Given the two vectors A =i+ j and B = j + k, find the following.
(a) A + B and |A + B|
(b) 3A - 2B
(c) A • B
(d) AxB and |AxB|
a) The value of the vector A + B is i + 2j + k and |A + B| is √6 units.
Given:
A = i + j
B = j + k
A + B = i + j + j + k
= i + 2j + k
|A + B| is the magnitude of the vectors A and B
|A + B| = √(1² + 2² + 1²)
= √(1 + 4 +1)
= √6
b) The vector 3A - 2B has the value of 3i + j -2k
Given:
A = i + j
3A = 3i + 3j
B = j + k
2B = 2j + 2k
3A - 2B = 3i + 3j - 2j - 2k
= 3i + j -2k
c) The scalar product of the vector that is A • B is equal to 3 units
The scalar product is calculated as follows:
A • B = i + j • j + k
= 1 + 1 * 1 + 1
= 1 + 1 + 1
= 3 units
d) The vector product or A x B is i - j + k and the magnitude of the same is √3 units.
i j k
A 1 1 0
B 0 1 1
The vector product is calculated as follows:
A x B = (1 * 1 - 1 * 0) i + (0 * 0 - 1 * 0) j + (1 * 1 - 0 * 1) k
= (1 - 0) i + (0 - 1) j + (1 - 0) k
= i - j + k
| A x B | = √(1² + (-1)² + 1²
= √1 + 1 + 1
= √3 units
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explain the appearance of a white cloth with blue stripes in a yellow light
Yellow light is produced by combining red and green lights. The counterpart of blue light with yellow light is blue light combined with red lights. Three basic colours of light are combined to produce white light.
Does blue light absorb yellow light?Blue light can be absorbed by a yellow pigment. So light that shines just on paper is therefore depleted of blue. Red light is now gone. The paper will seem red if it reflects the red light.
Why does anything appear blue in white light?White light causes an object to seem blue since it reflects just blue light and absorbs all other colours. Since there isn't any colour to reflect whenever that object is placed in red light, it absorbs all of the red light that strikes it and appears black.
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can you please tell me where does 1-14 i really need help thanks :) god bless you all
The above has to do with the study of the earth's lithospheric plates. See the attached image and the explanation below.
What are the processes of the movement of lithospheric plates?The movement of lithospheric plates is a geological process that occurs due to the motion of hot, molten material in the Earth's mantle. The lithosphere, which is the rigid outer layer of the Earth's surface, is divided into several large plates that move relative to each other.
These movements are caused by the convection of material in the mantle and the forces that arise at the boundaries between the plates.
There are three main types of plate boundaries: divergent, convergent, and transform. Divergent boundaries occur where plates move apart from each other, creating new oceanic crust. Convergent boundaries arise where plates collide, leading to subduction, volcanic activity, and the formation of mountains. Transform boundaries occur where plates slide past each other.
The movement of lithospheric plates gives rise to various geological phenomena, such as earthquakes, volcanic activity, and the formation of mountain ranges and ocean basins.
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Kindly please tell me the answer to this question...
Following are the answers:
Pressure = \(force/area = 1.92 N/0.196 m^2 = 9.79 N/m^2\) or 9.79 PaDensity = Pressure/ (acceleration due to gravity x height) =\(9.79 N/m^2 / (9.8 m/s^2 x 0.92 m)\) = 1060 kg/m^3.What is the pressure and density?1. To calculate the pressure exerted by the water column on the surface of the mercury, we can use the formula:
Pressure = force/area
The force is the weight of the water column and the area is the cross-sectional area of the container.
The weight of the water column is given by the mass of the water times the acceleration due to gravity:
mass = density x volume
volume = area x height
So, mass = density x area x height = 1000 kg/m^3 x pi x (0.025 m)^2 x 0.25 m = 0.196 kg
Weight = mass x acceleration due to gravity =\(0.196 kg * 9.8 m/s^2\) = 1.92 N
The cross-sectional area of the container is pi x (0.025 m)^2 =\(0.196 m^2.\)
So, Pressure = force/area = \(1.92 N/0.196 m^2 = 9.79 N/m^2\) or 9.79 Pa
2. To calculate the density of the oil, we can use the formula:
density = mass/volume
Since the height of the oil column is 0.92 m and the cross-sectional area of the container is 0.196 m^2, the volume of the oil column is 0.196 m^2 x 0.92 m = 0.18012 m^3.
We do not know the mass of the oil, but we can calculate it using the pressure exerted by the oil column on the surface of the mercury:
Pressure = force/area = density x acceleration due to gravity x height
So, density = Pressure/ (acceleration due to gravity x height) = 9.79 N/m^2 / (9.8 m/s^2 x 0.92 m) = 1060 kg/m^3.
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Help me please
What is the shape of the velocity vs time graph for an object going at constant speed?
A- A curve that rises gradually to the right
B- A curve that decreases gradually to the right
C- A straight line that rises gradually to the right
D- A strait line that decreases gradually to the right
E- A straight line that is level and neither increases or decreases going to the right
Answer: a or b can u pls give me brainlest
Explanation:A straight line is a curve with constant slope. Since slope is acceleration on a velocity-time graph, each of the objects represented on this graph is moving with a constant acceleration.
vertical columns in the periodic table represent a group or family of elements that have the same number of valence electrons by having similar electron configurations elements in the same group have similar properties list elements that have similar properties
The examples of elements with similar properties in the same group are fluorine, chlorine, bromine and iodine.
What is group of elements in a periodic table?
The groups in periodic table is the same as vertical columns in the periodic table
These groups represent a group or family of elements that have the same number of valence electrons by having similar electron configurations elements in the same group have similar properties.
Examples of elements found in the same group with similar chemical properties include;
group 7 elements.
FluorineChlorineBromineIodineThe valence electron of elements found in the group 7 of periodic table is negative 1 (-1).
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A water trough is 10 m long and has a cross-section in the shape of an isosceles trapezoid that is 40 cm wide at the bottom, 100 cm wide at the top, and has height 60 cm. If the trough is being filled with water at the rate of 0.1 m3/min, how fast (in m/min) is the water level rising when the water is 10 cm deep
Answer:
0.238 m/min
Explanation:
The volume of water in the trough V =Ah' where A = area of cross-section = area of isosceles trapezoid = 1/2(a + b)h where a = length of bottom of isosceles trapezoid = 40 cm = 0.4 m, b = length of top of isosceles trapezoid = 100 cm = 1 m and h = height of isosceles trapezoid = 60 cm = 0.6 m. So,
A = 1/2(a + b)h = 1/2(0.4 m + 1 m)0.6 m = (1.4 m)0.3 m = 0.42 m² and h' = height of water level in trough = H - h" where H = length of trough = 10 m and h" = depth of water level in trough = 10 cm = 0.1 m
So, V = Ah'
V = A(H - h") = A(10 - h")
Now, the rate of change of volume of the trough with respect to time dV/dt = d[A(10 - h")]/dt
dV/dt = -Adh"/dt
dh"/dt = -dV/dt/A
Since dV/dt = 0.1 m³/min, substituting the other variables into the equation, we have
dh"/dt = -dV/dt/A
dh"/dt = -0.1 m³/min/0.42 m²
dh"/dt = -0.238 m/min
This is the rate at which the depth is decreasing
Since the height h' = 10 - h"
dh'/dt = d(10 - h")/dt
= -dh"/dt
= -(-0.238 m/min)
= 0.238 m/min
So the water level is increasing at a rate of 0.238 m/min
What’s the right answer for 2
The little number you see to the right of the symbol for an element is called a subscript. That number indicates the number of atoms of that element present in the compound.
3. Each croquet ball in a set has a mass of 0.50 kg. The green ball, traveling at 12.0 m/s,
strikes the blue ball, which is at rest. Assuming that the balls slide on a frictionless
surface and all collisions are head-on, find the final speed of the blue ball in each of
the following situations:
a. The green ball stops moving after it strikes the blue ball.
b. The green ball continues moving after the collision at 2.4 m/s in the
same direction.
Answer:
a) v_{2f} = 12.0 m / s, b) v_{2f} = 9.6 m / s
Explanation:
This is an exercise in conservation of momentum, let's start by defining a system formed by the two balls, so that the forces during the collision have been internal and the moment is preserved.
Initial instant. Before the crash
p₀ = m v₁₀ + m v₂₀
where we use index 1 for the green ball and index 2 for the blue ball.
Final moment. After the crash
p_f = m v_{1f} + m v_{2f}
the moment is preserved
p₀ = p_f
m v₁₀ + m v₂₀ = m v_{1f} + m v_{2f}
they tell us that the blue ball is at rest before the crash
m v₁₀ = m v_{1f} + m v_{2f}
a) it is indicated that the green ball stops after the collision v1f = 0
m v₁₀ = m v_{2f}
v_{2f} = v₁₀
v_{2f} = 12.0 m / s
b) the speed of the green ball is v_{1f} = 2.4 m / s
m v₁₀ = m v_{1f} + m v_{2f}
v_{2f} = v_{1o}- v_{1f}
v_{2f} = 12.0 - 2.4
v_{2f} = 9.6 m / s
Write a paragraph describing Newton's 2nd Law in YOUR OWN WORDS.
Explanation:
if u apply more force to an object, it accelerates at a higher rate
Which of the following cars have the most kinetic energy
\(\huge\boxed{OptionA}\)
_____________________________________DATA:Blue Car:mass = 4 kg
velocity = 5 m/s^2
Orange truck:Mass= 2kg
Velocity = 7m/s^2
Grey Car:mass = 6 kg
velocity = 4m/s^2
Green Car:Mass = 8 kg
Velocity = 3 m/s^2
_____________________________________SOLUTION:By the equation of kinetic energy,
K.E = \(\frac{1}{2} mv^2\)
Where,
K.E is kinetic energy
m is mass
v is velocity
_____________________________________Kinetic energy of Blue car:
Directly substitute the variables in the equation,
K.E = \(\frac{1}{2}x4x5^2\)
Simplify the equation,
K.E = 50 J
_____________________________________Kinetic Energy of Silver Car:
Directly substitude the variable in the equation,
K.E = \(\frac{1}{2}x6x4^2\)
Simplify the equation,
K.E = 48J
_____________________________________Kinetic Energy of Green Car:Substitute the variables in the equation,
K.E = \(\frac{1}{2}x8x3^2\)
Simplify the Equation,
K.E = 36J
_____________________________________Kinetic Energy of Orange Truck:Substitute the variable,
K.E = \(\frac{1}{2}x 2x7^2\)
Simplify the equation,
K.E = 49J
_____________________________________As you can see that the highest value of kinetic energy is of Blue SUV thus it will be out answer.
_____________________________________Best Regards,'Borz'A child playing by the side of a well is throwing a stone vertically into the well with an initial velocity of 5 m/s. if the stone falls into the water 3 s later, find the height, h, of the point which the stone is thrown from the surface of the water. (g=10 m/s2)
a) 15m
b)30m
c)45m
d)60m
The height of the well is 30m
We are given that ,
The initial velocity of the stone = u = 5m/s
The stone fall into well in time = t = 3s
The acceleration due to gravity = g = 10 m/s²
To get the height of the stone which is thrown from the surface of the water in the well by equation,
h = ut + 1/2 gt²
Where, u is initial velocity in m/s, h is height in m, t is time in secs, g is acceleration due to gravity which is vertically upward direction i.e. (-10m/s²)
h = (5m/s) (3s) + 1/2 (-10m/s²)(3s)²
h = (15m) - (5m/s²)(9s²)
h = -30 m
Therefore, the height of the stone which is thrown upward is 30 m . And here, we can neglect the negative sign due to height is never be in negative.
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Please help The half-life of argon-44 is 12 minutes. Suppose you start with 20 atoms of
argon-44 and wait 12 minutes. How many atoms of argon-44 will be left?
A. 20 atoms
B. 40 atoms
C. 5 atoms
D. 10 atoms
Answer:
D. 10 atoms
Explanation:
5. Brownian motion of smoke particles a) State and explain the observation. b) State and explain what will be observed if the temperature surrounding the smoke cell is increased: What is the function of Microscope,Lens and Smoke
(A) When an air particle bombards a smoke particles, the smoke particle moves to the same direction as the air particle that hit it. When another air particle hits the smoke particle, it changes its direction to that of the second air particle, and so on. This is called Brownian motion.
(B) If the temperature of the surroundings is increased the air particles would gain kinetic energy resulting in more frequent collisions with the smoke particles. Therefore, the smoke particles would be seen moving haphazardly.
(C) The microscope helps magnify the smoke particles for a much more better observation. The lens help in focusing. As the air particles are really small and cannot be observed; therefore, smoke particles are used as they are easy to see and observe; thus, ensuring a better experiment.
EASY POINTS:Which element is magnetic? calcium chromium carbon cobalt
Answer:
D
Explaination:
I got it right on the quiz
Answer:
The answer is D. cobalt! :)
Which ball moved at the same speed as Ball 3?
Answer:
you forgot to attach the image
If the dresser is sliding with a constant velocity, find FFF, the magnitude of the force that the person applies.
Complete Question
The complete question is shown on the first uploaded image
Answer:
The force is \(F = mg \mu_k\)
Explanation:
Generally the magnitude of the force the person applied is equivalent to the kinetic frictional force acting on the dresser
So
\(F = \mu_k * N\)
Here N is the normal force acting on the dresser at point P and Q that is
\(N_p \ and \ N_Q\)
So
\(F = \mu_k [N_p + N_Q]\)
Generally this normal force is equal to the mass of the dresser multiplied by the acceleration due to gravity so
\(N_P + N_Q = mg\)
Hence
\(F = mg \mu_k\)
5. A car travels at a constant speed of 20 m/s around a horizontal circular track of radius 80 m. The tires roll without slipping. We can conclude that the coefficient of static friction between the tires and track:
Answer:
.5
Explanation:
First we define the variables
v=20
r=80
Next we construct our formula, since our tires do not slip, our centripetal force is equal to our friction
ma=mg*u
We calculate our centripetal force through finding our acceleration (v^2/r) and assuming our mass is a variable m (we cancel it out later).
mv^2/r = mg*u
All we have left is substitution and simplification
400m/80=mg*u
5=g*u
G is our gravitational acceleration (I will use 10, use 9.8 instead if you prefer)
5=10*u
u=.5
Calculate the quantity of heat energy which must be transferred to 2.25 kg of brass to raise its temperature from 20°C to 240°C if the specific heat of brass is 394 J/kgK.
The quantity of heat energy that must be transferred to 2.25 kg of brass to raise its temperature from 20 °C to 240 °C is 195030 J
How do i determine the quantity of heat energy?First, we shall list out the given parameters from the question. This is shown below:
Mass of brass (M) = 2.25 Kg Initial temperature of brass (T₁) = 20 °CFinal temperature of brass (T₂) = 240 °CChange in temperature of brass (ΔT) = 240 - 20 = 220 °CSpecific heat capacity of brass (C) = 394 J/kgKQuantity of heat energy (Q) =?The quantity of heat energy that must be transferred can be obtained as follow:
Q = MCΔT
= 2.25 × 394 × 220
= 195030 J
Thus, we can conclude quantity of heat energy that must be transferred is 195030 J
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A pumpkin is launched in the air and travels at a horizontal velocity of 25 meters per second for 5 seconds. How far does it travel horizontally?
Answer:
30.3 meters, 172 degrees
Explanation:
To insure the most accurate solution, this problem is best solved using a calculator and trigonometric principles. The first step is to determine the sum of all the horizontal (east-west) displacements and the sum of all the vertical (north-south) displacements.
Horizontal: 2.0 meters, West + 31.0 meters, West + 3.0 meters, East = 30.0 meters, West
Vertical: 12.0 meters, North + 8.0 meters, South = 4.0 meters, North
The series of five displacements is equivalent to two displacements of 30 meters, West and 4 meters, North. The resultant of these two displacements can be found using the Pythagorean theorem (for the magnitude) and the tangent function (for the direction). A non-scaled sketch is useful for visualizing the situation.
Applying the Pythagorean theorem leads to the magnitude of the resultant (R).
R2 = (30.0 m)2 + (4.0 m)2 = 916 m2
R = Sqrt(916 m2)
R = 30.3 meters
The angle theta in the diagram above can be found using the tangent function.
tangent(theta) = opposite/adjacent = (4.0 m) / (30.0 m)
tangent(theta) = 0.1333
theta = invtan(0.1333)
theta = 7.59 degrees
This angle theta is the angle between west and the resultant. Directions of vectors are expressed as the counterclockwise angle of rotation relative to east. So the direction is 7.59 degrees short of 180 degrees. That is, the direction is ~172 degrees.
launch a rocket ship weighs 4.5 million pounds. When it is launched from rest, it takes 8.00 s to reach 164 km/h ; at the end of the first 1.00 min , its speed is 1640 km/h . Part APart complete What is the average acceleration (in m/s2) of the rocket during the first 8.00 s? aav = 5.69 m/s2 Previous Answers Correct Part BPart complete What is the average acceleration (in m/s2) of the rocket between 8.00 s and the end of the first 1.00 min? aav = 7.88 m/s2 Previous Answers Correct Part CPart complete Assuming the acceleration is constant during each time interval (but not necessarily the same in both intervals), what distance does the rocket travel during the first 8.00 s? Express your answer with the appropriate units. x = 182 m Previous Answers Correct Part D Assuming the acceleration is constant during each time interval (but not necessarily the same in both intervals), what distance does the rocket travel during the interval from 8.00 s to 1.00 min? Express your answer with the appropriate units.
Answer:
A) a_av = 5.695 m/s²
B) a_av = 7.88 m/s
C) x_f = 182.24 m
D) x_f = 13029.12 m
Explanation:
A) Initial velocity at 0 seconds; v_i = 0 m/s
Final velocity at 8 seconds;v_f = 164 km/h = 45.56 m/s
So, formula for average acceleration is;
a_av = change in velocity/change in time.
Thus, for the first 8 seconds;
a_av = (45.56 - 0)/(8 - 0)
a_av = 45.56/8
a_av = 5.695 m/s²
B) From part A, we now want to find the average acceleration from 8 seconds to 1 minute (60 seconds)
After the 60 seconds, the speed is 1640 km/h = 455.56 m/s
Thus;
Average acceleration is;
a_av = (455.56 - 45.56)/(60 - 8)
a_av = 7.88 m/s
C) Now we are told the acceleration is constant and we want to find the distance the rocket travels during the first 8 seconds.
So, distance formula we will use is;
x_f = x_i + ½(v_i + v_f)t
So at this stage, v_i = 0 m/s and v_f = 45.56 m/s and x_i = 0 m
Thus;
x_f = 0 + ½(0 + 45.56)8
x_f = 182.24 m
D) Now we want to find the distance the rocket travels during the 8 seconds to 1 minute (60 seconds) interval.
From part B, v_i = 45.56 m/s and v_f = 455.56 m/s
Also, x_i = 0 while time interval(t) = 60 - 8 = 52 s
Thus;
x_f = 0 + ½(45.56 + 455.56)52
x_f = 13029.12 m
In Niels Bohr's 1913 model of the atom, the electron is in a circular orbit around a nucleus. An electron is 7.60 10-11 m from a nucleus traveling in a circular orbit with a speed of 2.45 106 m/s. (a) What is the magnitude of the magnetic moment due to the electron's motion?
Answer:
14.896 x 10^-24
Explanation:
Mo = (evr)/2e = electron = 1.6 x 10^-19v = velocityr = radius =7.60 x 10^-11Select the correct answer.
A boat moves 60 kilometers east from point A to point B. There, it reverses direction and travels another 45 kilometers toward point A. What are the total
distance and total displacement of the boat?
O A.
OB.
O C.
O D.
The total distance is 105 kilometers and the total displacement is 45 kilometers east.
The total distance is 60 kilometers and the total displacement is 60 kilometers east.
The total distance is 105 kilometers and the total displacement is 15 kilometers east.
The total distance is 60 kilometers and the total displacement is 45 kilometers east.
The total distance is 105 kilometers and the total displacement is 15 kilometers east. Option C
How to solve for the total distanceTo calculate the total distance, we add the distances traveled in each leg of the journey: 60 kilometers (from A to B) + 45 kilometers (from B back to A) = 105 kilometers.
However, displacement refers to the change in position of an object in a straight line from its starting point to its ending point. In this case, since the boat starts and ends at the same point (A), the total displacement is zero.
Hence The total distance is 105 kilometers and the total displacement is 15 kilometers east.
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If the speed of an object in uniform circular motion is quadrupled while the radius remains constant, the centripetal force is multiplied by a factor of____.
Answer:
F1 = M a1 = M V1^2 / R
F2 = M a2 = M V2^2 / R
F2 / F1 = (4 V1)^2 / V1^2 = 16
Need a 5 paragraph essay in the eartsh layers and how they function/ benefit the earth!
There is more to the Earth than what we can see on the surface. In fact, if you were able to hold the Earth in your hand and slice it in half, you'd see that it has multiple layers. But of course, the interior of our world continues to hold some mysteries for us. Even as we intrepidly explore other worlds and deploy satellites into orbit, the inner recesses of our planet remains off limit from us.
However, advances in seismology have allowed us to learn a great deal about the Earth and the many layers that make it up. Each layer has its own properties, composition, and characteristics that affects many of the key processes of our planet. They are, in order from the exterior to the interior – the crust, the mantle, the outer core, and the inner core. Let's take a look at them and see what they have going on.
Like all terrestrial planets, the Earth's interior is differentiated. This means that its internal structure consists of layers, arranged like the skin of an onion. Peel back one, and you find another, distinguished from the last by its chemical and geological properties, as well as vast differences in temperature and pressure.
Explanation:
22. A rock is thrown straight up with an initial velocity of 24.5 m/s. What maximum height will the rock reach before starting to fall downward? (Take acceleration due to gravity as 9.80 m/s².)
a. 9.80 m
b. 19.6 m
c. 24.5 m
d. 30.6 m
What are the two categories that the planets are divided into?
Answer:Hi how are you doing today Jasmine
A rocket sled accelerates to 50 m/s. When the rocket engine stips, the sled skids along its rails. If the coefficient of friction is 0.5, how fast is the sled moving after 2.50 s?
The sled's speed can be calculated by considering the acceleration, frictional force, and time. After substituting the given values and performing the calculations, the final speed is determined to be 12.25 m/s.
To calculate the speed of the sled after 2.50 seconds, we can use the equations of motion and consider the forces acting on the sled.
Let's denote the initial speed of the sled as v0, the final speed as vf, the acceleration as a, the time as t, and the coefficient of friction as μ.
Initially, the rocket sled is accelerating, so we can use the equation:
vf = v0 + at
Since the sled is skidding along its rails after the rocket engine stops, the only horizontal force acting on the sled is the force of friction. The frictional force can be calculated using the equation:
frictional force = coefficient of friction * normal force
Since the sled is moving horizontally, the normal force is equal to the weight of the sled, which can be calculated as:
weight = mass * gravity
Now, we can determine the acceleration of the sled using Newton's second law:
frictional force = mass * acceleration
Combining the equations and substituting the values, we have:
vf = v0 + (frictional force / mass) * t
To find the frictional force, we need to calculate the weight of the sled and then multiply it by the coefficient of friction:
frictional force = (mass * gravity) * coefficient of friction
Substituting this back into the previous equation, we get:
vf = v0 + ((mass * gravity * coefficient of friction) / mass) * t
Simplifying further, we have:
vf = v0 + (gravity * coefficient of friction) * t
Now we can substitute the given values into the equation. Assuming the acceleration due to gravity is approximately 9.8 m/s², the coefficient of friction is 0.5, the initial speed is 0 m/s (since the sled starts from rest), and the time is 2.50 s, we can calculate the final speed:
vf = 0 + (9.8 * 0.5) * 2.50
vf = 12.25 m/s
Therefore, the sled is moving at a speed of 12.25 m/s after 2.50 seconds.
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How long does it take for the total energy stored in the circuit to drop to 10% of that value?
Express your answer with the appropriate units.A cylindrical solenoid with radius 1.00 cm
and length 10.0 cm
consists of 150 windings of AWG 20 copper wire, which has a resistance per length of 0.0333 Ω/m
. This solenoid is connected in series with a 10.0 μF
capacitor, which is initially uncharged. A magnetic field directed along the axis of the solenoid with strength 0.160 T
is switched on abruptly.
How long does it take for the total energy stored in the circuit to drop to 10% of that value?
Express your answer with the appropriate units.
The energy stored in the circuit at any time t is given by \(U = (1/2)L*I^{2} + (1/2)Q^{2} /C = (1/2)L*(V_{0} /R)^{2} *e^{(-2t/(R*C))} + (1/2)C*V_{0} ^{2} *(1 - e^{(-2t/(R*C)})).\)The units are in seconds.
The total energy stored in the circuit can be calculated using the formula: U = (1/2)L*I² + (1/2)Q²/C, where L is the inductance, I is the current, Q is the charge on the capacitor, and C is the capacitance.
Initially, the capacitor is uncharged, so the second term is zero.
Therefore, the initial energy stored in the circuit is U₀ = (1/2)L*I₀², where I₀ is the initial current, which is zero.
When the magnetic field is switched on, a current begins to flow in the solenoid.
This current increases until it reaches its maximum value, given by I = V/R, where V is the voltage across the solenoid and R is its resistance.
Since the solenoid is connected in series with the capacitor, the voltage across the solenoid is equal to the voltage across the capacitor, which is given by V = Q/C, where Q is the charge on the capacitor.
The charge on the capacitor is given by Q = C*V, where V is the voltage across the capacitor at any time t.
Therefore, we have I = V/R = Q/(R*C) = dQ/dt*(1/R*C), where dQ/dt is the rate of change of charge on the capacitor.
This is a first-order linear differential equation, which can be solved to give \(Q(t) = Q_{0} *(1 - e^{(-t/(R*C)}))\), where Q₀ is the maximum charge on the capacitor, given by Q₀ = C*V₀, where V₀ is the voltage across the capacitor at t=0.
The current in the solenoid is given by I(t) = \(dQ/dt*(1/R*C) = (V_{0} /R)*e^{(-t/(R*C)}).\)
The energy stored in the circuit at any time t is given by\(U = (1/2)L*I^{2} + (1/2)Q^{2} /C = (1/2)L*(V_{0} /R)^{2} *e^{(-2t/(R*C))} + (1/2)C*V_{0} ^{2} *(1 - e^{(-2t/(R*C)})).\)
The time t at which the energy stored in the circuit drops to 10% of its initial value can be found by solving the equation U(t) = U₀/10, or equivalently, \((1/2)L*(V_{0} /R)^{2} *e^{(-2t/(R*C)}) + (1/2)C*V_{0} /R)^{2}*(1 - e^{(-2t/(R*C)})) = (1/20)L*I_{0} /R)^{2}.\)
This equation can be solved numerically using a computer program, or graphically by plotting U(t) and U₀/10 versus t on the same axes and finding their intersection point.
The solution is t = 1.74 ms.
The units are in seconds.
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find the electrical energy expended by an electric lamp labelled 100w in 8 seconds
Answer:800 Joules
Explanation: Energy = Power × Time = 100∗8100∗8 = 800 joules