a)6.8 + j8.2 ohms=10.65(50.33 deg) ohms
b)12-j6 ohms=13.41(-26.57 deg) ohms
c)6.6+j10 kohms=11.98(56.575 deg) ohms
RECTANGULAR FORMS
a) 6.8+j8.2 ohm
b)12-j6 ohm
c) 6.6+j10 kohm
POLAR FORMS
a) 10.65 angle 50.332 deg
b) 13.416 angle -26.565 deg
c) 11.982 angle 56.575 deg
Impedance, sometimes referred to as a "AC circuit," is the resistance to electricity passing through a circuit that uses an alternating current. Impedance, which is measured in Ohms, is the result of the combined action of the circuit's current-limiting elements, such as Resistance (R), Inductance (L), and Capacitance (C) (C).
The resistance to current flow in a Direct Current (DC) circuit is known as impedance, but in an AC circuit, impedance is the consequence of both the resistive (R) and reactive (X) components of the circuit. A DC circuit's electrical resistance is indicated by the letter "R," but an alternating AC circuit's resistance to current flow is indicated by the letter or symbol "Z."
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water flows at a rate of 2.5 m/s over a flat plate that is parallel to the oncoming flow. (a) determine the approximate location downstream of the front of the plate where the boundary layer becomes turbulent. assume that this occurs at a reynolds number rex
The boundary layer thickness is 7.92 x 10^(-3) m.
A) Formula for Reynolds number to reflect this question is given as;
R_e = Vx/v
Where;
V is velocity
x is polar coordinate where the boundary layer will become turbulent.
R_e is reynolds number.
Reynolds number for turbulent flow of flat plates has a value of 5 x 10^(5)
v is kinematic velocity
From water tables at standard conditions, kinematic viscosity is 1.12 x 10^(-6) m²/s
Now, R_e = Vx/v
We are looking for x, so let's make x the subject.
x = v•R_e/V
Thus, plugging in the relevant values to obtain;
x = [1.12 x 10^(-6) x 5 x 10^(5)]/0.5
x = 1.12m
The boundary layer thickness at the point where the flow becomes turbulent is given by;
δ = 5√(vx/V)
Where;
V is velocity
x is polar coordinate where the boundary layer will become turbulent.
R_e is reynolds number.
Reynolds number for turbulent flow of flat plates has a value of 5 x 10^(5)
v is kinematic velocity
Plugging in the relevant values to get ;
δ = 5√(1.12 x 10^(-6)•1.12/0.5)
δ = 5√(2.5088 x 10^(-6)
δ = 7.92 x 10^(-3) m
Therefore, the boundary layer thickness is 7.92 x 10^(-3) m.
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a fixed amount of a monatomic ideal gas is taken through the following cyclic process. originally (a) it is near atmospheric pressure (100 kpa) with a volume of 0.50 m3 and a temperature of 375 k. then it is isothermally compressed to half its volume (b). then its volume triples (c) during an isobaric expansion.
The qualitative sketch of the process on the PV plane is attached below:
What is ideal gas equation?The ideal gas law (also called the perfect gas law), the relationship between pressure P, volume V, and temperature T of a gas in the boundary region between low pressure and high temperature. Gas molecules move almost independently. each other. In such cases, all gases obey an equation of state known as the ideal gas law.
PV = nRT, where n is the number of moles of gas and R is the universal (or complete) gas constant, 8.31446261815324 joules/kelvin/mol (the universal gas constant is defined as Avogadro's number NA multiplied by Boltzmann's constant k) . ) In the International System of Units, energy is measured in joules, volume in cubic meters (m3), force in newtons (N), and pressure in pascals (Pa). where 1 Pa = 1 N/m2. A force of 1 Newton moving a distance of 1 meter does 1 Joule of work. Therefore, the dimension of both products PV and nRT is work (energy).
The qualitative sketch of the process is drawn on the PV plane.
At A,
P₁ = 100 kPa
V₁ = 0.50 m³
T₁ = 375 K
At B,
T₂ = 375 K
V₂ = 0.25 m³
P₁V₁ = P₂V₂
So, P₂ = (100× 10³ ×0.50)/ 0.25
= 200 kPa
At C,
P₃ = 200 kPa
V₃ = 0.75 m³
We know, V₂/ T₂ = V₃/ T₃
So,
T₃ = V₃T₂/ V₂
= (0.75 × 375)/ 0.25
= 1125 K
At D,
P₄ = 100 kPa
T₄ = 1125 K
V₄ = P₃V₃/ P₄
= (200 × 0.75)/ 100
= 1.5 m³
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The complete question is:
If the bond energies of X2 and C-X are 151.9 kJ and 2763 kJ, respectively, and the bond energies of C-C and C-c are 338 kJ and 604 kJ, respectively, give an estimate of AE for the reaction below: C2H4 + X2 => c2H4X2 (in kJ). (Enter your answer as the nearest whole number without units)
= [ 4 (C-H) + 1 (C=C) + 1 (X2) ] - [ 2 (C-X) + 1 + (C-C) + 4 (C-H)
= [ (C=C) +(X2) ] - [ 2 (C-X) + 1 (C-C) ]
= [ 604 +151.9 ] - [ 2 (276.3) + (338)]
= -134.7 KJ
= -135 KJ
Bond energies depend on the number of bonds between atoms. Even though π bonds are weaker than σ bonds, a double bond, which consists of a σ and π is bond, is stronger than a single bond because there are bonds.
A quantity of strength, equal to the distinction among the energies of the Bond energies depend on the number of bonds between atoms. Atoms and the energies of the separated atoms, is released, usually as warmness. that is, the bonded atoms have a lower strength than the character atoms do.
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Keeping constant speed , 0.8 m/s a marble rolls back and forth inside a shoebox. Make an order-ofmagnitude estimate of the probability of its escaping through the wall of the box by quantum tunneling. State the quantities you take as data and the values you measure or estimate for them.
The answer is e-10³⁰.
Solution:
The expression for find transmission co-efficient is
[tex]T = e^{-2CL}[/tex]
Substitute values in the above equation
T = e -2(2.5-10³²)(2×10⁻³)
= e -10×10²⁹
≈ e-10³⁰
Magnitude estimation is a psychophysical method by which participants judge the perceived strength of a stimulus and assign a numerical estimate. Magnitude is an exponential change of plus or minus one in the value of a quantity or unit. The term is commonly used in relation to the 10th power scientific notation.
Magnitude is used to make the size of numbers and the scale of things more intuitive and understandable. A rough estimate of size helps the project manager estimate a rough figure for the total cost of the project. The approximate financial impact of future projects. Change the precision as needed.
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You are asked to find the index of refraction for an unknown fluid, using only a laser and a Michelson interferometer. A Michelson interferometer consists of two arms--paths that light travels down, which end in mirrors--attached around a beam splitter. (Figure 1)The beam splitter separates the incoming light into two separate beams and then recombines them once they return from the ends of the arms. The recombined beams are sent to a telescope, where their interference pattern may be observed in detail.
Part A
Part complete
First, you must find the wavelength of the laser. You shine the laser into the interferometer and then move one of the mirrors until you have counted 100.0 fringes passing the crosshairs of the telescope. The extremely accurate micrometer shows that you have moved the mirror by 0.03164 millimeters. What is the wavelength λ of the laser?
Express your answer in nanometers, to four significant figures.
Part B
You now immerse the interferometer in a tank filled with some unknown liquid and carefully align the laser into the interferometer. You move the mirror until you count 100.0 fringes passing the crosshairs of the telescope. The micrometer indicates that the mirror has moved 0.02381 millimeters. What is the index of refraction for this mystery fluid?
Express your answer to four significant figures.
The wavelength λ of the laser is 632.8 nm and the index of refraction for this mystery fluid is 1.329.
What is wavelength and index of refraction?Refraction can change the wavelength of light or sound as it travels through a medium with a density greater than that of air. The wavelength reduces as a result of the wave's decreasing speed as it moves through the denser medium. If the wave travels faster when in a rarer medium, the wavelength rises. Because the speed and, consequently, the wavelength of the wave are lowered while traveling through the denser media, the index of refraction is greater for the denser medium than the rarer one.
An interferometer is a device that employs the interference phenomena to calculate the wavelength of light in terms of a standard length or the distance in terms of the wavelength of light that is already known. Through partial reflection and transmission, light from an extended source is split into two halves. The two directions in which these two beams are transmitted are at right angles to one another. They are noticed and studied as interference fringes that are created when they are reflected off the mirror.The angle between mirrors M1 and M2 determines how the fringes in the Michelson interferometer are shaped. Let S' be the replica of the source S, and let M2 be the virtual replica of mirror M1.Let D represent the distance between the mirror M1 and its virtual counterpart, M2'. As a result, the air film that is enclosed between M1 and M2' is what will cause the interference pattern to appear. Let S1' and S2' represent the respective Saras M1 and M2's photos.As a result of the amplitude division process, we are able to produce two coherent sources, S1' and S2'.
A)
2D = nλ
2 x 0.03164 x 10^-3 = 100 x λ
Wavelength, λ = 632.8 nm
B)
2D = n λ
2 x 0.02381 x 10^-3 = 100 λ
Wavelength,λ = 476.2 nm
Refractive index =λ₁/λ₂
Thus,
Refractive index = 632.8/476.2 = 1.329
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a slender, uniform metal rod of mass m and length l is pivoted without friction about an axis through its midpoint and perpendicular to the rod.
The metal rod's moment of inertia, Ml²/12, has a midway that is equal to 3k/M.
Explain about the midpoint and perpendicular?A line that cuts through the intersection of two adjacent line segments at a right angle is known as a perpendicular bisector. In other words, a perpendicular bisector always splits a line segment through its middle. The word "bisect" itself refers to an even or uniform division.
A line that is perpendicular to a triangle's side and travels through the middle of that side is said to be the perpendicular bisector of that side. The circumcenter of a triangle is where the triangle's three perpendicular bisectors meet (the center of the circle through the three vertices). A line segment is a perpendicular bisector when it encounters another line segment at a straight angle and divides it into two equal halves at its halfway.
When the rod is twisted at an angle, the spring is squeezed by a tiny distance.
springs, which add force
= k x
torque moment about an axis
= k x l /2
= k θ( l /2 )² ( x / .5 l = θ )
Angular acceleration times the rod's moment of inertia equals the moment of torque.
M l2 / 12 d2 / dt2 = k (l /2)2
d²θ/dt² = 3 k/M θ
acceleration equals two.
ω² = 3 k/M
ω = √ 3 k / M
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the vectors d and e are [-4,1] and [1,4] respectively .find the angle b/n and e (Use dot product )
The angle between the two vectors is 59 degrees.
What is the angle between the two vectors?
The angle between the two vectors is calculated by applying the following equation as shown below.
Mathematically, the formula for angle between two vectors is given as;
tan θ = Vy / Vx
where;
Vy is the resultant vertical vectorVx is the resultant horizontal vectorThe resultant vertical vector is calculated as follows;
∑ Vy = ( 1 + 4 ) = 5 y
The resultant horizontal vector is calculated as follows;
∑ Vx = ( -4 + 1 ) = -3x
The angle between the two vectors is calculated as follows;
tan θ = Vy / Vx
tan θ = 5 / 3
tan θ = 1.667
θ = arc tan ( 1.667 )
θ = 59 degrees
Thus, the angle between the two vectors is a function of the resultant vertical and horizontal vector.
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What does gravitational potential energy depend on?; What three things does gravitational potential depend on?; Does the gravitational potential energy of an object depends on the path?
The Gravitational potential energy depends on mass , acceleration and path and potential also as G=V/m.
a). As you can imagine, gravitational energy is the energy associated with gravity. This is the potential energy stored due to an object being in a higher position compared to a lower position. (e.g. far from the ground or close to the ground).
Gravitational energy is the energy associated with gravity.
b). Systems can increase their gravitational energy as their mass moves away from the center of the Earth and away from other bodies (such as the sun, planets, and stars) large enough to generate a large amount of gravity. there is.
Potential energy can be defined as the amount of work expended to lift an object from the ground to its current height. In another sense, this is the amount of work the body does before returning to ground level (the position of maximum stability) when translated from its current level. Therefore, potential energy can be related to work.
c). A negative sign indicates that gravity is doing positive work as the mass approaches. A negative potential indicates a bonding state. That is, when mass is placed under its influence, it becomes trapped in its potential, and if something can provide the energy to escape the clutches of gravity, it must not escape.
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Using the method of joints, determine the force in each member of the truss shown. Indicate whether the member is in tension or compression. Support A is a roller and Support Cis a pin.
The value of F CA = 4500 NC. The fundamental difference between compression and tension is that compression represents forces meant to shorten a body, whereas tension describes forces designed to prolong a body.
What is the difference between tension and compression?A rope bridge may support both the load it carries and itself by using tension forces that pull and stretch material in opposing directions. A rock arch bridge carries a load by pressing its rocks against one another due to compression forces that squeeze and push material inward.
One that pulls materials apart is a tension force. When material is compressed, it gets squeezed together. Some materials are more resilient to compression, some to tension, and some are suitable for usage when both compression and tension are present.
Efforts to distort an object are called tension and compression forces. The basic distinction between tension and compression is that tension describes forces intended to lengthen a body, whereas compression describes forces intended to shorten the body.
Support Reactions: Entire truss:
[tex]+\Sigma F_x=0: \quad B_x=0 \\[/tex]
[tex]+{ }_{+} \Sigma M_C=0: \quad-B_y(400 \mathrm{~mm})-(1200 \mathrm{~N})(500 \mathrm{~mm})=0 \\[/tex]
[tex]B_y=-1500 \mathrm{~N} \quad \mathbf{B}_y=1500 \mathrm{~N} \\[/tex]
[tex]+\left\lceil\Sigma F_y=0: \quad-1500 \mathrm{~N}-1200 \mathrm{~N}+C=0\right. \\[/tex]
C = 2700 N
Free body: Joint B:
[tex]$\frac{F_{B A}}{13}=\frac{F_{B C}}{12}=\frac{1500 \mathrm{~N}}{5} .[/tex]
[tex]$\begin{array}{r}F_{B A}=3900 \mathrm{~N} \\ \\ F_{B C}=3600 \mathrm{~N} C\end{array}$[/tex]
simplifying the above equation, we get
[tex]$\frac{F_{C A}}{5}=\frac{2700 \mathrm{~N}}{3}$[/tex]
Therefore, the value of [tex]$F_{C A}=4500 \mathrm{~N} C$[/tex]
The complete question is:
Using the method of joints, determine the force in each member of the truss shown. State whether each member is in tension or compression.
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A wave on a string is described by y(x, t) = (3.0 cm) × cos[2π(x/(2.4 m + t/(0.20 s))], where x is in m and t in s.
In what direction is this wave traveling?
What are the wave speed, frequency, and wavelength?
At t = 0.50 , what is the displacement of the string at x = 0.20 m?
The wave is traveling in the negative x direction.The wave speed is 12.0m/s.The frequency is 5HzThe wavelength is 2.4m.The displacement is 0.029m.
What is the displacement of the string ?The general wave equation is given by;
y(x, t) = y cos (2(x/λ) - 2ft) --------------------------------(i)
y(x, t) is the displacement of the wave at position x and a given time t
y(x, t) = (3.0 cm) × cos[2π(x/(2.4 m) + t/(0.20 s))] ------------------(ii)
Which can be re-written as;
y(x, t) = (3.0 cm) × cos[2π(x/(2.4 m)) + 2π(t/(0.20 s))] -------------(iii)
Comparing equations (i) and (iii) we have that;
=> 2π(x/(2.4 m) = 2π(x/λ)
=> λ = 2.4m
Therefore the wavelength of the wave is 2.4m
Also, still comparing the two equations;
=> 2π(t/(0.20 s) = 2πft
=> f = 1 / 0.20
=> f = 5Hz
Therefore the frequency of the wave is 5Hz
To get the wave speed (v), it is given by;
v = f x λ
Where f = 5Hz and λ = 2.4m
=> v = 5 x 2.4
=> v = 12.0m/s
Therefore, the speed of the wave is 12.0m/s
At t = 0.50s and x = 0.20m;
The displacement, y(x,t) of the string wave is given by
y(x, t) = (3.0 cm) × cos[2π(x/(2.4 m) + t/(0.20 s))]
Convert the amplitude of 3.0cm to m
=> 3.0cm = 0.03m
Carefully solve the equation
=> y(x, t) = (0.03m) × cos[2π((0.20)/(2.4 m)) + 2π(0.50/(0.20 s))]
=> y(x, t) = (0.03m) × -0.9684
=> y(x, t) = 0.029m
Therefore the displacement at those points is -0.029m
Also, the sign of the displacement shows that the direction of the wave is in the negative x direction.
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Please help me
Jadee S. pulls a piece of luggage across the airport, notices that she is lagging behind her parents and wants to catch up for a short amount of time. She pulls with a horizontal force of 33.3 N. The force due to friction for the luggage is 12.3 N and the mass of the luggage is 50.0 kg.
Answer:
The acceleration of the luggage is:
a = (33.3 N - 12.3 N) / 50.0 kg = 0.466 m/s²
Explanation:
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A cannon is mounted on a cart and carrying a cannonball. The total mass of the cart, cannon, and ball is M, and the cart is rolling with no friction at a velocity v in the positive x-direction as shown in the figure. The ball, of mass m, is fired with a velocity of Vball in the positive x-direction. What is the velocity of the cart (with cannon) after the ball is fired?
The velocity of the cart (with cannon) after the ball is fired is u₂ = Mv - mVb / ( M - m(ball) ).
option A is the correct answer.
What is the law of conservation of linear momentum?The law of conservation of linear momentum states that the sum of the initial momentum is equal to sum of the final momentum, provided that the system is Isolated.
Mathematically, the law of conservation of linear momentum is given as;
Pi = Pf
where;
Pi is the sum of the initial momentumPf is the sum of the final momentumm₁u₁ + m₂u₂ = v ( m₁ + m₂ )
where;
m₁ is the mass cannon ballu₁ is the final velocity of the cannon ballm₂ is the mass of the cartu₂ is the final velocity of the cartv is the initial velocity of the cannoncart-systemMass of the cannon ball + cart = M
mass of the cart only, m₂ = M - m(ball)
The final velocity of the cart is calculated as follows;
mVb + m₂u₂ = Mv
m₂u₂ = Mv - mVb
u₂ = Mv - mVb / m₂
u₂ = ( Mv - mVb ) / ( M - m(ball) )
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In what type of wave interaction does the wave bounce off of an object?; What happens when waves bounce off objects and interact?; What is it called when waves interact with each other?; What is the reflection of waves?
The wave bounces off an item when there is a reflection kind of wave interaction.
When waves strike a surface and rebound, reflection occurs. When a wave encounters a surface, it might reflect in all directions. For instance, visible light bounces off the surfaces it encounters. Your eyes receive the reflected light, which makes it possible for you to see the objects.
What occurs when waves come into contact?When two or more waves collide, interaction occurs. Wave interference is the term used to describe how waves interact with one another. Wave interference may occur when two waves that are traveling in opposite directions meet. The two waves pass through each other, and this affects their amplitude.
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What is the bouncing back of a ray of light sound or heat when the ray hits a surface that it does not go through?; Is the bouncing back of a light ray from a surface?; What is the bouncing back of the wave as it hits a surface?; What is the bouncing back of light when it hits?
When something bounces off a wall or other object, reflection happens. Following the rule of reflection, waves are reflected off of straight boundaries.
When light strikes a surface without being absorbed, what is the term used to describe the bounce back?Reflection
When light reflects off of something, it does so. The light will reflect at the same angle it struck the surface if the surface is smooth and shining, such as glass, water, or polished metal.
What does it mean for a sound to repeatedly bounce back to its source?Echoes. A sound that is repeated as a result of sound waves being reflected back is known as an echo. The way a rubber ball bounces off the ground is the same way that sound waves can bounce off smooth, hard things.
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find the sae's for the assigned nodal voltages. solve for the nodal voltages. round answer to 2 decimal places, add 0's as needed. express magnitude as positive number and angle between . ±180∘.
v1 = ∠ ∘ V
v2 = ∠ ∘ V
v3 = ∠ ∘ V
An express magnitude is a positive number and angle between. ±180∘. v1 = ∠ ∘ V.
Nodal stress analysis complements the previous mesh analysis in that it is equally powerful and based on the same concepts as matrix analysis. As the name suggests, node voltage analysis uses Kirchhoff's first law node equation to find the voltage potential around a circuit.
When we use the term node voltage, we are referring to the potential difference between two nodes in a circuit. Select one of the nodes in your circuit as the reference node. All other node voltages are measured with respect to this one reference node.
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Each metal is illuminated with 400 nm (3.10 eV) light. Rank the metals on the basis of the maximum kinetic energy of the emitted electrons. (If no electrons are emitted from a metal, the maximum kinetic energy is zero, so rank that metal as smallest.) Rank from largest to smallest. To rank items as equivalent, overlap them. The correct ranking cannot be determined.
The metals on the basis of the maximum kinetic energy of the emitted electrons is Cesium > potasium > Beryllium = aluminum = Megnesium = Platinum.
From Einstein photo electic effect equation
E = 3.1 eV
E = Wo+ Kmax
Kmax = E - Wo
For Be: E< Wo, Kmax = 0
There is no emission of photo electron
For K : Kmax = 3.1 -2.3 = 0.8 eV
For Al : E< Wo, Kmax = 0
There is no emission of photo electron
For Mg: E< Wo, Kmax = 0
There is no emission of photo electron
For Cs: Kmax = 3.1 -2.1 = 1.0 eV
For Pt : E< Wo, Kmax = 0
There is no emission of photo electron
Cesium > potasium > Beryllium = aluminum = Megnesium = Platinum
Einstein photo electic effect:
Einstein said that energy is a beam of particles related to frequency according to Planck's formula. When this beam is directed at a metal, photons collide with atoms. If the photon frequency is high enough to knock off an electron, the collision will produce the photoelectric effect.
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Why does a light ray bend when it travels from the air to water?
Answer: When the light hits the water, it causes the light to slow down, and when it does this, the light changes directions.
Explanation:
Let r be the separation vector from a fixed point (x’, y’, z’) to the point (x, y, z), and let r be its length. That is
(a) Refer to part (a)
(b) Refer to part (b)
(c) [tex]nr^{n-1}\bold{\hat{r}}[/tex]
Step-by-step Explanation:To answer the questions, I will show the derivations of each part by applying vector calculus methods such as the gradient operator, denoted by '▽', to the given quantities. I will use the definition of the gradient operator in Cartesian coordinates to solve each part.
Given:
[tex]\bold{r} = (x-x')\hat{\imath}+ (y-y')\hat{\jmath} + (z-z')\hat{k}; \ r=\sqrt{ (x-x')^2+(y-y')^2+(z-z')^2}[/tex]
Show that:
(a) [tex]\nabla (r^2)=2 \bold{r}[/tex]
(b)[tex]\nabla (1/r)=- \bold{\hat{r}}/r^2[/tex]
(c) What is the general formula for [tex]\nabla (r^n)[/tex]?
[tex]\hrulefill[/tex]
Part (a): Proving [tex]\Large{\nabla (r^2)=2\hat{r}}[/tex][tex]\hrulefill[/tex]The gradient operator in Cartesian coordinates is:
[tex]\nabla = \hat{\imath}\dfrac{\partial}{\partial x}+ \hat{k}\dfrac{\partial}{\partial y}+ \hat{\jmath}\dfrac{\partial}{\partial z}[/tex]
The square of the vector length is given by:
[tex]\Longrightarrow r^2 = [(x-x')\hat{\imath}+ (y-y')\hat{\jmath} + (z-z')\hat{k}]^2\\\\\\\\\Longrightarrow r^2 = (x-x')^2\hat{\imath}+ (y-y')^2\hat{\jmath} + (z-z')^2\hat{k}[/tex]
Now, applying the gradient operator to 'r²':
[tex]\Longrightarrow \nabla r^2 = \\\\\bullet \dfrac{\partial}{\partial x}[(x-x')^2\hat{\imath}+ (y-y')^2\hat{\jmath} + (z-z')^2\hat{k}]=2(x-x')\\\\\bullet \dfrac{\partial}{\partial y}[(x-x')^2\hat{\imath}+ (y-y')^2\hat{\jmath} + (z-z')^2\hat{k}] = 2(y-y')\\\\\bullet\dfrac{\partial}{\partial z}[(x-x')^2\hat{\imath}+ (y-y')^2\hat{\jmath} + (z-z')^2\hat{k}] =2(z-z')\\\\\\\\\therefore \nabla r^2=2(x-x')\hat{\imath}+ 2(y-y')\hat{\jmath} + 2(z-z')\hat{k}\\\\\\\\[/tex]
Factoring out a 2:
[tex]\Longrightarrow \nabla r^2=2[(x-x')\hat{\imath}+ (y-y')\hat{\jmath} + (z-z')\hat{k}]\\\\\\\\\therefore \boxed{\nabla r^2=2 \bold{r}}[/tex]
[tex]\hrulefill[/tex]
Part (b): Proving [tex]\nabla (1/r)=- \bold{\hat{r}}/r^2[/tex][tex]\hrulefill[/tex]The length 'r' is given by:
[tex]r=\sqrt{ (x-x')^2+(y-y')^2+(z-z')^2}[/tex]
Now applying the gradient operator to 1/r:
[tex]\dfrac{1}{r}=((x-x')^2+(y-y')^2+(z-z')^2)^{-1/2}[/tex]
[tex]\nabla \dfrac{1}{r} =\\\\\bullet \dfrac{\partial}{\partial x}[((x-x')^2+(y-y')^2+(z-z')^2)^{-1/2}]=-(x-x')[(x-x')^2+(y-y')^2+(z-z')^2]^{-3/2}\\\\\bullet\dfrac{\partial}{\partial y}[((x-x')^2+(y-y')^2+(z-z')^2)^{-1/2}]=-(y-y')[(x-x')^2+(y-y')^2+(z-z')^2]^{-3/2} \\\\\bullet\dfrac{\partial}{\partial z}[((x-x')^2+(y-y')^2+(z-z')^2)^{-1/2}]=-(z-z')[(x-x')^2+(y-y')^2+(z-z')^2]^{-3/2}[/tex]
[tex]\Longrightarrow \nabla \dfrac{1}{r} = - \dfrac{(x-x')\hat{\imath}+ (y-y')\hat{\jmath} + (z-z')\hat{k}}{[(x-x')^2+ (y-y')^2 + (z-z')^2]^{3/2}}[/tex]
[tex]\Longrightarrow \nabla \dfrac{1}{r} = -\dfrac{\bold{r}}{r^3}; \ \text{Where:} \ \bold{\hat{r}} =\dfrac{\bold{r}}{r} \rightarrow \bold{r}=\bold{\hat{r}}\cdot r \\\\\\\\\Longrightarrow \nabla \dfrac{1}{r} = -\dfrac{ \bold{\hat{r}} \cdot r}{r^3}\\\\\\\\\therefore \boxed{\nabla \dfrac{1}{r} = \dfrac{\bold{-\hat{r}}}{r^2}}[/tex]
[tex]\hrulefill[/tex]
Part (c): General formula for [tex]\nabla (r^n)[/tex][tex]\hrulefill[/tex]The general formula is as follows:
[tex]\Longrightarrow \nabla(r^n)=\dfrac{\partial}{\partial r}[r^n] \bold{\hat{r}} \\\\\\\\\therefore \boxed{\nabla(r^n)=nr^{n-1}\bold{\hat{r}}}[/tex]
Stars are defined to be on the main sequence if they are burning hydrogen in their cores (hydrogen is combining into helium through nuclear fusion). Eventually, a star will run out of hydrogen fuel in its core, nuclear fusion will stop, and the star will enter a new stage of its lifetime. What is true?
More massive stars emit more energy and run out of hydrogen fuel in their cores faster. More massive stars have shorter main-sequence lifetimes because they burn through their fuel at a faster rate
More massive stars emit more energy and run out of hydrogen fuel in their cores faster.
What is the role of hydrogen in stars?In order to create helium and energy, hydrogen nuclei must fuse. The procedure uses hydrogen as its fuel. As the hydrogen is depleted, the star's core condenses and warms up further. As a result, all the elements up to iron are formed through the fusing of heavier and heavier elements.
Nuclear fusion is how stars generate their energy. This process, known as the "proton-proton chain," which involves a series of events that turns four hydrogen atoms into one helium atom, dominates for the majority of stars.
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What physical quantity does the slope of the v2/2 v/s h graph represent: #Hint: The algebraic statement of the conservation of Mechanical Energy is FEequals 1 half m v squared plus m g h where E,m, and g are constants. Solve for v2/2and determine what the slope would be if 'h' was on the horizontal axis. slope--g slope=-Elm slope-E slope-h
The physical quantity that does the slope of the v2/2 v/s h graph represent is the slope-g. The slope is representing the change in v over the change in time.
By definition, that is the acceleration. Also extra than only a gimmick, it is geometry's manner of displaying us how variables now no longer most effective relate to 1 another, however how they are able to constitute the fee of alteration among the variables.
The slope of the displacement time graph offers us the bodily amount called pace. The slope of a pace graph represents an object's acceleration. As a result, the cost of the slope at a given time represents the object's acceleration at that time. The fee of changing of an object's pace with admire to time is described as acceleration.
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What’s the temperature 32 degrees Fahrenheit in degrees Celsius?
A proton accelerates from rest in a uniform electric field of 640 N/C. At some later time, its speed is 1.20 x 10% m/s (nonrelativistic, because v is much less than the speed of light).
(a) Find the acceleration of the proton.
(b) How long does it take the proton to reach this speed?
(c) How far has it moved in this time?
(d) What is its kinetic energy at this time?
a) The acceleration of the proton is 6.4×10¹⁰ m/s².
b) Time taken is 1.875×10⁻⁵ second .
c) It moved in this time is 11.25 meter.
d) its kinetic energy is 1.15×10⁻¹⁵ Joule.
What is electric field?The force per unit charge exerted on a positive test charge that is at rest at a given position is the force per unit charge that is used to define the electric field analytically.
uniform electric field: E = 640 N/C.
its speed is: v= 1.20 x 10⁶ m/s
a) the acceleration of the proton = qE/m
= (1.6×10⁻¹⁹×641)/(1.6×10⁻²⁷) m/s²
= 6.4×10¹⁰ m/s²
b) Time taken = (1.20 x 10⁶ / 6.4×10¹⁰)second = 1.875×10⁻⁵ second .
c) it moved in this time = ( 1.20 x 10⁶×1.875×10⁻⁵ )/2 meter = 11.25 meter.
d) its kinetic energy = mv²/2 = (1.6×10⁻²⁷)(1.20 x 10⁶)²/2 joule = 1.15×10⁻¹⁵ Joule.
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which of the following graphs represents the relationship between the gravitational force that two objects exert upon one another and the distance between the two objects?
The graph that best represents the relationship between the gravitational potential energy of a freely falling object and the object's height above the ground is the last graph.
What is Gravitational potential energy ?
Gravitational potential energy is the energy possessed by a body by virtue of its position. The formula is expressed as:
E = mgh
m is the mass of the object
g is the acceleration due to gravity
h is the height of the object
Note that the formula shows that the gravitational potential energy is directly proportional to the height of the object
This direct proportionality shows that the required graph will have a positive slope.
This shows that the graph that best represents the relationship between the gravitational potential energy of a freely falling object and the object's height above the ground is the last graph.
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What are acceleration vs. time graphs?
The acceleration versus time graphs are the mathematical representation of the variation of acceleration with time.
There are several methods of representing the variation of two quantities one such method is graphs.
When the variation of the acceleration of the object and the time taken by the object to reach at acceleration are plotted on a graph the provides us with a few certain quantities.
For an example the slope of the acceleration-time graph tell us about the sudden change in acceleration at that particular instant.
Also the area under the acceleration vs. time graph represents the change in velocity of the object.
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you have a horizontal grindstone (a disk) that is 87 kg, has a 0.35 m radius, is turning at 95 rpm (in the positive direction), and you press a steel axe against the edge with a force of 15 n in the radial direction. (a) Assuming the kinetic coefficient of friction between steel and stone is 0.20, calculate the angular acceleration of the grindstone. (b) How many turns will the stone make before coming to rest?
The angular acceleration is 0.19 and the number of turns is 41.07 revolutions.
m = 87 kg be the mass of the grindstone,
r = 0.35 m be the radius of the grindstone,
I = mr²/2 be the grindstone's moment of inertia,
ω = 95 revolution/min = 95 2π/60s = 9.9 s-1 be initial angular velocity,
F = 15 N be the force pressing radially,
f = 0.2 be the coefficient of friction,
α (unknown) be the grindstones angular acceleration.
a)The tangential force slowing the grindstone is
fF
Its torque is
rfF
By Newton's Second Law
rfF = Iα
Therefore
α = rfF/I = 2rfF/mr² = 2fF/mr
Substituting actual numbers
α = 2×0.2×15/(87×0.35) = 0.19 s-2
Note that the above is the absolute value of the acceleration.
Depending on which direction is considered positive or negative,
the answer may need to be negated.
b)
Let
t be the time it takes the grindstone to come to rest,
θ be the total angle by which the grindstone will turn before coming to rest.We have the following identities, which follow from the definition of acceleration:
ω = αt
θ = αt²/2
From the first
t = ω/α
Plug into the second
θ = α(ω/α)²/2 = ω²/2α
Substitute actual numbers
θ = 9.9²/2×0.19 = 257.92
The above result is in radians.
Since the result is supposed to be given in revolutions,
we need to divide by the number of radians per revolutions:
257.92/2π = 41.07 revolutions
Therefore, the angular acceleration is 0.19 and the number of turns is 41.07 revolutions.
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1. A compact car, with mass 725 kg, is moving at 115 km/h toward t
on the east. Sketch the moving car.
a. Find the magnitude and direction of its momentum.
Draw an arrow on your sketch showing the momentum.
b. A second car, with a mass of 2175 kg, has the same
momentum. What is its velocity?
The magnitude and direction of its momentum are 23259.5 kg*m/s. and eastward direction respectively.
Its velocity is 10.66 m/s
How can one determine the magnitude and the direction of the momentum?To find the magnitude and direction of the momentum of the compact car, we can use the formula for momentum, which is defined as the product of an object's mass and velocity:
momentum = mass * velocity
In this case, the mass of the compact car is 725 kg and its velocity is 115 km/h. Converting the velocity to meters per second (m/s) by dividing by 3.6 gives us a velocity of 31.94 m/s.
Plugging these values into the formula above gives us a momentum of 23259.5 kg*m/s.
The direction of the momentum is the same as the direction of the velocity, which in this case is toward the east. Therefore, we can represent the momentum of the compact car as an arrow pointing eastward.
To find the velocity of the second car, we can use the formula for momentum and solve for velocity:
velocity = momentum / mass
In this case, the mass of the second car is 2175 kg and its momentum is the same as that of the compact car, which we calculated to be 23259.5 kg*m/s. Plugging these values into the formula above gives us a velocity of 10.66 m/s.
Note that the velocity of the second car is much lower than that of the compact car, despite having the same momentum. This is because the second car has a much larger mass, so it takes a larger force to change its momentum by a given amount.
Therefore, the correct answer is as given above
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momentum of the system is constant for elastic collisions, but momentum of the system is lost during inelastic collisions. t/f
Momentum of any system is constant in all over elastic collisions, but momentum for the system doesn't lost during inelastic collisions. so it's False.
An elastic collision is a collision wherein there's no internet loss in kinetic power withinside the gadget because of the collision. Both momentum and kinetic power are conserved portions in elastic collisions. Suppose comparable trolleys are journeying closer to every different with identical speed. They collide, bouncing off every different without a loss in speed. This collision is flawlessly elastic due to the fact no power has been misplaced.
In reality, examples of flawlessly elastic collisions aren't a part of our ordinary experience. Some collisions among atoms in gases are examples of flawlessly elastic collisions. However, there are a few examples of collisions in mechanics wherein the power misplaced may be negligible. These collisions may be considered elastic, despite the fact that they may be now no longer flawlessly elastic. Collisions of inflexible billiard balls or the balls in a Newton's cradle are such examples.
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A hopper jumps straight up to a height of 1.3 m. With what velocity did he leave the floor
The velocity with which the jumper leaves the floor is 5.1 m/s.
What is the initial velocity of the jumper?
The initial velocity of the jumper or the velocity with which the jumper leaves the floor is calculated by applying the principle of conservation of energy as shown below.
Kinetic energy of the jumper at the floor = Potential energy of the jumper at the maximum height
¹/₂mv² = mgh
v² = 2gh
v = √2gh
where;
v is the initial velocity of the jumper on the floorh is the maximum height reached by the jumperg is acceleration due to gravityv = √(2 x 9.8 x 1.3)
v = 5.1 m/s
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ight with a frequency of 5.9 x1014 hz travels in a block of glass that has an index of refraction of 1.51. A) What is the wavelength of the light in vacuum ? | B) What is the wavelength of the light in the glass?
The wavelength in a vacuum is 5.08 x 10-7 m and the wavelength of the light in the glass is 3.367 x 10-7 m
As we know that
Velocity = wavelength x frequency
So, wavelength = Velocity / frequency
If the light travels in a vacuum,
Then, wavelength = speed of light/frequency
Here the frequency is 5.9 x1014 Hz
And the speed of light = 3 x 10 8 m/s
So, wavelength in vacuum = 3 x 10 8 / 5.9 x1014
= 5.08 x 10-7 m
Again, the wavelength of the light in the glass
λ=c/μn where μ is a block of glass that has an index of refraction
Then λ=(3 x 10 8 )/(1.51x5.9 x1014 )
= 3.367 x 10-7 m
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find the speed se of the spacecraft when it crashes into the earth. express the speed in terms of me , re , and the universal gravitational constant g .
Speed of the spacecraft right before the collision:[tex]\sqrt{} \frac{2G.Me}{Re}[/tex] .
Assumption: the earth is exactly spherical with a uniform density.
What is the magnitude of the gravitational force acting on the Earth due to the sun?The magnitude of the gravitational force acting on the Sun due to the Earth is 3.52×1022 3.52 × 10 22 Newtons.This question could be solved using the conservation of energy.
The mechanical energy of this spacecraft is the sum of:
the kinetic energy of this spacecraft, and
the (gravitational) potential energy of this spacecraft.
Let denote the mass of this spacecraft. At a distance of from the center of the earth (with mass ), the gravitational potential energy () of this spacecraft would be:[tex]GPE=\frac{G.Me.M}{R}[/tex]
Initially, (the denominator of this fraction) is infinitely large. Therefore, the initial value of will be infinitely close to zero.On the other hand, the question states that the initial kinetic energy () of this spacecraft is also zero. Therefore, the initial mechanical energy of this spacecraft would be zero.Right before the collision, the spacecraft would be very close to the surface of the earth. The distance between the spacecraft and the center of the earth would be approximately equal to , the radius of the earth.The of the spacecraft at that moment would be:Subtract this value from zero to find the loss in the of this spacecraft:Assume that gravitational pull is the only force on the spacecraft. The size of the loss in the of this spacecraft would be equal to the size of the gain in its .Therefore, right before collision, the of this spacecraft would be:On the other hand, let denote the speed of this spacecraft. The following equation that relates and to :Rearrange this equation to find an equation for :It is already found that right before the collision, . Make use of this equation to find at that moment:[tex]\sqrt{} \frac{2KE}{M}[/tex]=[tex]\sqrt{\frac{2G.Me}{Re} }[/tex]
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