivot, it will experience a torque due to the force of gravity acting on its center of mass.
This torque will cause the object to rotate until it reaches a position where its center of gravity is directly below the pivot point. In this position, the torque becomes zero, and the object is in a stable equilibrium.
This is because any deviation from this position will cause a restoring torque that brings the object back to its equilibrium position, thus ensuring that it comes to rest with its center of gravity below the pivot.
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Which of the following is not a change that occurred to organisms over time in the fossil record?
A. Decrease in in the number of species
B. Greater complexity
C. Increase in size
D. More diversity
Answers
Answer:A
Explanation: my teacher told me lol
The option that is not a change that occurred to organisms over time in the fossil record is A. Decrease in the number of species
A fossil record simply refers to the history of the life of the fossils. It means the remains of imprints of the organisms that are from earlier geological periods.
It should be noted that the fossil record is important to archeologists as it helps in placing events in their appropriate eras.
A change that didn't occur to the organisms over time in the fossil record is the decrease in the number of species.
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50. In your own words: Explain how the Doppler Radar works and why it is an important tool for today's Meteorologists ?
Answers
A sample of sewage from a town is found to have a B 5
of 176mg/L and a k value of 0.11/ day. Estimate the ultimate biological demand for this sewage
Answers
The ultimate biological demand for this sewage is 974 mg/L.
B5 or BOD5 is a test of the quantity of oxygen consumed by microorganisms as they break down organic matter in wastewater.
K is the rate constant of oxidation of biodegradable organic compounds.
Let's use these terms to estimate the ultimate biological demand for the given sewage. It is given that the B5 of sewage from a town is 176mg/L and a k value of 0.11/ day.
B5 can be used to calculate ultimate biological demand (UBD) as follows:
\(UBD = B5 / (1 - e {}^{(-kT)} )\)
where T is the time in days
Let's assume that the time is one year, i.e.,
T = 365 days
UBD = 176 / (1 - e⁽⁻⁰·¹¹ˣ³⁶⁵⁾) = 974 mg/L
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8\%) Problem 9: The maximum horizontal distance that Jean can throw a baseball is 58.2 m : D Assuming she can throw with the same initial speed at all launch angles, to what maximum height (measured in meters above the release point) will the ball rise if she throws it straight upward?
Answers
The maximum height the ball will rise if Jean throws it straight upward is 0 meters above the release point.
To find the maximum height the ball will rise if Jean throws it straight upward, we can use the concept of projectile motion. The key idea is that the vertical component of the initial velocity is zero at the highest point of the trajectory.
Given that the maximum horizontal distance is 58.2 m, we can assume that the initial speed (also known as the magnitude of the initial velocity) is constant for all launch angles.
Let's denote the maximum height as "h" (measured in meters above the release point). At the highest point of the trajectory, the vertical velocity component will be zero.
Using the equation for the vertical displacement in projectile motion, we have:
Vertical displacement = (Initial vertical velocity * time) + (0.5 * acceleration due to gravity * time^2)
Since the ball is thrown straight upward, the initial vertical velocity is positive (upward). The acceleration due to gravity is negative (downward). Therefore, the equation becomes:
h = (0 * t) + (0.5 * (-9.8 m/s^2) * t^2)
Simplifying the equation, we have:
h = -4.9t^2
To find the time it takes for the ball to reach its highest point, we need to use the equation for the vertical component of the velocity:
Vertical velocity = Initial vertical velocity + (acceleration due to gravity * time)
Since the vertical velocity is zero at the highest point, we have:
0 = 0 + (-9.8 m/s^2) * t
Solving for t, we find:
t = 0
This means that at the highest point of the trajectory, the ball has been in the air for 0 seconds.
Substituting t = 0 into the equation for h, we find:
h = -4.9(0)^2
h = 0
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When Jean throws the ball straight upward, it will not rise to any height above the release point. It will only reach the same level as the release point. To determine the maximum height the ball will reach when thrown straight upward, we need to use the concept of projectile motion. When the ball is thrown straight upward, it experiences a vertical motion under the influence of gravity.
First, we need to find the initial vertical velocity of the ball. Since the ball is thrown straight upward, the initial vertical velocity is equal to zero.
Next, we need to find the time it takes for the ball to reach its maximum height. We can use the equation:
t = (final velocity - initial velocity) / acceleration
The final velocity is zero, the initial velocity is zero, and the acceleration due to gravity is -9.8 m/s^2 (taking downward as negative). Therefore, the time taken to reach the maximum height is zero.
Now, we can use the equation for vertical displacement:
Δy = (initial velocity * t) + (0.5 * acceleration * t^2)
Substituting the values, we get:
Δy = (0 * 0) + (0.5 * -9.8 * 0^2)
=> Δy = 0
This means that the maximum height the ball will rise is 0 meters above the release point. The ball will reach the same level from where it was thrown, but not higher.
In conclusion, when Jean throws the ball straight upward, it will not rise to any height above the release point. It will only reach the same level as the release point.
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A patient has a temperature of 38.5 °C. What is the temperature in degrees Fahrenheit? Group of answer choices a.311 °F b.126.9 °F c.70.5 °F d.11.7 °F e.101.3 °F
Answers
A patient has a temperature of 38.5 °C. The temperature in degrees Fahrenheit is e.101.3 °F.
To convert a temperature from degrees Celsius (°C) to degrees Fahrenheit (°F), you can use the following formula:
°F = (°C × 9/5) + 32
In this case, the patient has a temperature of 38.5 °C. Applying the formula, we have:
°F = (38.5 × 9/5) + 32
°F ≈ (69.3) + 32
°F ≈ 101.3
So, the patient's temperature in degrees Fahrenheit is approximately 101.3 °F. Therefore, the correct answer among the choices provided is option e.101.3 °F.
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The half-life of Po-214 is 0.001 seconds. How much of a 10-gram sample is left after 0.003 seconds?
Answers
You would be left with 1.25 g.
After 0.001 second, 10 grams decays to 5 grams.
After another 0.001 second, 5 g decays to 2.5 g. (Total time 0.002 s; two half-lives have passed)
After another 0.001 second, 2.5 g decays to 1.25 g. (Total time 0.003 s; 3 half-lives have passed)
em 6
Which provides a definition for
matter?
Anything that
A.
B.
C.
D.
is solid and has weight
takes up space and is heavy
takes up space and has mass
is a liquid or solid
Answers
Answer:
definition of matter is that matter takes up space and has mass
how to calculate time in uniform electric field using acceleration and distance only
the formula
Answers
Answer:
\(t = \sqrt{\frac{s}{a} }\)
Explanation:
In order to find the time in a uniform magnetic field in terms of acceleration and time we will use the general formula of acceleration, which is given as follows:
\(a = \frac{v}{t} \\\\where,\\\\a = acceleration\\v = velcoity = \frac{s}{t} \\\\t = time\ taken\\s = distance\ covered\ in\ the\ field\\\\therefore,\\\\a = \frac{s}{t^2}\\\\t^2 = \frac{s}{a}\\\\t = \sqrt{\frac{s}{a} }\)
This formula can be used to measure time in a uniform electric field in terms of distance and time only.
A golfer strikes a golf ball of mass 0.05 kg, and the time of impact between the golf club and the ball is 1.0 ms. If the ball acquires a velocity of magnitude 70 m/s, calculate the average force exerted on the ball.
Answers
The average force exerted on the golf ball during the time of impact is 3500 Newtons.
To calculate the average force exerted on the golf ball, we can use Newton's second law of motion, which states that the force exerted on an object is equal to the rate of change of its momentum. Mathematically, it can be expressed as:
Force (F) = Change in momentum / Time
Given:
Mass of the golf ball (m) = 0.05 kg
Time of impact (t) = 1.0 ms = 1.0 × 10^(-3) s
Final velocity of the golf ball (v) = 70 m/s
First, we need to calculate the change in momentum:
Change in momentum = Final momentum - Initial momentum
The initial momentum is assumed to be zero since the ball was initially at rest. Therefore, the change in momentum is equal to the final momentum.
Change in momentum = m * v
Plugging in the values:
Change in momentum = 0.05 kg * 70 m/s = 3.5 kg·m/s
Now we can calculate the average force exerted on the ball:
Force = Change in momentum / Time
Force = 3.5 kg·m/s / 1.0 × 10^(-3) s = 3500 N
Therefore, the average force exerted on the golf ball is 3500 Newtons.
The average force exerted on the golf ball during the time of impact is 3500 Newtons.
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Can someone please help me it's urgent!
Answers
Answer:
6.8 x 10^6
Explanation:
Look up M A T H W A Y it will help you with all of your algebra needs
When you pick something up do you do work to it
Answers
Answer:
Simple answer: Yes
Explanation:
Even if you touch an item with a stick you re still doing work to it, most of the time something sitting on a table not being disturbed is having work done to it. Everything has the force of gravity working on it to essentially keep the items from floating away so workis being done to it.
Work done can be something so small (e.g) a pencil sitting on a table) or as big as an earthquake or kicking a ball through a window and smashing the glass.
the position of an object in simple harmonic motion is defined by the funciton y = (0.50 m) sin (πt/2). determine the maximum speed of the object.
Answers
The maximum speed of the object in simple harmonic motion, defined by the function \(\(y = (0.50 \, \text{m}) \sin \left(\frac{\pi t}{2}\right)\)\), is 0.50 m/s.
In simple harmonic motion, the velocity of an object is given by the derivative of the position function with respect to time. In this case, the position function is \(\(y = (0.50 \, \text{m}) \sin \left(\frac{\pi t}{2}\right)\)\). Taking the derivative of this function with respect to time, we find \(\(\frac{{dy}}{{dt}} = \frac{\pi}{2} \cdot (0.50 \, \text{m}) \cdot \cos \left(\frac{\pi t}{2}\right)\)\).
To find the maximum speed, we need to determine the maximum value of the absolute value of the derivative. The maximum value of the cosine function is 1, so the maximum speed is \(\(\frac{\pi}{2} \cdot (0.50 \, \text{m}) \cdot 1 = 0.50 \pi \, \text{m/s}\)\). Using the approximation \(\(\pi \approx 3.14\)\), we can calculate the maximum speed to be approximately 1.57 m/s.
However, it is important to note that the original position function \(\(y = (0.50 \, \text{m}) \sin \left(\frac{\pi t}{2}\right)\)\) does not explicitly involve time in seconds. Therefore, the given maximum speed of 0.50 m/s might be an error or an assumption made based on the specific problem context.
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find all the second partial derivatives. f(x, y) = (sin(mx + ny))2
Answers
The second partial derivatives of f(x, y) = (sin(mx + ny))² are:
∂²f/∂x² = -m²(sin(mx + ny))²∂²f/∂y² = -n²(sin(mx + ny))²∂²f/∂x∂y = ∂²f/∂y∂x = mn(sin(mx + ny))²Second Derivatives PartialThe reason these are the second partial derivatives is because we are taking the derivative of the function twice with respect to x and y.
The first partial derivative with respect to x is:∂f/∂x = ∂/∂x (sin(mx + ny))² = ∂/∂x (sin²(mx + ny)) = 2sin(mx + ny)cos(mx + ny) * ∂/∂x (mx + ny) = m cos(mx + ny)sin(mx + ny)
The second partial derivative with respect to x is:∂²f/∂x² = ∂/∂x (m cos(mx + ny)sin(mx + ny)) = -m²sin²(mx + ny)
Similarly, the second partial derivative with respect to y is :∂²f/∂y² = ∂/∂y (n cos(mx + ny)sin(mx + ny)) = -n²sin²(mx + ny)
And lastly, the partial derivative with respect to x and y is :∂²f/∂x∂y = ∂/∂y(m cos(mx + ny)sin(mx + ny)) = mn cos(mx + ny)sin(mx + ny)
So, all these second derivatives are the result of applying the chain rule and the product rule several times.
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You exert a force of 100N on a rope that pulls a sled across a flat icy surface. The rope is oriented 30 degrees above the horizontal. The sled has a mass of 10kg and it is carrying a 30kg child. The sled starts at rest. You can assume friction is negligible.
You reach a hill and start going up, still applying a constant 100N force at 30 degrees from the slope. The sled is now moving at constant speed. What is the angle of the slope?
Answers
The angle of slope if the sled carrying a 30 kg child which is moving at a constant speed is 12.7°
Given that,
Force, F = 100 N
Mass, m = 10 + 30 = 40 kg
Angle at which the force is applied, θ = 30°
Resolving the force applied into its horizontal and vertical components,
cos θ = \(F_{x}\) / F
\(F_{x}\) = 100 * cos 30° = 100 * 0.866
\(F_{x}\) = 86.6 N
A force due to the weight of the cart is applied on the ground which is directly vertical downwards. But the slope is at an angle. So resolving this force into its horizontal and vertical components.
sin θ = \(w_{x}\) / w
w = m g
w = 40 * 9.8
w = 392 N
Given that the velocity when travelling up the slope is constant. So the acceleration is zero.
Σ \(F_{x}\) = m a = 0
\(F_{x}\) - \(w_{x}\) = 0
\(w_{x}\) = 86.6 N
θ = \(sin^{-1}\) ( 86.6 / 392 ) = \(sin^{-1}\) ( 0.22 )
θ = 12.7°
Therefore, the angle of slope if the sled carrying a 30 kg child which is moving at a constant speed is 12.7°
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