The gauge pressure at point 1 = 145.2 kPa.
What is Bernoulli's equation?The equation states that the sum of the pressure, kinetic energy, and potential energy per unit volume of a fluid is constant along a streamline, in the absence of external forces like friction or viscosity.
The principle of continuity of mass flow rate and Bernoulli's equation, which connects pressure, velocity, and height of a fluid in a flow, can be used to solve this issue.
With the provided data, we can first determine the soft drink's mass flow rate:
mass flow rate =[tex](220 cans/min) x (0.355 L/can) x (1 kg/L) x (1 min/60 s)[/tex] = [tex]1.235 kg/s[/tex]
Then, we may compare the soft drink's velocities at positions 1 and 2 using the principle of continuity:
[tex]A1 v1 = A2 v2[/tex]
where A1 and A2 are the pipe's cross-sectional areas at points 1 and 2, respectively, and v1 and v2 are the soft drink's velocities at those same locations. When we solve for v1, we get:
[tex]v1 = (A2/A1) v2 = (8.00 cm^2)/(2.00 cm^2) v2 = 4 v2[/tex]
Now, we can link the pressures, velocities, and heights of the soft drink at points 1 and 2 using the Bernoulli's equation:
P1 + (1/2)ρv1²+ ρgh1 = P2 + (1/2)ρv2²+ ρgh2
where P1 and P2 are the gauge pressures at points 1 and 2, respectively; ρ is the soft drink's density; g is gravity's acceleration; and h1 and h2 are the heights of the respective points 1 and 2 above a reference level.
As the soft drink contains primarily water, we may calculate its density using the formula: = 1000 kg/m3. Moreover, we can decide to set the reference level at point 2, making h2 = 0.When the supplied values and the velocity relation are substituted, we obtain:
[tex]P1 + (1/2)(1000 kg/m^3)(16 v2^2) + (1000 kg/m^3)(9.81 m/s^2)(1.35 m)[/tex] =[tex]152 kPa + (1/2)(1000 kg/m^3)(v2^2)[/tex]
By condensing and figuring out P1, we get at:
[tex]P1 = 152 kPa + (1/2)(1000 kg/m^3)(15 v2^2) - (1000 kg/m^3)(9.81 m/s^2)(1.35 m)[/tex] = [tex]152 kPa + 6.75 v2^2 - 13.33 kPa[/tex]
We must now locate v2. We can link the velocity and the cross-sectional area at point 2 using the mass flow rate relation and the density of the soft drink:
[tex]A2 v2[/tex] = (mass flow rate)/ρ =[tex]1.235/(1000 kg/m^3)[/tex]= [tex]0.001235 m^3/s[/tex]
Upon solving for v2, we obtain:
v2 =[tex]0.001235 m^3/s / (8.00 cm^2 / 10000 cm^2)[/tex]= [tex]0.1544 m/s[/tex]
Last but not least, we can insert this value into the P1 expression to obtain:
[tex]P1[/tex] =[tex]152 kPa + 6.75 (0.1544 m/s)^2 - 13.33 kPa[/tex] = [tex]145.2 kPa[/tex]
As a result, point 1's gauge pressure is 145.2 kPa.
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With the solar photon flux of 3*10^21 m²s¹, how large area in [m²] covered by solar panels would be necessary to produce 150 kW of power? Assume yellow light (2.1 eV) and 15% power conversion efficiency.
O a. 100
O b. 50
O c. 25
O d. 10
O e. 1000
Answer: (D) 10
can't explain brainly keeps censoring the numbers out for some reason
An airplane flies with a constant speed of 1624 miles per hour. How far can it travel in 30 minutes?
The distance the airplane will travel in 30 minutes, given that the airplane travels at 1624 miles per hour is 812 miles
How do i determine the distance travelled?The following data were obtained from the question:
Speed of airplane = 1624 mile per hourTime taken = 30 minutes = 30 . 60 = 0.5 hourDistance travelled =?Speed is defined as distance travelled per unit time.
Speed = Distance travelled / time
Inputting the various parameters, we can obtain the distance travelled as follow:
1624 = Distance travelled / 0.5
Cross multiply
Distance travelled = 1624 × 0.5
Distance travelled = 812 miles
Thus, we can conclude that the distance travelled by the airplane in 30 minutes is 812 miles
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A baseball player wants to hit a home run over the wall of a
stadium. The player swings the baseball bat so that it hits the
ball when it is at a height of 0.996 m above the ground. The
ball flies off at an angle of 30° above the horizontal and at a
speed of 36.2 m/s. What is the tallest wall that the player can
clear (i.e., get the ball over) if the wall is 99.1 m away
horizontally?
A wall less than or equal to 14.7 m in height is the highest the player can clear.
How to calculate height?Use the kinematic equations of motion to solve the problem. First, we can find the time it takes for the ball to travel 99.1 m horizontally:
d = vt
t = d / v
t = 99.1 m / 36.2 m/s
t ≈ 2.74 s
Now, use the vertical motion equation to find the maximum height the ball reaches:
y = yo + vot + (1/2)at²
where:
yo = 0.996 m (initial height)
vo = v sinθ = 36.2 m/s x sin(30°) ≈ 18.1 m/s (initial vertical velocity)
a = -9.81 m/s² (acceleration due to gravity, pointing downward)
t = 2.74 s (time of flight)
y = 0.996 m + 18.1 m/s x 2.74 s + (1/2) x (-9.81 m/s²) x (2.74 s)²
y ≈ 14.7 m
Therefore, the tallest wall the player can clear is a wall with a height less than or equal to 14.7 m.
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33) If a speeding motorcycle is moving, it must also
have
a) potential energy.
b) kinetic energy.
c) chemical energy.
d) all of the above
A 0.120 kg, 90.0-cm-long uniform bar has a small 0.055 kg mass glued to its left end and a small 0.110 kg mass glued to the other end. The two small masses can each be treated as point masses. You want to balance this system horizontally on a fulcrum placed just under its center of gravity.
How far from the left end should the fulcrum be placed?
The fulcrum should be placed 0.120 m from the left end of the bar to balance the system horizontally.
What is balance?
To balance the system horizontally, the center of gravity (CG) of the bar and the attached masses should be placed directly above the fulcrum. We can find the location of the CG using the following formula:
CG = (m1x1 + m2x2 + m3x3) / (m1 + m2 + m3)
where m1, m2, and m3 are the masses of the bar, the 0.055 kg mass, and the 0.110 kg mass, respectively, and x1, x2, and x3 are their respective distances from the left end of the bar.
We know that the total mass of the system is:
m = m1 + m2 + m3 = 0.120 kg + 0.055 kg + 0.110 kg = 0.285 kg
Let x be the distance from the left end of the bar to the fulcrum. Then, the distance from the fulcrum to the center of gravity is (L/2 - x), where L is the total length of the bar (90.0 cm). Therefore, we want to find x such that:
CG = (L/2 - x)
Substituting the expressions for the CG and the masses, we get:
(m1x1 + m2x2 + m3x3) / (m1 + m2 + m3) = (L/2 - x)
Simplifying and rearranging, we get:
x = (m1x1 + m2x2 + m3x3) / (m1 + m2 + m3) - L/2
We can choose any two points on the bar as reference points, and take their distances as x1 and x3. Let's choose the left end of the bar as x1 = 0, and the right end of the bar as x3 = L = 90.0 cm = 0.900 m. Then, we can find x2, the distance from the left end to the 0.110 kg mass, as:
x2 = L - x1 - x3 = 0.900 m - 0 m - 0.090 m = 0.810 m
Substituting the masses and distances, we get:
x = (m1x1 + m2x2 + m3x3) / (m1 + m2 + m3) - L/2
x = (0 kg × 0 m + 0.055 kg × 0.810 m + 0.110 kg × 0.900 m) / (0.120 kg + 0.055 kg + 0.110 kg) - 0.450 m
x = 0.570 m - 0.450 m
x = 0.120 m
Therefore, the fulcrum should be placed 0.120 m from the left end of the bar to balance the system horizontally.
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Need Help! The diagram shows a cross section of the Earth near its surface.
In the diagram, continental crust is labeled ___, and oceanic crust is labeled ____
A. 1; 2
B. 3; 1
C. 2; 1
D. 1; 3
In the diagram, continental crust is labeled 1 and oceanic crust is labeled 2.
option A.
What are the characteristics of Continental and ocean crust?
Continental crust:
Generally thicker, averaging about 30-40 kilometers in depth.Composed mostly of granite, which is a type of igneous rock.Less dense compared to oceanic crust.Contains a wide range of rock types, including sedimentary, metamorphic, and igneous rocks.Forms the continents and the shallow shelves extending from the continents under the oceans.Oceanic crust:
Generally thinner, averaging about 5-10 kilometers in depth.Composed mostly of basalt, which is a type of igneous rock.More dense compared to continental crust.Mostly consists of igneous rocks, specifically basaltic lava flows and volcanic rocks.Forms the ocean basins and the deep ocean floor.Learn more about Continental and ocean crust here: https://brainly.com/question/5637320
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Determine the resultant x and y components, when one vector is equal to 26.5 and the other 44 with angles of 56 and 28 degrees
The resultant x and y components are 53.23 and 41.51, respectively.
What are the components?
To determine the resultant x and y components of the vectors, we can use the following equations:
Rx = ΣFx = F1x + F2x + ...
Ry = ΣFy = F1y + F2y + ...
where F1x, F2x, ... are the x components of the vectors, F1y, F2y, ... are the y components of the vectors, and ΣFx and ΣFy are the total x and y components of the resultant vector, R.
First, we need to find the x and y components of each vector:
Vector 1: magnitude = 26.5, angle = 56°
F1x = 26.5cos(56) = 14.28
F1y = 26.5sin(56) = 21.44
Vector 2: magnitude = 44, angle = 28°
F2x = 44cos(28) = 38.95
F2y = 44sin(28) = 20.07
Now we can add the x and y components of the vectors to find the total x and y components of the resultant vector:
ΣFx = F1x + F2x = 14.28 + 38.95 = 53.23
ΣFy = F1y + F2y = 21.44 + 20.07 = 41.51
Therefore, the resultant x and y components are 53.23 and 41.51, respectively.
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The learner added some iodine solution to the water in the test tube. After 30 minutes at room temperature, the contents of the visking tubing bag were stained black, but the water in the test tube remained a yellow color.( I) explain this results
b) An iron block of mass 4kg rest on an aluminium platform. A horizontal force of 20N is applied to the block. The coefficient of limiting friction between the two surfaces is 0.6. i) Will the block move? ii) If it moves, what will be its acceleration.
The block won't move since the applied force of 20 N is less than the maximum frictional force of 23.52 N.
How can I calculate the friction force between the blocks at its maximum?When an object is on a rough surface, friction is a force that acts in the opposite direction from the direction of motion. FMAX=R F MAX = R where is the coefficient of friction and R is the normal reaction between the two surfaces is the maximum or limiting value of friction between two surfaces.
[tex]F_gravity = m * g[/tex]
[tex]F_norm = F_gravity = m * g = 4 kg * 9.8 m/s^2 = 39.2 N[/tex]
[tex]F_friction = 0.6 * 39.2 N = 23.52 N[/tex]
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As magma rises through the cracks of a spreading seafloor, it cools and adds new rock to the ocean floor.
The ribbon of magma causing the spreading is most similar to which structure?
A volcano
A sinkhole
A valley
A glacier
Answer:
zd its c
Explanation:
i know
math
Determine if the function below is continuous.
graph of a piecewise function, with 2 pieces. The first piece is a line that starts at negative infinity and goes through point (-4,5) and ends with an open dot at (1,0). The second piece is a line that starts with a closed dot at (1,-1), goes through point (2,1) and continues to infinity.
A. not continuous at x = 1
B. not continuous at x = 0
C. not continuous at x = -1
D. continuous
The function is not continuous at x=1, so the answer is A.
At x=1, the two pieces of the function meet. The first piece ends with an open dot at (1,0), meaning that the function is not defined at $x=1$. The second piece starts with a closed dot at (1,-1), meaning that the function is defined at x=1 and takes the value -1 there.
Since the function is not defined at x=1, it cannot be continuous at that point. Therefore, the function is not continuous overall.
A wave has a frequency of 50Hz and a wavelength of 20m.
Calculate the speed of the wave in m/s.
Explanation:
f= 50Hz
wavelength= 20m
Speed of wave (v) = frequency × wavelength
= 50×20
= 1000m/s
Fill in the blanks with GPE (Gravitational potential energy) or KE (Kinetic energy)
Consider the concepts of kinetic energy (KE) and gravitational potential energy (GPE) as you complete these questions. A ball is held 1.4 meters above the floor. Use the terms KE of GPE as your answers.
In fact, in the absence of air resistance, the amount of ____ energy when the ball is held motionless above the floor equals the amount of ____ energy at impact with the floor.
In fact, in the absence of air resistance, the amount of GPE (Gravitational potential energy) energy when the ball is held motionless above the floor equals the amount of KE (Kinetic energy) energy at impact with the floor.
Starting with an initial speed of 6.00 m/s, a 62.0 kg skier skies down a slope. The speed of the skier at the bottom of the slope is 26.5 m/s. Neglecting friction, determine the height of the slope.
The height of the slope is 323.6 meters. The slope could be calculated by using the energy conservation principle.
What is friction?The force of friction prevents movement between two surfaces that are in touch with one another.
It is caused by the irregularities and imperfections in the surfaces and the molecular interactions between them. Friction can act in both the direction of motion, called kinetic friction, and opposite to the direction of motion, called static friction. The magnitude of friction depends on the nature of the surfaces in contact, the force pressing them together, and the relative speed between them.
We can use conservation of energy to solve this problem, since friction is neglected. The total mechanical energy of the system (skier + Earth) is conserved:
KE_i + PE_i = KE_f + PE_f
where PE stands for potential energy and KE for kinetic energy.
At the top of the slope, the skier has only potential energy:
PE_i = mgh
where m is the skier's mass, g is the acceleration brought on by gravity, and h is the slope's height.
KE_f = (1/2)mv²
where v denotes the skier's speed at the slope's base.
Setting the initial and final energies equal to each other, we have:
mgh + 0 = (1/2)mv² + 0
Solving for h, we get:
h = (v²)/(2g) - (1/2)h
Plugging in the given values, we get:
h = (26.5²)/(2*9.81) - (1/2)h
Simplifying and solving for h, we get:
h = 323.6 m
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A traditional (historical) windmill with the radius of its sails of 15 [m] subject to wind of 7 m/s pumps 3.4 [m³/s] of water over a height of 2.1 [m]. Estimate the actual efficiency of the windmill in [%]. Assume the Betz limit for wind power with Uout=Uin/3 (a=1/3), the air density of 1.225 kg/m³, and the water density of 1000 kg/m³.
O a. 70
O b. 40
O c. 50
Od. 80
O e. 60
According to the question the efficiency of the windmill is 60%.
What is efficiency?Efficiency is the ability to achieve maximum productivity with minimal effort, waste, or expense. It is the ratio of output to input in any system or process, and it is the measure of how well resources are used to achieve desired outcomes. Efficiency is important in all aspects of life, from businesses to households, as it helps to ensure resources are used wisely and without waste. Increased efficiency can lead to improved productivity, higher profits, and lower costs.
The actual efficiency of the windmill can be calculated using the following equation:
Efficiency = (Power Output / Power Input) x 100
Power Output = (Flow rate x Height x Density of Water) / Time
Power Input = (Area of the Windmill x Density of Air x U³) / 2
Therefore, the efficiency of the windmill is:
Efficiency = [(3.4 m³/s x 2.1 m x 1000 kg/m³) / 1] / [(π x 152 m² x 1.225 kg/m³ x 7 m/s³) / 2] x 100
Efficiency = 60%.
Therefore, the correct option is E
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An experiment is conducted in which a cart travels across a horizontal surface and collides with a wall. Data collected from the experiment are used to create the graph of the cart’s velocity as a function of time. All frictional forces are considered to be negligible. Which data from the graph should the student use to determine the direction of the net force exerted on the cart and the direction of the change in momentum of the cart from the time intervals of A to B ?
The detail that the student will use to determine the direction of the net force exerted on the cart is The velocity.
How to get the detail from the graphWhen examining the velocity-time graph of a cart during time intervals A to B, it is crucial that we analyze it to determine both the net force's direction and the change in momentum's direction.
By observing the slope of the velocity-time graph, one can conclude the acceleration of the vehicle; if positive, then the acceleration moves mostly in a forward direction, but when negative, the opposite holds true, indicating the acceleration's backward motion towards initial position.
To get the direction of the change in momentum, you would have to analyze the slope of the velocity vs. time graph at the various time intervals.
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Solve pls:
a) What are the maina dvantages of an epicyclicgearbox? b) Figure Q4 overleaf shows a diagram for an epicyclic gear train. Power is supplied to
wheel 3 and is delivered to a load attached to the epicyclic arm, body 2. Wheel 5 is fixed to the gear case, body 1.
i) Determine T4 if t3 = 30, t4 = 40 and t5 = 60.
a) The main advantages of an epicyclic gearbox are:
High gear ratios can be achieved in a small space, making it a compact design.It can provide a smooth and efficient transfer of power due to the multiple contact points between gears.It can be used for different applications, such as increasing torque or speed, reversing direction, and providing a neutral point.How to solve a gearbox?b) i) Using the formula for the gear ratio of an epicyclic gear train:
T4/T3 = (t2/t1) x (t5/t2) x (t4/t5)
T4/30 = (1/2) x (60/20) x (40/60)
T4 = 40 Nm
ii) From the law of gearing for an epicyclic gear train:
w21 = (t3/t2) x (t5/t4) x w31 - (t3/t2) x w2
Substituting the given values:
w21 = (30/20) x (60/40) x 200 - (30/20) x 100
w21 = 150 rad/s
iii) The fixing couple that must be applied to wheel 5 can be found from the power transmitted by the gear train:
P = w3 x T3 = w2 x T2 = w1 x T1
Substituting the given values:
9 kW = 200 rad/s x 30 Nm = w2 x T2 = w2 x 20 Nm
w2 = 450 rad/s
T2 = (9 kW) / (450 rad/s) = 20 Nm
The fixing couple that must be applied to wheel 5 is equal in magnitude and opposite in direction to T2, so it is -20 Nm.
iv) The tangential force at the pitch point between wheels 3 and 4 can be found from the formula:
Ft = (2 x Pd) / (m x z3)
where Pd is the diametral pitch, m is the module, and z3 is the number of teeth on wheel 3.
Substituting the given values:
Pd = 25.4 / 5 = 5.08 teeth/inch
z3 = t3 / m = 30 / 5 = 6 teeth
Ft = (2 x 5.08) / (5 x 6) = 0.846 N
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Help!! I need to get this done before 5pm!! I will give brainiest to whoever gets it done quickly!
Blood typing is a process that determines the type of antigens and antibodies present on the surface of red blood cells.
How to explain the bloodThere are four major blood types: A, B, AB, and O. Each blood type is characterized by the presence or absence of certain antigens and antibodies. For example, type A blood has A antigens on the surface of red blood cells and B antibodies in the plasma, while type B blood has B antigens on the surface of red blood cells and A antibodies in the plasma.
Blood transfusions are medical procedures in which blood or blood products are transferred from one person to another. Blood transfusions can be life-saving in situations where a person has lost a significant amount of blood due to injury or surgery, or has a medical condition that affects the production or function of their own blood cells. It is important to match the blood type of the donor and the recipient to prevent adverse reactions, such as hemolysis (breakdown of red blood cells) or an immune response.
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when a pure water is heated , it increases in volume
As water freezes below 4°C, the volume of the liquid shrinks and water molecule motion slows. When the temperature rises over 4 °C, the water molecules spread out and take up more space, increasing the volume.
How does cooling water from 4 C to 0 C affect its volume?Due to the peculiar property of water known as "Anomalous Expansion of Water," when 1 liter of water is cooled from 4°C to 0°C, the volume of the water starts to grow. Between 4°C and 0°C, water expands abnormally.
What happens to water's specific volume when it is heated from 0 degrees Celsius?Water volume reduces as it is heated from 0°C to 4°C because the water's density will be maximum
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Explain why sound wave travel faster in liquids than in gases
Answer:
Sound travels faster in liquids than in gases because molecules are more tightly packed. In fresh water, sound waves travel at 1,482 meters per second (about 3,315 mph). That's well over 4 times faster than in air!
Explanation:
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A block of wood weighing 15 N rest on top of the table. How mush force is exerted by the table on the blockso that it will remainat rest. Explain your answer
A wave has a frequency of 11 Hz and a wavelength of 9 meters
what is the speed of the wave
The speed of the wave is 99 meters per second.
What is Wave?
A wave is a disturbance or variation that travels through a medium or space, transferring energy without the net movement of matter. Waves can occur in various forms, such as mechanical waves that require a medium for propagation (e.g., sound waves, water waves) or electromagnetic waves that can travel through a vacuum (e.g., light waves, radio waves).
The speed of a wave can be calculated using the formula:
speed (v) = frequency (f) x wavelength (λ)
Given that the frequency (f) is 11 Hz and the wavelength (λ) is 9 meters, we can plug these values into the formula to calculate the speed of the wave:
v = 11 Hz x 9 meters
v = 99 meters per second
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similarities and differences between a lunar eclipse and the disappearing sun. HELP FAST
Answer: A solar eclipse results when the moon passes in between the earth and the sun hiding the sun fully or partly for some time. A lunar eclipse occurs when the earth passes in between the moon and the sun casting its shadow on the moon and thus hiding it fully or partly for some time.
Using the Left Hand Rule, if motion is toward you and the field is up, which way is the current?
A. Left
B. Right
C. Toward you
D. Away from you
what’s the answer for this
On Earth the gravitational field strength is 10N/kg. What would be the weight of someone on Earth who had a mass of 75kg?
750N/kg
750N
75N
7500kg
The weight of someone on Earth with a mass of 75kg is 750N.
option B.
What would be the weight of someone on Earth?The weight of someone on Earth is calculated using Newton's second law of motion.
W = mg
where;
m is the massg is acceleration due to gravityFor the mass is 75kg and the gravitational field strength on Earth is 10N/kg.
W = 75 kg x 10 N/kg
W = 750 N
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A weather balloon used by meteorologists is made of a flexible bag that allows the gas inside freely expand. If a weather balloon containing 25.0 m^3 of helium gas  is released from sea level, what is the volume of gas when the balloon reaches a height of 2100 m, where the pressure is 0.82×10^5 PA? Assume the temperature is unchanged. 
The volume of the helium gas in the weather balloon, when it reaches a height of 2100 m, is 30.85 m^3.
To solve this problem, we can use the ideal gas law, which states that the product of the pressure (P), volume (V), and temperature (T) of an ideal gas is proportional to the number of particles of the gas (n) and the universal gas constant (R). Since the temperature is assumed to be constant, we can write:
P1V1 = P2V2
where P1 and V1 are the initial pressure and volume of the helium gas, and P2 and V2 are the pressure and volume at a height of 2100 m.
At sea level, the pressure is approximately 1.01×10^5 Pa, so we have:
P1 = 1.01×10^5 Pa
V1 = 25.0 m^3
At a height of 2100 m, the pressure is 0.82×10^5 Pa, so we can solve for V2:
P1V1 = P2V2
V2 = (P1*V1)/P2
V2 = (1.01×10^5 Pa * 25.0 m^3) / (0.82×10^5 Pa)
V2 = 30.85 m^3
Therefore, the volume of the helium gas in the weather balloon when it reaches a height of 2100 m is approximately 30.85 m^3.
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why does pure water increase in volume when heated
Answer:
Pure water increases in volume when heated due to the phenomenon of thermal expansion. When water is heated, the kinetic energy of its molecules increases, causing them to move faster and spread out. This increased molecular movement leads to an increase in the average distance between water molecules, resulting in an expansion of the water.
Explanation:
.......
Increasing the number of loops in a solenoid will cause the strength of its magnetic field to
As long as the current is steady, a solenoid's magnetic field will become stronger as the number of loops increases.
Was the electromagnet made stronger or weaker by adding more loops?An electromagnet's strength can be increased by increasing the current or voltage as well as the quantity of wire loops encircling the iron core.
What number of loops will result in the strongest magnetic field?It follows that increasing the number of turns or the current flow (by raising the voltage because I = V/R) will produce a stronger magnetic field for a given coil length. Therefore, a coil with N of 1000 turns would generate a stronger magnetic field than one with N of only 100 turns.
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The 4.00 kg block is attached to a vertical rod by means of two strings. When the system rotates about the axis of the rod, the strings are extended as shown in the (Figure 1) and the tension in the upper string is 78.0 N. Find the number of revolutions per minute at which the lower cord just goes slack.?
The number of revolutions per minute at which the lower cord just goes slack is approximately 38.2 rpm.
To solve this problem, we need to use the concept of centripetal force and tension in the strings. When the lower string just goes slack, the tension in the string is zero and the centripetal force required for circular motion is provided only by the weight of the block.
We can start by calculating the weight of the block using the formula:
[tex]F_g[/tex] = m * g
Where [tex]F_g[/tex] is the weight, m is the mass, and g is the acceleration due to gravity. Substituting the given values, we get:
[tex]F_g[/tex] = (4.00 kg) * (9.81 m/s²) = 39.24 N
Since the tension in the upper string is 78.0 N, the net force acting on the block is:
[tex]F_{net}[/tex] = T - [tex]F_g[/tex] = 78.0 N - 39.24 N = 38.76 N
This net force provides the centripetal force required for circular motion, given by the formula:
[tex]F_{cp}[/tex] = m * v² / r
Where [tex]F_{cp}[/tex] is the centripetal force, m is the mass, v is the velocity, and r is the radius of rotation. Since we are asked to find the number of revolutions per minute, we can convert this to radians per second using the conversion factor:
1 revolution = 2π radians
1 minute = 60 seconds
Substituting the given values, we get:
Fcp = (4.00 kg) * v² / r
38.76 N = (4.00 kg) * v² / r
Solving for v²/r, we get:
v² / r = 38.76 N / 4.00 kg
v² / r = 9.69 m/s²
Next, we can use the fact that the length of the upper string is twice the length of the lower string to find the radius of rotation:
2l + l = 3l = r
Where l is the length of the lower string. Substituting the given values, we get:
3l = r = 0.600 m
Finally, we can substitute this value into the equation for v²/r to get:
v² / (0.600 m) = 9.69 m/s²
v² = 5.814 m²/s²
v = 2.41 m/s
To convert this to radians per second, we multiply by 2π and divide by the circumference of the circle:
ω = 2πv / (2πr) = v / r = 2.41 m/s / 0.600 m = 4.02 rad/s
To find the number of revolutions per minute, we can multiply this by the conversion factor:
1 radian per second = 60 / 2π revolutions per minute
N = (4.02 rad/s) * (60 / 2π) = 38.2 rpm
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