part a: N= 4.39 x 10^10 photons/second. part b: N = 8.25 photons/second are the answers.
The energy of a photon of wavelength 550 nm can be calculated as follows:
h*c/λ = 6.626 x 10^-34 x 3 x 10^8 / (550 x 10^-9)
wavelength = 3.6 x 10^-19 J.
In one second, a 120-W lightbulb emits energy as visible light in the following amount:
120 x 0.035 = 4.2 W.
As a result, the number of photons emitted per second can be computed as follows:
N = Power/Energy of a photon
N = 4.2/3.6 x 10^-19
N = 1.17 x 10^16 photons/second
PART A: The area of a 4.0-mm diameter pupil is given by the following formula:
A = πr^2
A = π(2.0 mm)^2
A= 3.14 x 4.0 x 10^-6 m^2
The distance from the bulb to the pupil is 2.2 m.
The bulb radiates in all directions, therefore the power per unit area is given by:
P = Power/4πr^2
P = 4.2/(4π(2.2)^2)
P = 0.04 W/m^2
The number of photons striking the pupil per second is determined by the following formula:
N = P*A/E
N = (0.04)(3.14 x 4.0 x 10^-6)/3.6 x 10^-19
N = 4.39 x 10^10 photons/second
PART B: The bulb is now 1.5 km distant from the pupil.
The power per unit area at this distance is:
P = Power/(4πr^2)
P = 4.2/(4π(1500)^2)
P = 2.98 x 10^-10 W/m^2.
Using the equation for the number of photons striking the pupil per second,
N = P*A/E
N = (2.98 x 10^-10)(3.14 x 4.0 x 10^-6)/3.6 x 10^-19
N = 8.25 photons/second
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how long would it take a leopard, running at an average speed of 20 m/s to travel 500 m?
Answer:
25 seconds
Explanation:
500/20
A block of ice at 0∘C is dropped from a height that causes it to completely melt upon impact. Assume that there is no air resistance and that all the energy goes into melting the ice.
What is the height necessary for this to occur. [ Hint: Equate the joules of gravitational potential energy to the product of the mass of ice and its heat of fusion (Lf = 335,000 J/kg).].
A height of about 34,184.49 meters is required for the block of ice to totally melt upon impact.
To determine the height necessary for the block of ice to completely melt upon impact, we need to equate the gravitational potential energy to the energy required to melt the ice.
The gravitational potential energy is given by the formula:
PE = m * g * h
Where PE is the gravitational potential energy, m is the mass of the ice, g is the acceleration due to gravity, and h is the height.
The energy required to melt the ice is the product of the mass of the ice and its heat of fusion (Lf = 335,000 J/kg). Therefore, the energy required to melt the ice is:
E = m * Lf
By equating the gravitational potential energy to the energy required to melt the ice, we have:
m * g * h = m * Lf
The mass of the ice cancels out on both sides, leaving:
g * h = Lf
Now we can solve for the height h:
[tex]\begin{equation}h = \frac{Lf}{g}[/tex]
Substituting the values:
Lf = 335,000 J/kg
g = 9.8 m/s²
h = 335,000 J/kg / 9.8 m/s²
Calculating the height h:
h ≈ 34,184.49 meters
Therefore, the height necessary for the block of ice to completely melt upon impact is approximately 34,184.49 meters.
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Question
If you are tracking the energy in a system, but the total energy seems to be going down, where has the energy gone ?
What is the relationship between large amplitude waves and small amplitude waves?
A Both waves have variable energy.
B Large amplitude waves have more energy than small amplitude waves.
C Large amplitude waves have less matter than small amplitude waves.
D Both waves have constant matter.
Answer:
c.large amplitude waves have less matter than small amplitude waves
15 points!
Five good work habits are:
be trustworthy
let others start
get by
be dependable
be efficient
slow and steady
punctuality
courtesy
Developing good work habits is essential for professional success and personal growth. Here are five commendable work habits:
1. Trustworthiness: Being trustworthy builds strong relationships with colleagues and clients. It involves fulfilling commitments, maintaining confidentiality, and being honest and transparent.
2. Initiating Collaboration: Allowing others to start and actively participating in collaborative efforts fosters teamwork and encourages diverse perspectives. It shows respect for colleagues' expertise and encourages collective problem-solving.
3. Dependability: Demonstrating reliability and consistency in meeting deadlines and delivering quality work builds trust and credibility. Being dependable ensures that others can rely on you and reduces stress within the workplace.
4. Efficiency: Striving for efficiency maximizes productivity and minimizes wasted time and resources. Effective time management, prioritization, and streamlining processes contribute to a productive work environment.
5. Punctuality and Courtesy: Being punctual demonstrates respect for others' time and sets a positive example. Additionally, practicing courtesy, such as using polite language, active listening, and showing appreciation, promotes a harmonious and respectful workplace.
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An oil drop whose mass is 2.9×10^−15 kg is held at rest between two large plates separated by 1.2 cm (Figure 1) when the potential difference between the plates is 360 V . Part A How many excess electrons does this drop have? Express your answer as an integer.
The oil drop has an excess of 4 electrons when the potential difference between the plates is 360 V.
The excess charge on the oil drop can be determined by equating the electric force on the oil drop with the gravitational force acting on it. The electric force is given by Coulomb's law:
Felectric = qE,
where Felectric is the electric force, q is the charge, and E is the electric field.
The gravitational force on the oil drop is given by:
Fgravity = mg,
where Fgravity is the gravitational force, m is the mass of the oil drop, and g is the acceleration due to gravity.
Since the oil drop is at rest, the net force on it is zero:
Felectric - Fgravity = 0.
Substituting the expressions for the electric and gravitational forces:
qE - mg = 0.
Solving for the charge q:
q = mg/E.
We are given that the mass of the oil drop is 2.9 × 10^(-15) kg, the potential difference between the plates is 360 V, and the electric field E is related to the potential difference by E = V/d, where d is the separation between the plates (1.2 cm = 0.012 m).
Substituting the values:
q = (2.9 × 10^(-15) kg × 9.8 m/s^2) / (360 V / 0.012 m).
q ≈ 6.4 × 10^(-19) C.
Since the charge of a single electron is approximately 1.6 × 10^(-19) C, we can calculate the number of excess electrons:
Number of excess electrons = q / (1.6 × 10^(-19) C).
Substituting the value of q:
Number of excess electrons ≈ (6.4 × 10^(-19) C) / (1.6 × 10^(-19) C).
Number of excess electrons ≈ 4.
The oil drop has an excess of 4 electrons.
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wave is modeled with the function y(x,t)=0.25\cos(0.30x-0.90t+\frac{\pi}{3})y(x,t)=0.25cos(0.30x−0.90t+3 π ) where all lengths are in meters and all times in seconds.
- Find the wavelength of the wave.
- Find the period of the wave
- Find the wave speed (a positive number).
-What is the instantaneous velocity of one of the particles that make up the medium at position 0 at time 22 sec? Make sure to use the correct sign (plus or minus) for your answer.
The wavelength of the wave is 20π/0.30 meters, which simplifies to approximately 209.44 meters.
The period of the wave is 2π/0.90 seconds, which simplifies to approximately 6.98 seconds.
The wave speed is given by the ratio of the wavelength to the period, which is approximately 29.97 meters per second.
To find the instantaneous velocity of a particle at position 0 at time 22 seconds, we differentiate the displacement equation with respect to time and evaluate it at the given time and position.
The derivative of y(x,t) with respect to t is 0.25(0.90)sin(0.30x - 0.90t + π/3). Plugging in x = 0 and t = 22, we find the instantaneous velocity to be approximately 0.177 m/s in the positive direction.
Determine how to find the wavelength of the wave?The given wave equation is y(x,t) = 0.25cos(0.30x - 0.90t + π/3), where x represents the position and t represents the time. The coefficient of x, 0.30, corresponds to the angular wave number (k) of the wave.
The coefficient of t, -0.90, corresponds to the angular frequency (ω) of the wave. By comparing the equation with the general form y(x,t) = Acos(kx - ωt + φ), we can identify the values for k and ω.
Determine how to find the period of the wave?The wavelength (λ) of a wave is given by λ = 2π/k. In this case, k = 0.30, so the wavelength is 2π/0.30, which simplifies to approximately 209.44 meters.
Determine how to find the wave speed?The period (T) of a wave is given by T = 2π/ω. In this case, ω = -0.90, so the period is 2π/(-0.90), which simplifies to approximately 6.98 seconds.
Determine find the instantaneous velocity particles?The wave speed (v) is the ratio of the wavelength to the period, v = λ/T. Substituting the values, we get v = (2π/0.30) / (2π/(-0.90)), which simplifies to approximately 29.97 meters per second.
To find the instantaneous velocity of a particle at position 0 at time 22 seconds, we differentiate the displacement equation with respect to time.
The derivative of cos(0.30x - 0.90t + π/3) with respect to t is -0.90sin(0.30x - 0.90t + π/3).
Plugging in x = 0 and t = 22, we find the instantaneous velocity to be approximately 0.177 m/s in the positive direction.
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What I ferromagnetism??
Answer:
Ferromagnetism is a kind of magnetism that is associated with iron, cobalt, nickel, and some alloys or compounds containing one or more of these elements. It also occurs in gadolinium and a few other rare-earth elements.
Explanation:
Hope this helped Mark BRAINLIEST!!!
How many people does it take to change a lightbulb?
Which is a property of every
mixture?
An object is placed +15 cm in front of a concave mirror. A real image is formed +30 cm in front of the mirror. What is the focal distance, f, for this mirror? O +10 cm O +30 cm 0 -30 cm O -10 cm
From the given data the focal distance for the given concave mirror is +30 cm.
The focal distance, f, for a concave mirror, can be determined using the mirror formula:
1/f = 1/v - 1/u
Where:
f is the focal distance,
v is the image distance,
u is the object distance.
In this case, the object distance, u, is given as +15 cm (in front of the mirror), and the image distance, v, is given as +30 cm (in front of the mirror). We can substitute these values into the mirror formula:
1/f = 1/30 - 1/15
Simplifying the equation, we get:
1/f = (2 - 1) / 30
1/f = 1/30
f = 30 cm
Therefore, the focal distance, f, for this concave mirror is O +30 cm.
In conclusion, the focal distance for the given concave mirror is +30 cm.
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On the dot below, which represents the book, draw and label the forces (not components) that act on the book at the lowest point of its circular path. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot. ii. At the lowest point of the circular path, the book is moving only in the horizontal direction. In what direction, if any, is the net vertical force on the book? Up _No Down direction, since the net vertical force is equal to zero Without deriving any equations, briefly explain your reasoning in terms of the book's motion.
At the lowest point of its circular path, the book experiences three distinct forces: the gravitational force (downward), the normal force (upward), and the centripetal force (toward the center of the circular path).
At the lowest point of the circular path, the book is moving only in the horizontal direction. This means that the book has no vertical acceleration since its velocity in the vertical direction remains constant. According to Newton's second law, when there is no vertical acceleration, the net vertical force must be zero.
The gravitational force acts downward, pulling the book toward the center of the Earth. The normal force, exerted by the surface supporting the book, acts upward and balances the gravitational force. These two forces have equal magnitudes and opposite directions, resulting in a net vertical force of zero.
Therefore, at the lowest point of its circular path, the book experiences no net vertical force. The cancellation of the gravitational force and the normal force allows the book to move in a centripetal force circular path with only horizontal motion.
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2) What are the two main types of waves?
Answer:
The two types of waves are longitudinal and transverse
Explanation:
hope it helped! and i hope its correct
6. Let u(x, y) = xy. (a) Show that u is harmonic. (b) Find a harmonic conjugate of u.
a. u is harmonic for let u(x, y) = xy because ∂²u/∂x² + ∂²u/∂y² = 0.
b. A harmonic conjugate of u is v = (x² - y²)/2.
To determine that u is harmonic, we need to verify whether u satisfies Laplace's equation or not. Laplace's equation is given by:
∂²u/∂x² + ∂²u/∂y² = 0
On differentiating u(x, y) = xy partially, we get,
∂u/∂x = y∂u/∂y = x
On differentiating partially again, we get,
∂²u/∂x² = 0∂²u/∂y² = 0
Therefore, ∂²u/∂x² + ∂²u/∂y² = 0
Thus, u is harmonic.
To find a harmonic conjugate of u, we need to find v(x, y) such that u + iv is analytic. In other words, v must satisfy the Cauchy-Riemann equations. The Cauchy-Riemann equations are given by:
∂u/∂x = ∂v/∂y∂u/∂y = -∂v/∂x
On substituting u = xy in the above equations, we get,
∂v/∂y = x∂v/∂x = -y
On integrating the above equations, we get
v = (x² - y²)/2 + C
Thus, a harmonic conjugate of u is v = (x² - y²)/2.
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for another identical object initially at rest no frictional force is exerted during segment 2
The relative speeds of the two objects at the bottom of the incline will be zero.
In segment 1, the object is released from rest at the top of an inclined plane. As it rolls down the incline, the potential energy is converted into both kinetic energy and rotational energy.
Assuming no slipping occurs, the object's linear velocity increases while its angular velocity remains constant.
During segment 2, where no frictional force is exerted, there are no external forces acting on the object. As a result, there is no change in the object's kinetic energy or angular velocity. Therefore, the object's linear velocity remains constant.
At the bottom of the incline, both objects will have the same linear velocity. Since they are identical objects, the relative speed between them will be zero.
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--The complete Question is, For another identical object initially at rest, no frictional force is exerted during segment 2. If the first object is released from rest at the top of an inclined plane in segment 1, and both objects roll down the incline, what will be the relative speeds of the two objects at the bottom of the incline?--
In which of these can convection not occur
A. Gases
B.solids
C.liquids
D.fluids
I believe the answer would be B.solids.
Explanation:
Hope this helped!
In solids these can convection not occur and convection can occur in both liquids and gases, as they are both considered fluids. Thus, option B is correct.
However, convection cannot occur in solids, as the particles in solids are generally fixed in position and do not have the freedom to move and transfer heat through bulk motion as in convection.
A solid is one of the three fundamental states of matter, along with liquids and gases. It is characterized by having a definite shape and volume. Solids are composed of tightly packed particles, such as atoms, molecules, or ions, that are held together by strong intermolecular forces.
These particles vibrate in fixed positions but do not have the ability to move freely throughout the substance. As a result, solids maintain their shape and volume under normal conditions and exhibit relatively low compressibility compared to gases and liquids. Examples of solids include metals, rocks, wood, plastics, and ice.
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What makes water a natural resource?
A.
It is a natural material that can be easily replaced.
B.
It is a material found in nature that people can use.
C.
It is a material that people can easily make and use.
D.
It is a energy source that cannot be replaced.
Answer:
I belive is the answer is B. It is a material found in nature that people can use. Because water is technically a natural resource.
What is one way that testing helpes scientists determing which solution for a problem works best?
How fast would a 20 kg dog have to run, to reach a momentum of 120 kg x m/s? Include unit in your answer. Worth 40 points!
Answer:
6 m/s
Explanation:
momentum= mass times velocity
120 = 20v
v = 120/20
Answer:
7
Explanation:
7
a radio-controlled model airplane has a momentum given by [(−0.75kg⋅m/s3)t2 (3.0kg⋅m/s)]i^ (0.25kg⋅m/s2)tj^, where t is in seconds. What is the x -component of the net force on the airplane?
Express your answer in terms of the given quantities.
The x-component of the net-force acting on the radio-controlled model airplane is given by the expression -1.5 kg·m/s^2 * t.
The x-component of the net force (F_x) on an object can be calculated using Newton's second law, which states that the net force is equal to the rate of change of momentum:
F_x = dp_x/dt
Given that the momentum of the airplane is [(−0.75kg·m/s^3)t^2 (3.0kg·m/s)]i^, we can find the x-component of the momentum (p_x) as the coefficient of the i^ unit vector. Taking the derivative of p_x with respect to time (t), we obtain the x-component of the net force:
F_x = d/dt [(−0.75kg·m/s^3)t^2] = (-0.75kg·m/s^3) * 2t
Simplifying the expression, we find:
F_x = -1.5 kg·m/s^2 * t
Therefore, the x-component of the net force on the airplane is -1.5 kg·m/s^2 * t.
The x-component of the net force acting on the radio-controlled model airplane is given by the expression -1.5 kg·m/s^2 * t, where t is the time in seconds. This equation represents the rate of change of the x-component of the momentum with respect to time.
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Two long parallel wires are separated by 9.03 cm and carry currents 1.71 A and 4.67 A. Find the magnitude of the magnetic force that acts on a 3.17-m length of either wire.
The magnitude οf the magnetic fοrce that acts οn a 3.17 m length οf either wire is 0.0405 Newtοns.
What is magnetic fοrce?Magnetic fοrce refers tο the fοrce exerted οn a charged particle οr a current-carrying cοnductοr due tο the presence οf a magnetic field.
F = (μ₀ * I₁ * I₂ * L) / (2π * d)
Given:
I₁ = 1.71 A
I₂ = 4.67 A
L = 3.17 m
d = 9.03 cm = 0.0903 m
After applying the formula to the supplied values, we get:
F = (4π × 10⁻⁷ T·m/A) * (1.71 A) * (4.67 A) * (3.17 m) / (2π * 0.0903 m)
Simplifying the expressiοn:
F ≈ 4π × 10⁻⁷ * 1.71 * 4.67 * 3.17 / 2 * 0.0903
F ≈ 0.0405 N
Therefοre, the magnitude οf the magnetic fοrce that acts οn a 3.17 m length οf either wire is 0.0405 Newtοns.
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are positively charged ions form when atoms lose electrons
Answer: Atoms that lose electrons acquire a positive charge as a result because they are left with fewer negatively charged electrons to balance the positive charges of the protons in the nucleus. Positively charged ions are called cations. Most metals become cations when they make ionic compounds.
Explanation: hope this helps
Here is a list of ways of producing electricity
A ) Wind farms
B ) Coal fired power stations
C ) hydroelectric power stations
D ) nuclear power stations
E ) gas fired power stations
F ) tidal power stations
G) using geothermal energy
(1) State which of these produces clean electricity
(2) State which of these uses fossils fuel
(3) state which of these uses a renewable energy resources
Please help me
Please do not answer from links
Urgent !!!!!!
Will give the brainliest!
* please answer correctly
Answer:
1) A,B,F,G
2)B,D,E
3) A,C,F,G
Explanation:
in a pig-calling contest, a caller produces a sound with an intensity level of 87 db.
In a pig-calling contest, the caller produces a sound with an intensity level of 87 dB. Understanding the intensity level of the sound helps evaluate its potential impact on the surrounding environment and human hearing.
The intensity level of a sound is a measure of its loudness and is expressed in decibels (dB). The decibel scale is logarithmic, meaning that each increase of 10 dB represents a tenfold increase in sound intensity.
To calculate the intensity level in decibels, we use the formula:
dB = 10 * log10(I/I₀)
where I is the sound intensity and I₀ is the reference intensity (typically the threshold of hearing, which is approximately 10^(-12) W/m^2).
Given that the intensity level is 87 dB, we can rearrange the formula to solve for I:
87 = 10 * log10(I/I₀)
8.7 = log10(I/I₀)
I/I₀ = 10^(8.7)
I = (10^(8.7)) * I₀
In the pig-calling contest, the caller produces a sound with an intensity level of 87 dB. The calculation of the sound intensity requires knowledge of the reference intensity and the logarithmic relationship of the decibel scale.
By using the formula and considering the logarithmic nature of the scale, we can determine the actual sound intensity. Remember that the decibel scale is logarithmic, so even a small increase in decibels represents a significant increase in sound intensity. Understanding the intensity level of the sound helps evaluate its potential impact on the surrounding environment and human hearing.
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The purple arrow on the graph shows speed and direction.
What is this arrow called?
A. an acceleration arrow
B. a vector
C. a force arrow
D. a speed arrow
The purple arrow on the graph that shows speed and direction, this arrow called B. a vector.
Vectors have magnitude (size) and direction. They are represented graphically as arrows, in which the direction of the arrow denotes the direction of the vector, while the length of the arrow represents the vector's magnitude (size). Furthermore, the term "vector" is used in physics to refer to a quantity that has both magnitude and direction, such as velocity, force, and acceleration.
They are essential in physics since they enable scientists to precisely describe how objects move and interact. Moreover, vector quantities, such as velocity and force, are often depicted with arrows in physics diagrams, with the direction of the arrow representing the vector's direction and the arrow's length representing its magnitude. The use of vectors in physics and other fields has made a significant contribution to the advancement of science and technology. So therefore the correct answer is B. a vector, the purple arrow on the graph that shows speed and direction.
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What must the charge (sign and magnitude) of a particle of mass 1.46 g be for it to remain stationary when placed in a downward-directed electric field of magnitude 630 N/C?
Use 9.81 m/s^2 for the magnitude of the acceleration due to gravity.
The particle remains stationary in a downward-directed electric field, the charge must be negative to balance the gravitational force.
The two forces acting on the particle are the gravitational force and the electric force. The gravitational force is given by F_gravity = m * g, where m is the mass of the particle and g is the acceleration due to gravity (9.81 m/s^2). The electric force is given by F_electric = q * E, where q is the charge of the particle and E is the magnitude of the electric field.
For the particle to remain stationary, the electric force must exactly balance the gravitational force, i.e., F_electric = F_gravity. Therefore, we can equate the two expressions and solve for the charge.m * g = q * E
Given that the mass of the particle is 1.46 g (0.00146 kg) and the magnitude of the electric field is 630 N/C, we can substitute these values into the equation:0.00146 kg * 9.81 m/s^2 = q * 630 N/C
Solving for q, we have:
q = (0.00146 kg * 9.81 m/s^2) / 630 N/C ≈ 2.262 x 10^-5 C.
Therefore, the charge of the particle must be approximately 2.262 x 10^-5 C. Since the particle remains stationary in a downward-directed electric field, the charge must be negative to balance the gravitational force.
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What is the relationship between the kinetic energy of molecules in an object and the object's temperature? Responses As the kinetic energy of the molecules decreases, the temperature increases. The total kinetic energy of the molecules is not affected by a change in temperature. As the temperature increases, the kinetic energy of the molecules increases. The kinetic energy always increases whether the temperature increases or decreases.
As the temperature increases, the kinetic energy of the molecules also increases.
The relationship between the kinetic energy of molecules in an object and the object's temperature is that as the temperature increases, the kinetic energy of the molecules also increases.
This relationship is governed by the kinetic theory of gases, which states that the temperature of a gas is directly proportional to the average kinetic energy of its molecules.
Temperature is a measure of the average kinetic energy of the particles in a substance. When the temperature of an object increases, the molecules within it gain more energy and move faster.
As a result, their kinetic energy increases. Conversely, when the temperature decreases, the molecules lose energy and their kinetic energy decreases.
The kinetic energy of a molecule is directly related to its mass and velocity. As the temperature increases, the molecules move with greater speed and collide more frequently with each other and the walls of their container.
These collisions transfer energy, leading to an overall increase in kinetic energy.
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When the image of an object is seen in a plane mirror, the distance from the mirror to the image depends on?
A. The Wavelength of light used for viewing
B. The distance from the object to the mirror
C. The distance of both the observer and the object to the mirror.
When the image of an object is seen in a plane mirror, the distance from the mirror to the image depends on the distance from the object to the mirror.
It is because a Plane Mirror is the incident ray, the reflected ray, and the normal to the surface all lie in the same plane, and the angle of incidence is equal to the angle of reflection. So, the distance from the object to the mirror equals the distance from the image to the mirror. Thus whoever the person viewing this image must sight at this image location.
there is a 3-way switch in the bedroom hallway leading into the living room. show your calculation of how to determine the box size for this switch. the box contains cable clamps.
Therefore, the minimum box size for this switch will be 2.5 cubic inches, while the recommended size is at least 18 cubic inches for a 14-gauge wire. However, the calculated box volume is 13 cubic inches. Hence, the box size required will be greater than 13 cubic inches.
In order to determine the box size for a 3-way switch in the bedroom hallway leading into the living room, the calculation needs to be done. Also, the box contains cable clamps.
So, here is how to determine the box size for this switch:
Calculation:
Firstly, we need to calculate the box volume. For that, we need the formula:
Box volume = Number of wires x 2 + Box fill volume Box fill volume is 2.0 cubic inches for each 14-gauge wire or 2.25 cubic inches for each 12-gauge wire or larger plus clamps for each box.
The box size of switch can be found out as follows:
The minimum box size for a three-way switch is 2.5 cubic inches.
However, the National Electrical Code recommends at least 18 cubic inches for a 14-gauge wire or 20 cubic inches for a 12-gauge wire.
The volume of the cable clamps is also to be included in the calculation. So, let's say the wire gauge used is 14-gauge.
Then,
the box volume = (2 x number of wires) + (2.0 cubic inches for each 14-gauge wire x number of wires) + volume of cable clamps.
Let the number of wires be 3 and the volume of each cable clamp be 0.5 cubic inches.
The calculation will be as follows:
Box volume = (2 x 3) + (2.0 x 3) + (0.5 x 2) = 6 + 6 + 1 = 13 cubic inches.
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Following data are given for a direct shear test conducted on dry sand:
• Specimen dimensions: 63 mm X 63 mm X 25 mm (height)
• Normal stress: 105 kN/m2
• Shear force at failure: 300 N
a. Determine the angle of friction, Ø’
b. For a normal stress of 180 kN/m2 , what shear force is required to cause failure?
c. What are the principal stresses at failure for the condition given in (b)?
d. Locate the pole on this Mohr’s circle and find the angle of inclination of the major and minor principal plane with the horizontal?
The correct answer is Option a. The angle of friction, Ø', can be determined using the formula:
Ø' = tan^(-1)(τ'/σn)
where Ø' is the angle of friction, τ' is the shear stress, and σn is the normal stress. In this case, the shear stress can be calculated by dividing the shear force at failure (300 N) by the area of the specimen. The area is the product of the specimen's height (25 mm) and its average perimeter [(2 * 63 mm) + (2 * 25 mm)].
Ø' = tan^(-1)(300 N / (25 mm * [(2 * 63 mm) + (2 * 25 mm)]) / 105 kN/m^2
Simplifying the equation:
Ø' = tan^(-1)(0.0256) ≈ 1.47°
b. To determine the shear force required to cause failure for a normal stress of 180 kN/m^2, we can use the formula:
τ' = Ø' * σn
Given Ø' = 1.47° and σn = 180 kN/m^2:
τ' = 1.47° * 180 kN/m^2 = 2.646 kN/m^2
c. The principal stresses at failure can be calculated using the equation:
σ1 = σn + τ'
σ3 = σn - τ'
Given σn = 180 kN/m^2 and τ' = 2.646 kN/m^2:
σ1 = 180 kN/m^2 + 2.646 kN/m^2 = 182.646 kN/m^2
σ3 = 180 kN/m^2 - 2.646 kN/m^2 = 177.354 kN/m^2
d. The pole on the Mohr's circle represents the average normal stress (σn) and is located at (σn, 0). Since σn = 180 kN/m^2, the pole would be located at (180 kN/m^2, 0).
To find the angle of inclination of the major and minor principal plane with the horizontal, we can use the equation:
θ = (1/2) * tan^(-1)((2 * τ') / (σ1 - σ3))
Given τ' = 2.646 kN/m^2, σ1 = 182.646 kN/m^2, and σ3 = 177.354 kN/m^2:
θ = (1/2) * tan^(-1)((2 * 2.646 kN/m^2) / (182.646 kN/m^2 - 177.354 kN/m^2))
Simplifying the equation:
θ ≈ 1.05°
Therefore, the angle of inclination of the major and minor principal plane with the horizontal is approximately 1.05°.
In conclusion, the angle of friction (Ø') is approximately 1.47°. For a normal stress of 180 kN/m^2, the shear force required to cause failure is approximately 2.646 kN/m^2. The principal stresses at failure for this condition are σ1 = 182.646 kN/m^2 and σ3 = 177.354 kN/m^2. The pole on the Mohr's circle is located at (180 kN/m^2, 0), and the angle of inclination of the major and minor principal plane with the horizontal is approximately 1.05°.
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