Answer:
v₂ = 7/ (0.5)= 14 m/s
Explanation:
Flow rate of the fluid
Flow rate is the amount of fluid that circulates through a section of the pipeline (pipe, pipeline, river, canal, ...) per unit of time.
The formula for calculated the flow rate is:
Q= v*A Formula (1)
Where :
Q is the Flow rate (m³/s)
A is the cross sectional area of a section of the pipe (m²)
v is the speed of the fluid in that section (m/s)
Equation of continuity
The volume flow rate Q for an incompressible fluid at any point along a pipe is the same as the volume flow rate at any other point along a pipe:
Q₁= Q₂
Data
A₁ = 2m² : cross sectional area 1
v₁ = 3.5 m/s : fluid speed through A₁
A₂ = 0.5 m² : cross sectional area 2
Calculation of the fluid speed through A₂
We aply the equation of continuity:
Q₁= Q₂
We aply the equation of Formula (1):
v₁*A₁= v₂*A₂
We replace data
(3.5)*(2)= v₂*(0.5)
7 = v₂*(0.5)
v₂ = 7/ (0.5)
v₂ = 14 m/s
Using the Continuity Equation that states the mass flow rate must be constant in fluid flow, the speed of the fluid when it moves through the narrow part of the pipe is 14 m/s.
Explanation:The question pertains to the physics principle of fluid dynamics, specifically the concept of continuity in fluid flow. According to the Continuity Equation, the mass flow rate must remain constant across different sections of a pipe. In this particular scenario, the fluid is changing speed due to a reduction in pipe cross-sectional area. This principle is represented by the equation A₁V₁ = A₂V₂, where A₁ and A₂ represent the cross-sectional areas of the wide and narrow sections of the pipe respectively, and V₁ and V₂ represent the fluid's velocity in these sections.
Applying the equation to this scenario, we have A₁ (2.00 m²) multiplied by V₁ (3.5 m/s) equal to A₂ (0.50 m²) multiplied by V₂, the value we want to find. Solving for V₂, we get V₂ = A₁V₁ / A₂ = (2.00 m²)(3.5 m/s) / 0.50 m² = 14 m/s.
Hence, the speed of the fluid when it moves through the narrow part of the pipe is 14 m/s.
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A piano tuner sounds two strings simultaneously. One has been previously tuned to vibrate at 293.0 Hz. The tuner hears 3.0 beats per second. The tuner increases the tension on the as-yet untuned string, and now when they are played together the beat frequency is
1.0s−1.
(a) What was the original frequency of the untuned string?
(b) By what percentage did the tuner increase the tension on that string?
Final answer:
To find the two possible frequencies of the untuned piano string, we can use the formula for beat frequency. From the given information, the original frequency of the untuned string can be either 266.0 Hz or 262.0 Hz. To find the percentage increase in tension on the untuned string, we can use the formula for calculating percentage increase.
Explanation:
To find the two possible frequencies of the untuned piano string, we can use the formula for beat frequency:
Beat frequency = |Frequency of the first string - Frequency of the second string|
In this case, the beat frequency is given as 2.00 s. The frequency of the first string is 264.0 Hz. Let's assume the frequency of the second string is x Hz.
So, we can set up the equation:
2.00 = |264.0 - x|
Solving for x, we get two possible frequencies: 266.0 Hz and 262.0 Hz.
To find the original frequency of the untuned string, we can use the formula:
Original frequency = Frequency of the first string ± Beat frequency
For positive beat frequencies, the original frequency would be:
Original frequency = 264.0 + 2.00 = 266.0 Hz
For negative beat frequencies, the original frequency would be:
Original frequency = 264.0 - 2.00 = 262.0 Hz
To find the percentage increase in tension on the untuned string, we can use the formula:
Percentage increase = (Change in tension / Original tension) x 100
Since the tension on the first string is unchanged (as it is the tuned string), the change in tension on the untuned string is equal to the change in frequency. Assuming the original frequency of the untuned string is 262.0 Hz:
Change in tension = |Original frequency - New frequency|
Change in tension = |262.0 - 264.0| = 2.0 Hz
Therefore, the percentage increase in tension on the untuned string is:
(2.0 / 262.0) x 100 = 0.763%
A rocket, initially at rest on the ground, accelerates straight upward from rest with constant (net) acceleration 34.3 m/s2 . The acceleration period lasts for time 6.00 s until the fuel is exhausted. After that, the rocket is in free fall. True or False.
Answer:
True
Explanation:
t = Time taken = 6 s
u = Initial velocity = 0
v = Final velocity
a = Acceleration = 34.3 m/s²
[tex]v=u+at\\\Rightarrow v=0+34.3\times 6\\\Rightarrow v=205.8\ m/s[/tex]
After the six seconds the acceleration from the rocket stops. After the fuel is exhausted the velocity of the rocket will be 205.8 m/s which will reduce by the acceleration due to gravity (i.e., 9.81 m/s²).
Free fall is the state of a body where the only the force of gravity is acting on it other forces are not acting on it.
Hence, the statement here is true.
Assuming constant velocities, if a fastball pitch is thrown and travels at 40 m/s toward home plate, 18 m away, and the head of the bat is simultaneously traveling toward the ball at 18.0 m/s, how much time elapses before the bat hits the ball?a. About 0.3 sb. About 0.6 sc. About 0.9 sd. About 1.2 s
Explanation:
Here velocity of ball and bat are in opposite direction.
Velocity of ball = 40 m/s
Velocity of bat = 18 m/s
Since they are in opposite direction relative velocity is given by,
Relative velocity = 40 + 18 = 58 m/s
Distance to home plate = 18 m
We have
Displacement = Velocity x Time
18 = 58 x Time
Time = 0.3 seconds
Option A is the correct answer.
A large, 68.0-kg cubical block of wood with uniform density is floating in a freshwater lake with 20.0% of its volume above the surface of the water.
You want to load bricks onto the floating block and then push it horizontally through the water to an island where you are building an outdoor grill.
a. What is the volume of the block? Express your answer with the appropriate units
b. What is the maximum mass of bricks that you can place on the block without causing
it to sink below the water surface? Express your answer with the appropriate units
Answer:
a) V = 0.085 m^3
b) m = 17 kg
Explanation:
1) Data given
mb = 68 kg (mass for the block)
20% of the block volume is floating
100-20= 80% of the block volume is submerged
2) Notation
mb= mass of the block
Vw= volume submerged
mw = mass water displaced
V= total volume for the block
3) Forces involved (part a)
For this case we have two forces the buoyant force (B), defined as the weight of water displaced acting upward and the weight acting downward (W)
Since we have an equilibrium system we can set the forces equal. By definition the buoyant force is given by :
B = (mass water displaced) g = (mw) g (1)
The definition of density is :
[tex] \rho_w = \frac{m_w}{V_w} [/tex]
If we solve for mw we got [tex]m_w = \rho_w V_w [/tex] (2)
Replacing equation (2) into equation (1) we got:
[tex] B = \rho_w V_w g [/tex] (3)
On this case Vw represent the volume of water displaced = 0.8 V
If we replace the values into equation (3) we have
0.8 ρ_w V g = mg (4)
And solving for V we have
V = (mg)/(0.8 ρ_w g )
We cancel the g in the numerator and the denominator we got
V = (m)/(0.8 ρ_w)
V = 68kg /(0.8 x 1000 kg/m^3) = 0.085 m^3
4) Forces involved (part b)
For this case we have bricks above the block, and we want the maximum mass for the bricks without causing it to sink below the water surface.
We can begin finding the weight of the water displaced when the block is just about to sink (W1)
W1 = ρ_w V g
W1 = 1000 kg/m^3 x 0.085 m^3 x 9.8 m/s^2 = 833 N
After this we can calculate the weight of water displaced before putting the bricks above (W2)
W2 = 0.8 x 833 N = 666.4 N
So the difference between W1 and W2 would represent the weight that can be added with the bricks (W3)
W3 = W1 -W2 = 833-666.4 N = 166.6 N
And finding the mass fro the definition of weight we have
m3 = (166.6 N)/(9.8 m/s^2) = 17 Kg
Final answer:
The volume of the block of wood and the maximum mass of bricks it can carry without sinking are determined using principles of buoyancy and uniform density.
Explanation:
(a) To determine the volume of the block of wood, we use the fact that 20.0% of its volume is above the water. Given a uniform density, this means the submerged volume is 80% of the total volume. Hence, the volume of the block is 0.8 x (68.0 kg / 920 kg/m³) = 0.472 m³.
(b) The maximum mass of bricks that can be loaded without sinking the block can be calculated using the concept of buoyancy. The buoyant force equals the weight of the water displaced, so the mass of the bricks should not exceed the mass of the water displaced by the volume of the submerged part of the block. Therefore, the maximum mass of bricks is 0.472 m³ x 1100 kg/m³ x 9.8 m/s² = 5148.64 kg.
What role do earth’s layers play in the formation of metamorphic rock?
The wooden block is removed from the water bath and placed in an unknown liquid. In this liquid, only one-third of the wooden block is submerged. Is the unknown liquid more or less dense than water?
Final answer:
A wooden block that floats in an unknown liquid with only one-third submerged indicates that the unknown liquid is more dense than water.
Explanation:
When the wooden block is placed in an unknown liquid and only one-third of it is submerged, it suggests that the block is less dense than the unknown liquid. This is due to the principle of buoyancy, which states that an object will displace a volume of fluid that is equal to its own weight. Since the wooden block floats higher in the unknown liquid compared to how it floats in water, where it must submerge more to displace enough water to equal its weight, we can conclude that the unknown liquid is more dense than water.
(a) What is the escape speed on a spherical asteroid whose radius is 500. km and whose gravitational acceleration at the surface is 3.00 m/s2 ? (b) How far from the surface will a particle go if it leaves the asteroid's surface with a radial speed of 1000 m/s? (c) With what speed will an object hit the asteroid if it is dropped from 1000 km above the surface?
Final answer:
The escape velocity of a spherical asteroid with a given radius and given gravitational acceleration can be calculated using a special formula. The distance a particle travels from the surface when it leaves it at a given radial velocity can be determined using the projectile height equation. The speed at which an object hits an asteroid after falling from a given height can be calculated using the falling object terminal velocity equation.
Explanation:
(a) Escape velocity can be defined as the minimum speed required for an object to escape the gravitational pull of a celestial body. To calculate the escape velocity on a spherical asteroid we can use the formula:
escape velocity = sqrt(2 * acceleration due to gravity * radius).
Using the given values, the escape velocity of a spherical asteroid is:
exhaust rate = sqrt(2 * 3.00 m/s2 * 500 000 m) = 6928 m/s.
(b) To calculate the distance from the surface that a particle will travel when it leaves the surface of the asteroid with a radial velocity of 1000 m/s, we can use the formula for the height of the projectile:
Height = (radial velocity)2 / (2 * acceleration due to gravity).
After entering the specified values, the particle travels the distance:
Altezza = (1.000 m/s)2 / (2 * 3,00 m/s2) = 166.666,67 m.
(c) To calculate the speed at which an object hits an asteroid as it falls 1000 km above the surface, we can use the equation for the final velocity of the falling object:
Final velocity = sqrt (initial velocity2 + 2 * acceleration due to gravity * height).
Substituting the given values, the object hits the asteroid with a speed of:
Final velocity = square(0 + 2 * 3.00 m/s2 * 1,000,000 m) = square(6,000,000 m2/s2) = 2,449 m/s.
A batter hits a 0.140-kg baseball that was approaching him at 19.5 m/s and, as a result, the ball leaves the bat at 44.8 m/s in the reverse of its original direction. The ball remains in contact with the bat for 1.7 ms. What is the magnitude of the average force exerted by the bat on the ball?
Answer:
5295.3 N
Explanation:
According to law of momentum conservation, the change in momentum of the ball shall be from the momentum generated by the batter force
mv + P = mV
P = mV - mv = m(V - v)
Since the velocity of the ball before and after is in opposite direction, one of them is negative
P = 0.14(44.8 - (-19.5)) = 9 kg m/s
Hence the force exerted to generate such momentum within 1.7ms (0.0017s) is
F = P/t = 9/0.0017 = 5295.3 N
The half-life of Actinium 227 decays in 20 years. Calculate the mass of the element left when a 2kg sample was left for 160 years.
The mass of Actinium 227 left is 0.0078 kg
Explanation:
The amount of mass left of a radioactive isotope after time t is given by the equation:
[tex]m(t) = m_0 (\frac{1}{2})^{-\frac{t}{\tau_{1/2}}}[/tex]
where
[tex]m_0[/tex] is the initial amount of the sample
t is the time
[tex]\tau_{\frac{1}{2}}[/tex] is the half-life of the isotope
For the sample of Actinium 227 in this problem,
[tex]m_0 = 2 kg[/tex]
[tex]\tau_{1/2}=20 years[/tex]
t = 160 years
Substituting into the equation,
[tex]m(160) = (2 kg) (\frac{1}{2})^{-\frac{160}{20}}=0.0078 kg[/tex]
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Which of the following statements are true as a block of ice melts? Check all that apply.
a. The temperature of the ice/water system remains constant.
b. Heat energy leaves the ice/water system.
c. Heat energy enters the ice/water system.
d. The temperature of the ice/water system decreases.
e. The temperature of the ice/water system increases.
Answer:
a. The temperature of the ice/water system remains constant.
c. Heat energy enters the ice/water system.
Explanation:
As we know that when ice coverts into the water then it is known as phase transfer. In the phase transfer process temperature and the pressure remains constant but the heat enters into the system. This heat is responsible for melting the ice. The heat is taken by ice is known as latent heat.
Therefore the option "a" and "c" are correct.
The true statements as a block of ice melts are that the temperature of the ice/water system remains constant, and heat energy enters the system. The temperature does not change, increase, or decrease during the melting process until all the ice is melted. So the correct options are a and c.
Explanation:As a block of ice melts, several key processes occur within the ice/water system:
Therefore, from the options provided:
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Confidence intervals for the population mean μ and population proportion p _____ as the size of the sample increases.
Answer:
becomes narrower
Explanation:
Confidence intervals for the population mean μ and population proportion p becomes narrower as the size of the sample increases.
As,the sample size increases,standard error decreases,so margin of error decreases and hence width of CI decreases.
108J of work was done on a closed sysem. During this phase of the experiment, 79J of heat was added to the system.What was the total change in the internal energy of the system.
The change in internal energy of the system is +187 J
Explanation:
According to the first law of thermodynamics, the change in internal energy of a system is given by the equation:
[tex]\Delta U = Q + W[/tex]
where
[tex]\Delta U[/tex] is the change in internal energy
Q is the heat absorbed by the system
W is the work done on the system
For the system in this problem, we have
W = +108 J is the work done on it
Q = +79 J is the hear added to it
So, the change in internal energy is
[tex]\Delta U = 108 + 79 = +187 J[/tex]
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What parts of the nucleotide make up the backbone of the dna molecule
The backbone of the DNA molecule is formed by alternating sugar and phosphate groups. The nitrogenous bases are located in the interior of the molecule.
Explanation:The backbone of the DNA molecule is made up of the alternating sugar and phosphate groups. The sugar and phosphate groups are bonded by covalent bonds, and they line up on the outside of each strand. The nitrogenous bases, which include adenine (A), thymine (T), cytosine (C), and guanine (G), are stacked in the interior of the DNA molecule.
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The backbone of a DNA molecule is formed from alternating sugar and phosphate groups of nucleotides. The nitrogenous bases, which are not part of the backbone, protrude from it and are involved in the formation of the double helix structure.
Explanation:The backbone of the DNA molecule consists of alternating sugar and phosphate groups. A nucleotide, which is the building block of DNA, consists of three components: a nitrogenous base, a pentose sugar, and a phosphate group. The backbone is formed by the bonding of the phosphate group of one nucleotide to the sugar of the next nucleotide, creating a chain of sugar-phosphate bonds.
For instance, the phosphate group is attached to the 5' carbon of one nucleotide and the 3' carbon of the next nucleotide. In this sense, the DNA molecule can be visualized as a twisted ladder where the backbone represents the rails of the ladder and the nitrogenous bases represent the steps.
It's important to note that the nitrogenous bases are not part of the backbone; They stick out from the backbone and are involved in hydrogen bonding with the nitrogenous bases of the complementary DNA strand, resulting in a double helix structure. The two strands of the DNA run in opposite directions making them antiparallel.
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Classify the given types of matter as either baryonic (ordinary matter that contains protons and neutrons) or as nonbaryonic ("extraordinary" matter that consists of more exotic subatomic particles)
(Select B - Baryonic, N - Nonbaryonic. If the first is B and the rest N, enter
BNNNNN).
a. matter in our bodies
b. dark matter consisting of weakly interacting subatomic particles
c. dark matter consisting of Jupiter-sized planets in galactic halos
d. matter in brown dwarfs
e. matter that probably makes up the majority of dark matter in the universe
Answer:
a) B
b) N
c) B
d) B
e) N
Explanation:
a) The matter in our bodies is the regular matter, basically, we are made of carbon molecules and water. So it involves baryonic matter.
b) In this case, the weakly interacting subatomic particles know as (WIMPs), is the primary candidate for dark matter and this kind of particle has not yet been discovered. We are talking about the nonbaryonic matter.
c) Planets as a Jupiter are made of baryonic matter, in the specific case of Jupiter, it is approximately 75% hydrogen and 24% helium by mass and they are baryonic matter.
d) By definition, brown dwarfs are objects which have a size between a giant gaseous planet like Jupiter and a smaller star, so using the definitions above they are made of baryonic matter.
e) The majority of dark matter is made of non-baryonic matter.
A block of mass 3 kg, which has an initial
speed of 4 m/s at time t = 0, slides on a
horizontal surface.
Find the magnitude of the work that must
be done on the block to bring it to rest.Answer in units of J If a constant friction force of magnitude 2 Newtons is exerted on the block by the surface, find the magnitude of the acceleration of the block.
How far does the block slide before it comes to rest? units of m
Answer:
Explanation:
Kinetic energy of the block
= 1/2 m v²
= .5 x 3 x 4 x 4
= 24 J
Negative work of - 24 J is required to be done on this object to bring it to rest.
magnitude of acceleration due to frictional force
= force / mass
2 / 3
= 0 .67 m /s²
Let the body slide by distance d before coming to rest so work done by force = Kinetic energy
= 2 x d = 24
d = 12 m
The magnitude of the work done to bring the block to rest is 2 times the distance the block slides. The magnitude of the acceleration of the block is 2/3 m/s^2. The block slides a distance of 8/3 m before it comes to rest.
Explanation:The work done on an object is given by the equation:
Work = Force x Distance
In this case, the work done on the block to bring it to rest is equal to the force applied multiplied by the distance the block slides.
Given that the force exerted by the surface is 2 Newtons, we can calculate the magnitude of the work done:
Work = 2 N x Distance
To determine the distance the block slides, we need to calculate its deceleration using Newton's second law:
Force = mass x acceleration
Since the friction force is constant and in the opposite direction of motion, we can write:
2 N = 3 kg x acceleration
Solving for acceleration, we find:
acceleration = 2 N / 3 kg
With the acceleration calculated, we can use the kinematic equation:
vf^2 = vi^2 + 2ad
Since the final velocity is 0 (block comes to rest), the equation simplifies to:
0 = (4 m/s)^2 + 2(-acceleration)d
Solving for distance, we find:
d = (4 m/s)^2 / (2 x -acceleration)
Now, we can substitute the calculated acceleration into the equation to find the distance the block slides.
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Soil tilth refers to ________. Select one:
A. ratio of bulk density to particle density
B. the moisture content at which a soil is best suited for tillage
C. the physical suitability of a soil for plant growth
D. the bearing strength of a soil under a given downward force
E. micro-aggregates produced as a by-product of tillage
Answer:
Option c
Explanation:
Soil tilth refers to the physical condition of the soil, particularly with regards to the suitability of the soil for the growth of crop.
The determinants of the tilth in the soil incorporates the arrangement and steadiness of aggregated particles of the soil, pace of water penetration, level of air circulation, dampness content and seepage.
Thus option C follows the definition of the soil tilth.
Answer:
Option (C)
Explanation:
The soil tilth is defined as one of the physical property that determines the necessary condition for the better growth of plants and crops.
This property depends upon various factors such as-
fertility of the soil amount of moisture content in the soil rate and type of drainage pattern rate of percolation of water into the deeper zone of the soil porosity and permeability of the soil.These are the main features that characterize the soil tilth, and fulfilling the above required necessary condition, a plant grows in a much better way.
Thus, the correct answer is option (C).
A sled having a certain initial speed on a horizontal surface comes to rest after traveling 10 m. If the coefficient of kinetic friction between the object and the surface is 0.20, what was the initial speed of the object
Answer:
the initial speed of the object is 6.26 m/s
Explanation:
given information:
distance, s = 10 m
the coefficient of kinetic friction, μ = 0.2
we use the equation where the kinetic energy is equal to the friction force.
kinetic energy, KE = [tex]\frac{1}{2} mv^{2}[/tex]
friction work, W = F(friction) s
KE = W
[tex]\frac{1}{2} mv^{2}[/tex] = F(friction) s
where, F(friction) = μ N, N is normal force (N = m g)
= μ m g
so,
[tex]\frac{1}{2} mv^{2}[/tex] = μ m g s
[tex]\frac{1}{2} v^{2}[/tex] = μ g s
[tex]v^{2}[/tex] = 2 μ g s
= 2 (0.2) (9.8) (10)
= 39.2
hence,
v = [tex]\sqrt{3.92}[/tex]
= 6.26 m/s
The initial speed [tex]\( v_i \)[/tex] of the sled was approximately[tex]\( 6.26 \, \text{m/s} \).[/tex]
The initial speed of the object can be found using the work-energy principle, which states that the work done by friction will be equal to the change in kinetic energy of the sled.
First, let's calculate the work done by friction. The force of friction [tex]\( F_{\text{friction}} \)[/tex] is given by the product of the normal force \( N \) and the coefficient of kinetic friction [tex]\( \mu_k \):[/tex]
[tex]\[ F_{\text{friction}} = \mu_k N \][/tex]
[tex]\[ W = F_{\text{friction}} d = \mu_k m g d \][/tex]
[tex]\[ \Delta KE = 0 - \frac{1}{2} m v_i^2 = -\frac{1}{2} m v_i^2 \][/tex]
According to the work-energy principle, the work done by friction is equal to the change in kinetic energy:
[tex]\[ \mu_k m g d = -\frac{1}{2} m v_i^2 \][/tex]
Solving for \( v_i \), we get:
[tex]\[ v_i^2 = -2 \mu_k g d \][/tex]
[tex]\[ v_i = \sqrt{-2 \mu_k g d} \][/tex]
Given that [tex]\( \mu_k = 0.20 \), \( g = 9.81 \, \text{m/s}^2 \), and \( d = 10 \, \text{m} \),[/tex] we can plug in these values:
[tex]\[ v_i = \sqrt{-2 \times 0.20 \times 9.81 \times 10} \][/tex]
[tex]\[ v_i = \sqrt{39.24} \][/tex]
[tex]\[ v_i \approx 6.26 \, \text{m/s} \][/tex]
Therefore, the initial speed \( v_i \) of the sled was approximately[tex]\( 6.26 \, \text{m/s} \).[/tex]
The answer is: [tex]6.26 \, \text{m/s}.[/tex]
Part A What is a radio galaxy? How can radio galaxies affect the gas surrounding them? Drag the items on the left to the appropriate blanks on the right to complete the sentences. (Not all terms will be used.)
Explanation:
A radio galaxy is a galaxy with a powerful radio luminance relative to the stars' visible and infrared luminosity. Radio galaxies can emit strong and speedy particle jets. They inject in their surroundings high amounts of kinetic energy. Many radio galaxies have jets of plasma shooting out in opposite direction. Thus they ionize the gases surrounding them.
A disk-shaped merry-go-round of radius 2.13 m and mass 175 kg rotates freely with an angular speed of 0.651 rev/s. A 55.4 kg person running tangentially to the rim of the merry-go-round at 3.51 m/s jumps onto its rim and holds on. Before jumping on the merry-go-round, the person was moving in the same direction as the merry-go-round's rim.
Calculate the final kinetic energy for this system.
Answer:
Explanation:
To find out the angular velocity of merry-go-round after person jumps on it , we shall apply law of conservation of ANGULAR momentum
I₁ ω₁ + I₂ ω₂ = ( I₁ + I₂ ) ω
I₁ is moment of inertia of disk , I₂ moment of inertia of running person , I is the moment of inertia of disk -man system , ω₁ and ω₂ are angular velocity of disc and man .
I₁ = 1/2 mr²
= .5 x 175 x 2.13²
= 396.97 kgm²
I₂ = m r²
= 55.4 x 2.13²
= 251.34 mgm²
ω₁ = .651 rev /s
= .651 x 2π rad /s
ω₂ = tangential velocity of man / radius of disc
= 3.51 / 2.13
= 1.65 rad/s
I₁ ω₁ + I₂ ω₂ = ( I₁ + I₂ ) ω
396.97 x .651 x 2π + 251.34 x 1.65 = ( 396.97 + 251.34 ) ω
ω = 3.14 rad /s
kinetic energy = 1/2 I ω²
= 3196 J
You are designing a rotating metal flywheel that will be used to store energy. The flywheel is to be a uniform disk with radius 23.0 cm. Starting from rest at t = 0, the flywheel rotates with constant angular acceleration 3.00 rad/s2 about an axis perpendicular to the flywheel at its center.If the flywheel has a density (mass per unit volume) of 8600 kg/m3, what thickness must it have to store 800 J of kinetic energy at t = 8.00 s?
Answer:
t = 0.0735 m
Explanation:
Angular acceleration of the flywheel is given as
[tex]\alpha = 3 rad/s^2[/tex]
now after t = 8 s the speed of the flywheel is given as
[tex]\omega = \alpha t[/tex]
[tex]\omega = 3 \times 8 [/tex]
[tex]\omega = 24 rad/s[/tex]
now rotational kinetic energy of the wheel is given as
[tex]K = \frac{1}{2}I\omega^2[/tex]
[tex]K = \frac{1}{2}(\frac{1}{2}mR^2)(24^2)[/tex]
[tex]800 = \frac{1}{4}m(0.23)^2(24^2)[/tex]
[tex]m = 105 kg[/tex]
now we have
[tex]m = \rho (\pi R^2) t[/tex]
[tex]105 = 8600(\pi \times 0.23^2) t[/tex]
[tex]t = 0.0735 m[/tex]
We have that for the Question, it can be said that thickness must it have to store 800 J of kinetic energy at t = 8.00 s
h=0.0735m
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You are designing a rotating metal flywheel that will be used to store energy. The flywheel is to be a uniform disk with radius 23.0 cm. Starting from rest at t = 0, the flywheel rotates with constant angular acceleration 3.00 rad/s2 about an axis perpendicular to the flywheel at its center.If the flywheel has a density (mass per unit volume) of 8600 kg/m3,what thickness must it have to store 800 J of kinetic energy at t = 8.00 s?Generally the equation for the is mathematically given as
[tex]N=\pir^2h*P\\\\N=3.14*(23*10^{-2}^2)*h*8600\\\\N=1428.5h\\\\[/tex]
Therefore
[tex]E=\frac{1}{2}Iw^2\\\\800=\frac{1}{2}*(\frac{NR^2}{2}w^2)\\\\800=\frac{1}{2}*(\frac{(1428.523*10^{-2})^2}{2}(2*8)^2)[/tex]
h=0.0735m
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A runner of mass 60.0kg runs around the edge of a horizontal turntable mounted on a vertical, frictionless axis through its center. The runner's velocity relative to the earth has magnitude 2.50m/s . The turntable is rotating in the opposite direction with an angular velocity of magnitude 0.190rad/s relative to the earth. The radius of the turntable is 3.60m , and its moment of inertia about the axis of rotation is 81.0kg*m2 .
A) Find the final angular velocity of the system if the runner comes to rest relative to the turntable. (You can treat the runner as a particle.)
answer in rad/s please
To find the final angular velocity of the system, we need to apply the principle of conservation of angular momentum. The final angular momentum can be obtained by equating the initial angular momentum and the final angular momentum of the system. Solving for the final angular velocity gives us a value of approximately 38.54 rad/s.
Explanation:To find the final angular velocity of the system, we need to apply the principle of conservation of angular momentum. The initial angular momentum of the system is given by:
Li = Itωt + Irωr
Where It and Ir are the moments of inertia of the turntable and the runner respectively, and ωt and ωr are their respective angular velocities.
Since the runner comes to rest relative to the turntable, ωr = 0. Therefore, the final angular momentum of the system is:
Lf = Itωt
Using the conservation of angular momentum principle, we can set Li equal to Lf:
Itωt = Itωt
Substituting the given values:
81.0kg × m² × 0.190rad/s = It × ωt
Solving for ωt, we find that the final angular velocity of the system is approximately 38.54 rad/s.
To find the final angular velocity of the runner and turntable system, we apply conservation of angular momentum. The final angular velocity is calculated to be 0.611 rad/s. This involves determining the initial angular momenta of both the runner and the turntable, then using the total moment of inertia to find the final velocity.
To find the final angular velocity of the system when the runner comes to rest relative to the turntable, we need to apply the principle of conservation of angular momentum. The initial angular momentum of the system (runner plus turntable) must equal the final angular momentum since there are no external torques acting.
Step-by-Step Calculation :
Determine the initial angular momentum of the runner:
The runner's linear velocity is 2.50 m/s, and they can be treated as a particle moving on a circular path with radius 3.60 m. So, the initial angular momentum (L_runner) is the product of the runner's mass (m), velocity (v), and radius (r):L_runner = m * v * r = 60.0 kg * 2.50 m/s * 3.60 m = 540 kg·m²/s.Determine the initial angular momentum of the turntable:
The initial angular velocity of the turntable (ω_t) is 0.190 rad/s, and its moment of inertia (I_t) is 81.0 kg·m². So, the initial angular momentum (L_turntable) is:L_turntable = I_t * ω_t = 81.0 kg·m² * 0.190 rad/s = 15.39 kg·m²/s.Calculate the total initial angular momentum:
The total initial angular momentum (L_initial) is:L_initial = L_runner - L_turntable = 540 kg·m²/s - 15.39 kg·m²/s = 524.61 kg·m²/s. (The minus sign indicates the turntable rotates in the opposite direction of the runner.)Find the final angular velocity:
When the runner comes to rest relative to the turntable, their combined angular momentum will be conserved. Let ω_final be the final angular velocity and I_total be the combined moment of inertia. The runner's moment of inertia can be treated as a point mass:I_runner = m * r² = 60.0 kg * (3.60 m)² = 777.60 kg·m².I_total = I_t + I_runner = 81.0 kg·m² + 777.60 kg·m² = 858.60 kg·m².Using conservation of angular momentum:L_initial = I_total * ω_finalω_final = L_initial / I_total = 524.61 kg·m²/s / 858.60 kg·m² = 0.611 rad/s.The final angular velocity of the system is 0.611 rad/s.
The Huka Falls on the Waikato River is one of New Zealand's most visited natural tourist attractions. On average the river has a flow rate of about 300,000 L/s. At the gorge, the river narrows to 18 m wide and averages 22 m deep.(a) What is the average speed (in m/s) of the river in the gorge?_______m/s.(b) What is the average speed (in m/s) of the water in the river downstream of the falls when it widens to 63 m and its depth increases to an average of 42 m________m/s.
Answer
Given,
Flow rate of river is equal to 300,000 L/s.
Width of river = 18 m
and depth of river = 22 m
a) Average speed of river
Q = 300,000 L/s
= 300 m³/s
Q = Av
[tex]v = \dfrac{Q}{A}[/tex]
[tex]v = \dfrac{300}{18 \times 22}[/tex]
[tex]v = \dfrac{300}{396}[/tex]
[tex]v = 0.757\ m/s[/tex]
b) Average speed when river is widen to 63 m and depth is increased to
[tex]v = \dfrac{Q}{A}[/tex]
[tex]v = \dfrac{300}{63 \times 42}[/tex]
[tex]v = \dfrac{300}{2646}[/tex]
[tex]v = 0.113\ m/s[/tex]
Final answer:
The average speed of the river in the gorge is 0.75 m/s, and downstream of the falls, when the river widens and its depth increases, the average speed decreases to 0.125 m/s.
Explanation:
To calculate the average speed of the river at Huka Falls, we can use the formula for flow rate, which is the volume of fluid passing a point in the river per unit of time:
Flow rate (Q) = Area (A) imes Velocity (V)
(a) Average speed in the gorge: Given that the flow rate (Q) is 300,000 liters per second (which is equal to 300 cubic meters per second, since 1,000 liters is equal to 1 cubic meter), and the cross-sectional area of the river in the gorge (A) is 20 meters wide imes 20 meters deep (400 square meters), we can solve for velocity (V) using the formula:
Q = A imes V
300 m³/s = 400 m² imes V
V = 300 m³/s \/ 400 m²
V = 0.75 meters per second
(b) Average speed downstream of the falls: Downstream, the river widens to 60 meters and deepens to an average of 40 meters, so the cross-sectional area is 2,400 square meters. Using the same flow rate, we can find the new velocity:
Q = A imes V
300 m³/s = 2,400 m² imes V
V = 300 m³/s \/ 2,400 m²
V = 0.125 meters per second
In order to sail through the frozen Arctic Ocean, the most powerful icebreaker ever built was constructed in the former Soviet Union. At the heart of the ship’s power plant is a nuclear reactor with a power output of 5.60* 10^7 W. How long will it take for this power plant to do 5.35* 10^10 J of work?
Answer:
955.36 seconds ≈ 16 minutes
Explanation:
Power(P) is the rate of doing work(W)
That is, P = W/t, where t is the time.
multipying both sides with 't' and dividing with 'P', we get: t=W/P
Here, W = 5.35 x 10^10 J and P = 5.6 x 10^7 W ( 1 W = 1 J/s).
Therefore , on dividing W with P, we get 955.36 seconds.
Russell drags his suitcase 15.0 M from the door of his house to the car at a constant speed with a horizontal force of 95.0 N. How much work does Russell do to overcome the force of
The work done is 1425 J
Explanation:
The work done by a force to move an object is given by
[tex]W=Fd cos \theta[/tex]
where
F is the magnitude of the force
d is the displacement
[tex]\theta[/tex] is the angle between the direction of the force and of the displacement
For the suitcase in this problem, we have:
F = 95.0 N is the force applied
d = 15.0 m is the displacement
[tex]\theta=0[/tex], since the force is parallel to the displacement
Substituting, we find
[tex]W=(95.0)(15.0)(cos 0)=1425 J[/tex]
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Tarzan swings on a 35.0 m long vine initially inclined at an angle of 44.0◦ with the vertical. The acceleration of gravity if 9.81 m/s2.
What is his speed at the bottom of the swing if he
a) starts from rest?
b) pushes off with a speed of 6.00 m/s?
Answer:
(A) Vf = 13.8 m/s
(B) Vf = 15.1 m/s
Explanation:
length of rope (L) = 35 m
angle to the vertical = 44 degrees
acceleration due to gravity (g) = 9.8 m/s^{2}
(A) from conservation of energy
final kinetic energy + final potential energy = initial kinetic energy + initial potential energy
0.5m(Vf)^{2} + mg(Hf) = 0.5m(Vi)^{2} + mg(Hi)
where
m = mass
Hi = initial height = 35 cos 44 = 25.17
Hf = final height = length of vine = 35 m
Vi = initial velocity = 0 since he starts from rest
Vf = final velocity
the equation now becomes
0.5m(Vf)^{2} + mg(Hf) = mg(Hi)
0.5m(Vf)^{2} = mg (Hi - Hf)
0.5(Vf)^{2} = g (Hi - Hf)
0.5(Vf)^{2} = 9.8 x (25.17 - 35)
0.5(Vf)^{2} = - 96.3 (the negative sign tells us the direction of motion is downwards)
Vf = 13.8 m/s
(B) when the initial velocity is 6 m/s the equation remains
0.5m(Vf)^{2} + mg(Hf) = 0.5m(Vi)^{2} + mg(Hi)
m(0.5(Vf)^{2} + g(Hf)) = m(0.5(Vi)^{2} + g(Hi))
0.5(Vf)^{2} + g(Hf) = 0.5(Vi)^{2} + g(Hi)
0.5(Vf)^{2} = 0.5(Vi)^{2} + g(Hi) - g(Hf)
0.5(Vf)^{2} = 0.5(6)^{2} + (9.8 x (25.17 - 35))
0.5(Vf)^{2} = -114.3 ( just as above, the negative sign tells us the direction of motion is downwards)
Vf = 15.1 m/s
Answer:
a) [tex]v_{f} \approx 0.328\,\frac{m}{s}[/tex], b) [tex]v_{f} \approx 6.009\,\frac{m}{s}[/tex]
Explanation:
Let consider that bottom has a height of zero. The motion of Tarzan can be modelled after the Principle of Energy Conservation:
[tex]U_{g,1} + K_{1} = U_{g,2} + K_{2}[/tex]
The final speed is:
[tex]K_{2} = U_{g,1} - U_{g,2} + K_{1}[/tex]
[tex]\frac{1}{2}\cdot m \cdot v_{f}^{2} = m\cdot g \cdot L\cdot (\cos \theta_{2}-\cos \theta_{1}) + \frac{1}{2}\cdot m \cdot v_{o}^{2}[/tex]
[tex]v_{f}^{2} = 2 \cdot g \cdot L \cdot (\cos \theta_{2} - \cos \theta_{1}) + v_{o}^{2}[/tex]
[tex]v_{f} = \sqrt{v_{o}^{2}+2\cdot g \cdot L \cdot (\cos \theta_{2}-\cos \theta_{1})}[/tex]
a) The final speed is:
[tex]v_{f} = \sqrt{(0\,\frac{m}{s} )^{2}+2\cdot (9.807\,\frac{m}{s^{2}} )\cdot (35\,m)\cdot (\cos 0^{\textdegree}-\cos 44^{\textdegree})}[/tex]
[tex]v_{f} \approx 0.328\,\frac{m}{s}[/tex]
b) The final speed is:
[tex]v_{f} = \sqrt{(6\,\frac{m}{s} )^{2}+2\cdot (9.807\,\frac{m}{s^{2}} )\cdot (35\,m)\cdot (\cos 0^{\textdegree}-\cos 44^{\textdegree})}[/tex]
[tex]v_{f} \approx 6.009\,\frac{m}{s}[/tex]
A quantity of N2 occupies a volume of 1.4 L at 290 K and 1.0 atm. The gas expands to a volume of 3.3 L as the result of a change in both temperature and pressure. find density of the gas
Answer:
[tex]\rho = 0.50 g/L[/tex]
Explanation:
As we know that
PV = nRT
here we have
[tex]P = 1.0 atm[/tex]
[tex]P = 1.013 \times 10^5 Pa[/tex]
so we have
[tex]V = 1.4 \times 10^{-3} m^3[/tex]
T = 290 K
now we have
[tex](1.013 \times 10^5)(1.4 \times 10^{-3}) = n(8.31)(290)[/tex]
[tex]n = 0.06 [/tex]
now the mass of gas is given as
[tex]m = n M[/tex]
[tex]m = (0.06)(28)[/tex]
[tex]m = 1.65 g[/tex]
now density of gas when its volume is increased to 3.3 L
so we will have
[tex]\rho = \frac{m}{V}[/tex]
[tex]\rho = \frac{1.65 g}{3.3 L}[/tex]
[tex]\rho = 0.50 g/L[/tex]
Final answer:
The density of a gas can be calculated using the ideal gas law, which states that PV = nRT. To find the density, we can rearrange the equation and substitute the given values.
Explanation:
The density of a gas can be calculated using the ideal gas law, which states that PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. To find the density, we can rearrange the equation as follows:
Density = mass/volume = (molar mass * n) / V
Using the given information, we can substitute the values into the equation and solve for the density.
George Of The Jungle's wife, Mrs. Of The Jungle, has been pestering him to go on a diet. He should have listened. During his commute home last Thursday afternoon, the Number 8 vine upon which he was swinging (along a circular path) snapped.
At the time of the incident, George was at the bottom of his swing, moving at a peppy 14.1 m/s. Given that the maximum tension that the vine (length 7.3 m) was able to tolerate was 4150 N, determine George's mass.
A) 110 kg
B) 120 kg
C) 130 kg
D) 140 kg
E) 150 kg
Answer:
112.06 kg - Thats heavy !
Explanation:
Let's do force balance here. Let the object of our interest be George. The forces acting on him are the tension in the upward direction, his weight in the downward direction and the centrifugal force in the downward direction. Considering the upward and downward directions on the y-axis and f=given the fact that George doesn't move up or down, the forces are balanced along the y-axis. Hence doing force balance:
magnitude of forces upward =magnitude of forces downward
i.e., Tension(T) = Weight(mg) + Centrifugal force (mv²/r)
where: 'm' is the mass of George, g is the acceleration due to gravity (9.8 m/s²). v is the speed with which George moves (14.1 m/s) and r is the radius of the circle in which he's moving at the instant (Here since he's swinging on the rope, he moves in a circle with radius as the length of the rope and hence r=7.3m).
therefore, T = m (9.8 + (14.1)²/7.3) = 4150 N
Therefore, m = 112.06 kg
A 12 inch telescope has an angular resolution how many times smaller when compared to a 4 inch telescope?
Answer:
θ₂ = 3 θ₁
Explanation:
given,
telescope of lens diameter = 12 inch
another telescope of lens diameter = 4 inch
comparison of resolution power.
Using the formula of resolution
[tex]\theta = \dfrac{1.22 \lambda}{D}[/tex]
for diameter = 12 inch
[tex]\theta_1 = \dfrac{1.22 \lambda}{D_1}[/tex].....(1)
for diameter = 4 inch
[tex]\theta_2 = \dfrac{1.22 \lambda}{D_2}[/tex].......(2)
dividing equation (2) from (1)
[tex]\dfrac{\theta_2}{\theta_1} = \dfrac{D_1}{D_2}[/tex]
now,
[tex]\dfrac{\theta_2}{\theta_1} = \dfrac{12}{4}[/tex]
[tex]\dfrac{\theta_2}{\theta_1} =3[/tex]
θ₂ = 3 θ₁
hence, we can say that resolution of telescope of 12 inch is 3 time smaller than the resolution of 4 inch telescope.
Linear Thermal Expansion (in one dimension)
1) The change in length ΔL is proportional to the original length L, and the change in temperature ΔT : ΔL = αLΔT, where ΔL is the change in length , and α is the coefficient of linear expansion.
a) The main span of San Francisco’s Golden Gate Bridge is 1275 m long at its coldest (–15ºC). The coefficient of linear expansion, α , for steel is 12×10−6 /ºC. When the temperatures rises to 25 °C, what is its change in length in meters?
2) The change in volume ΔV is very nearly ΔV ≈ 3αVΔT . This equation is usually written as ΔV = βVΔT, where β is the coefficient of volume expansion and β ≈ 3α . V is the original volume. ΔT is the change in temperature. Suppose your 60.0-L (15.9-gal) steel gasoline tank is full of gasoline, and both the tank and the gasoline have a temperature of 15.0ºC . The coefficients of volume expansion, for gasoline is βgas = 950×10−6 /ºC , for the steel tank is βsteel = 35×10−6 /ºC .
a) What is the change in volume (in liters) of the gasoline when the temperature rises to 25 °C in L?
b) What is the change in volume (in liters) of the tank when the temperature rises to 25 °C in L?
c) How much gasoline would be spilled in L?
Answer:
1) [tex]\Delta L= 0.612\ m[/tex]
2) a. [tex]\Delta V_G=0.57\ L[/tex]
b. [tex]\Delta V_S=0.021\ L[/tex]
c. [tex]V_0=0.549\ L[/tex]
Explanation:
1)
given initial length, [tex]L=1275\ m[/tex]initial temperature, [tex]T_i=-15^{\circ}C[/tex]final temperature, [tex]T_f=25^{\circ}C[/tex]coefficient of linear expansion, [tex]\alpha=12\times 10^{-6}\ ^{\circ}C^{-1}[/tex]∴Change in temperature:
[tex]\Delta T=T_f-T_i[/tex]
[tex]\Delta T=25-(-15)[/tex]
[tex]\Delta T=40^{\circ}C[/tex]We have the equation for change in length as:
[tex]\Delta L= L.\alpha. \Delta T[/tex]
[tex]\Delta L= 1275\times 12\times 10^{-6}\times 40[/tex]
[tex]\Delta L= 0.612\ m[/tex]
2)
Given relation:
[tex]\Delta V=V.\beta.\Delta T[/tex]
where:
[tex]\Delta V[/tex]= change in volume
V= initial volume
[tex]\Delta T[/tex]=change in temperature
initial volume of tank, [tex]V_{Si}=60\ L[/tex]initial volume of gasoline, [tex]V_{Gi}=60\ L[/tex]initial temperature of steel tank, [tex]T_{Si}=15^{\circ}C[/tex]initial temperature of gasoline, [tex]T_{Gi}=15^{\circ}C[/tex]coefficients of volumetric expansion for gasoline, [tex]\beta_G=950\times 10^{-6}\ ^{\circ}C[/tex]coefficients of volumetric expansion for gasoline, [tex]\beta_S=35\times 10^{-6}\ ^{\circ}C[/tex]a)
final temperature of gasoline, [tex]T_{Gf}=25^{\circ}C[/tex]
∴Change in temperature of gasoline,
[tex]\Delta T_G=T_{Gf}-T_{Gi}[/tex]
[tex]\Delta T_G=25-15[/tex]
[tex]\Delta T_G=10^{\circ}C[/tex]
Now,
[tex]\Delta V_G= V_G.\beta_G.\Delta T_G[/tex]
[tex]\Delta V_G=60\times 950\times 10^{-6}\times 10[/tex]
[tex]\Delta V_G=0.57\ L[/tex]
b)
final temperature of tank, [tex]T_{Sf}=25^{\circ}C[/tex]
∴Change in temperature of tank,
[tex]\Delta T_S=T_{Sf}-T_{Si}[/tex]
[tex]\Delta T_S=25-15[/tex]
[tex]\Delta T_S=10^{\circ}C[/tex]
Now,
[tex]\Delta V_S= V_S.\beta_S.\Delta T_S[/tex]
[tex]\Delta V_S=60\times 35\times 10^{-6}\times 10[/tex]
[tex]\Delta V_S=0.021\ L[/tex]
c)
Quantity of gasoline spilled after the given temperature change:
[tex]V_0=\Delta V_G-\Delta V_S[/tex]
[tex]V_0=0.57-0.021[/tex]
[tex]V_0=0.549\ L[/tex]
The blood plays an important role in removing heat from th ebody by bringing the heat directly to the surface where it can radiate away. nevertheless, this heat must still travel through the skin before it can radiate away. we shall assume that the blood is brought to the bottom layer of skin at a temperature of 37.0 degrees C and that its outer surface of the skin is at 30.0 degrees C. Skin varies in thickness from 0.500mm to a few millimeters on the palms and the soles so we shall assume an average thickness off 0.740mm. a 165lb, 6 ft person has a surface area of about 2.00 m^2 and loses heat at a net rate of 75.0 w while resting. On the basis of our assumptions, what is the thermal conductivity of this persons skin?
Answer: Thermal comductivity (K) is 3.964x 10 ^-3 W/m.k
Explanation:
Thermal comductivity K = QL/A∆T
Q= Amount of heat transferred through the material in watts = 75W
L= Distance between two isothermal planes = 0.740mm
A= Area of the surface in square metres = 2m^2
∆T= Temperature change = (37-30) °C.
Solving this : K =( 75 x 0.740 x 10^-3)/ 2 x (37-30)
K = 3.964x 10 ^-3 W/m.k
The question pertains to calculating the thermal conductivity of human skin, a Physics concept linked to heat transfer. By using Fourier's Law of heat conduction, and rearranging the formula to solve for thermal conductivity using given data, an approximate thermal conductivity can be obtained.
Explanation:The query is related to the determination of thermal conductivity of human skin based on the known parameters. This phenonmenon belongs to the domain of Physics, specifically heat transfer. Here, thermal conductivity is the measure of a material's ability to conduct heat. In this scenario, you have to consider the heat conduction through the skin, which relies on Fourier's Law of heat conduction. It can be represented as:
Q = (k*A*(T1 - T2))/d
Where, Q is the heat transfer rate, k is the thermal conductivity, A is the area of heat transfer, T1 and T2 are the initial and final temperatures, and d is the thickness of the material, in this case, the skin.
From the given data, you can plug in the values into this formula. However, our primary objective is to find out 'k'. Rearranging the formula to find k gives us:
k = (Q * d) / (A * (T1 - T2))
Now, if we put all the given values into the formula, we get:
k = (75 W * 0.00074 m) / (2 m^2 * (37°C - 30°C))
Solving this would provide us with the estimate of thermal conductivity of the skin.
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