The magnitude of normal force that floor exerts on chair is 120.24 N. the magnitude of the frictional force acting on the chair is 123.24 N
Simply put, "distance or quantity" is the definition of size. In terms of motion, it shows the absolute or relative size, direction, or movement of an object. It is used to describe something's size or scope. Magnitude in physics typically refers to a size or quantity.
Vertical component of the force = F sinθ
= 38X sin42°
= 38 X 0.669
= 25.42N
Total normal force acting on the chair = weight +Vertical component force
Total normal force acting on the chair = 95N + 25.42N
= 120.24 N
horizontal component of force= f cosθ.
= 38X cos 42°
= 38 X 0.743
= 28.24 N
Total normal frictional force on the chair
frictional force = weight+ horizontal component of force
frictional force= 95N + 28.24 N
frictional force= 123.24 N
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in short-track speed skating, the track has straight sections and semicircles 16 min diameter. assume that a 68 kgskater goes around the turn at a constant 12 m/s.(figure 1)
The track in relatively brief speed skating has straight stretches and semicircles with a 16-minute diameter. This 68 kg skate makes the maneuver at a steady 12 m/s while exerting 1224 N.
What is speed and what is its unit in physics?The pace at which a distance changes over time is referred to as speed. It has a dimension of time-distance. As a consequence, the fundamental unit of time as well as the basic measure of distances are combined to form the Special name of speed. Thus, the meter per second (m/s) is the Unit of measure of speed.
r = 8 m.
tangential velocity = 12 m/s.
angular speed is,
ω = v/r
= (12 m/s)/(8 m)
= 1.5 rad/s
centripetal acceleration,
a = r*ω²
= (8 m)*(1.5 rad/s)²
= 18 m/s².
force,
F = m*a
= (68 kg)*(18 m/s²)
= 1224 N
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Which is an example of kinetic energy being used?(1 point)
Answer:
A person walking, a soaring baseball, a crumb falling from a table and a charged particle in an electric field are all examples of kinetic energy at work.
Explanation:
Walking and running. Cycling. In a windmill, when the moving air hits the blades, it causes rotation which ultimately leads to the generation of electricity. In a hydropower plant, when the kinetic energy of the moving water hits the turbine the kinetic energy of the water gets converted to mechanical energy.
Adam lifts a book from the floor, carries it across the room, and places it on a high shelf. When is Adam doing work on the book?
W= FD
Work = Force x Distance
A. He does work from the moment he touches the book until he lets it go.
B. He does work only while he carries the book across the room.
C. He does work on the book at no time during this process.
D. He does work as he picks up the book and as he lifts it to the shelf.
Adam lifts a book from the floor, carries it across the room, and places it on a high shelf, he does work only while he carries the book across the room, which is in option B, as work = force x distance.
What is the work done by a person?According to the equation for work (W = FD), work is only done when a force is applied to an object and it moves some distance as a result of that force, when Adam lifts the book from the floor, he applies force to it, but the book does not move any distance, so no work is done when the object doesn't move, but when he moved the book, work was done.
Hence, he does work only while he carries the book across the room, which is in option B, as he lifts a book from the floor and carries it across the room.
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If a machine does work faster, what does that mean about the power rating?
If a machine does some kind of work faster, then this mean that the power rating of the machine is higher. Power rating is directly proportional to the work done.
What is Power rating?The power rating of an equipment is the highest power input which is allowed to flow through a particular equipment. According to the particular discipline, the power may be referred to as the electrical or mechanical power.
Work is proportional to the energy, and higher power means higher energy present in an object. If a machine does work at a faster rate, this means that it has high power rating.
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Learning Goal: To understand the meaning of the variables that appear in the equations for one-dimensional kinematics with constant acceleration.Motion with a constant, nonzero acceleration is not uncommon in the world around us. Falling (or threwn) objects and cars starting and stopping approximate this type of motion It is also the type of motion most frequently involved in introductory kinematics problems The kinematic equations for such motion can be written as z(t) = x + vt + at^2 v(t) = 1+ at where the symbols are defined as follows .A. X(t) is the position of the particle is the initial position of the particle . B. x(i) is the velocity of the particleC. v(t) is the initial velocity of the particle D. Vi is the acceleration of the particle
The variables in the kinematic equations for motion in one dimension with constant acceleration are time, initial position, initial velocity, initial acceleration, velocity, and position.
The following are accurate definitions for the variables in the kinematic equations for motion in one dimension with constant acceleration:
A. x(t) represents the particle's position at time t as determined by its initial position x. (0).
B. v(t) represents the particle's speed at time t as determined by its initial speed v. (0).
C. The particle's starting position, or the position at time t=0, is represented by the number x(0).
D. The initial velocity of the particle, or its speed at time t=0, is known as v(0).
E. an is the particle's constant acceleration.
It's vital to remember that the x and v subscripts denote the time at which the location or velocity is being measured. As an illustration, x(0) is the particle's position at time.
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a - ladder leans against the side of a house. the bottom of the ladder is from the side of the house. how high is the top of the ladder from the ground? if necessary, round your answer to the nearest tenth.
The top of the 12 ft ladder that leans against the house, and the bottom of the ladder is 9 ft from the side of the house = 7.9 ft
The triangle formed by the ladder must be a right angle with a hypotenuse (Side opposite to right angle) of 12 feet since the side of the house must be vertical to the ground to form a right angle.
Let x represent how high the ladder's top is above the ground.
We have the following using the Pythagoras theorem of right triangles:
c² = a² + b²
12² = 9² + x²
x² = 144 - 81
= 63
Hence,
x = √63
= 7.9 ft
The question is incomplete, it should be:
A 12 - ft ladder leans against the side of a house. the bottom of the ladder is 9 ft from the side of the house. how high is the top of the ladder from the ground? if necessary, round your answer to the nearest tenth.
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Calculate the height to which water at 4°C will rise in a capillary tube of 0.5 × 10^-3 m diameter. (Given g=9.8 m/s^2; angle of contact 0°; surface tension= 0.072 N/m.
Step-by-step Explanation
Given: Diameter (D) of the capillary tube = m
Surface Tension (T) =
The angle of contact () =
Acceleration due to gravity (g) =
The temperature of water =
To Find: The height (h) of rising of water in the capillary tube
Solution:
Formula to find the height of the rise
The following expression is used to find the height (h) of rising of water in the capillary tube;
Where, is the density of water, and is the radius of the capillary tube.
Calculating the height of the rise in the capillary tube
Since the diameter of the capillary tube is , the radius of the tube will be;
And, at , the density of water is
Substituting all the required values in the above formula, we get;
Hence, the water will rise in a capillary tube to a height of
According to Newton's third law, what is the reaction force in the previous question to the normal force of the person on the floor? A. The gravitational pull of the person on Earth B. The gravitational pull of the box on Earth C. The force of the box on the person D. The normal force of the floor on the box E. The weight of the box F. The weight of the person G. The normal force of the box on the floor H. The force of the person on the box I. The normal force of the floor on the person
Answer:
A
Explanation:
If 2 J of wok is done in raising a 180 g red delicious apple to bring it to your mouth to take a bite of the apple, how far is it lifted?
Answer:1.13m
Explanation:
Potential Energy= mgh
m= mass(kg)
g= gravitational field strength(9.81m/s^2)
h= height
PE=mgh
PE= 2J
therefore: 2J= 0.180kg x 9.81 m/s^2 x h
h= 2/(1.764)
h= 1.13m
George is pushing his 20kg son, Jake, on the swing at the park. He pulls his son up to a height of 2m, pauses, and lets him go. What is Jake's kinetic energy before his dad lets go of the swing? What is Jake's kinetic energy at the lowest point? What is Jake's velocity at the top of the lowest point?
Jake's velocity at the lowest point of the swing is 7 m/s.
What is kinetic and potential energy?Potential energy is the energy stored in any object or system due to the position or arrangement of its parts. It is, however, unaffected by factors outside of the object or system, such as air or height. Kinetic energy, on the other hand, is the energy of moving particles in an object or system.
To solve this problem, we need to use the conservation of energy principle, which states that the total energy in a system remains constant.
At the highest point, Jake has potential energy (due to his position above the ground), but no kinetic energy (since he is not moving).
As he swings down, his potential energy is converted to kinetic energy, and at the lowest point of the swing, he has the maximum kinetic energy and minimum potential energy. Then, as he swings back up, the process is reversed.
To calculate Jake's potential energy at the top of the swing, we use the formula:
PE = mgh
where m is the mass of Jake, g is the acceleration due to gravity (9.8 m/s²), and h is the height above the ground. Substituting in the values, we get:
PE = (20 kg)(9.8 m/s²)(2 m) = 392 J
This is Jake's potential energy at the top of the swing. At this point, he has no kinetic energy.
At the lowest point of the swing, Jake has converted all of his potential energy into kinetic energy. We can use the conservation of energy principle to find his kinetic energy at this point:
KE = PE
where KE is kinetic energy and PE is potential energy. Substituting in the values, we get:
KE = 392 J
This is Jake's kinetic energy at the lowest point of the swing.
To find Jake's velocity at the lowest point, we can use the formula for kinetic energy:
KE = (1/2)mv²
where v is velocity. Rearranging the formula to solve for v, we get:
v = √((2KE)/m)
Substituting in the values, we get:
v = √((2(392 J))/(20 kg)) = 7.0 m/s
This is Jake's velocity at the lowest point of the swing.
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When a weak stimulus is applied in rapid succession, it will often reach threshold due to
spatial summation.
excitatory summation.
voltage potential.
inhibitory summation.
temporal summation.
When a weak stimulus is applied in rapid succession, it will often reach threshold due to Temporal summation.
What is a stimulus ?Stimulus is the effect that cause a change in the external or internal environment which brings physiological responses.
Here,
The weak stimulus applied is converted into a large signal so that it could reach the threshold.
Temporal summation is the phenomenon that could convert a series of weak stimuli at a specific frequency into a large signal and to achieve action potential.
Hence,
When a weak stimulus is applied in rapid succession, it will often reach threshold due to Temporal summation.
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How can a spacecraft reduce its speed during a trip to Mars?
Answer options with 4 options
A.
eliminate all forces acting on the spacecraft
B.
fire rockets to increase force acting in the direction of travel
C.
fire reverse rockets to increase force acting opposite of the direction of travel
D.
open a parachute to increase force acting opposite to the direction of motion of the spacecraft
A spacecraft can reduce its speed during a trip to Mars by firing reverse rockets to increase force acting opposite of the direction of travel.
option C.
How can a spacecraft reduce its speed?To reduce the speed of a spacecraft during a trip to Mars, the most common method is to fire reverse rockets, also known as braking thrusters.
These thrusters are used to generate a force that acts in the opposite direction of the spacecraft's motion, slowing it down. This is called a deceleration burn, and it is an important step in the process of entering into orbit around a planet or landing on its surface.
The reverse thrust slows down the spacecraft, reducing its speed and allowing it to be captured by the planet's gravity. This is crucial for the success of the mission and ensures that the spacecraft can be safely guided into orbit or landed on the surface of the planet.
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57. You are driving through town at 12 m/s when suddenly a ball rolls out in front of you. You apply the brakes and begin decelerating at 3.5 m/s2. (a) how far do you travel before stopping? (b) When you have traveled only half the distance in part a, is your speed 6.0 m/s, greater than 6 m/s or less than 6 m/s? Support your answer with calculations.
a) Travelled 20.57 meters before coming to a stop.
b) When travelled half the distance, the speed is 2.6 m/s, which is less than 6 m/s.
What does speed mean?Speed is a measure of how fast an object is moving, without regard to the direction of its motion. It is defined as the distance travelled by an object per unit of time. Speed is a scalar quantity, which means it only has magnitude (i.e., numerical value) and not direction. The standard unit of speed is meters per second (m/s) in the International System of Units (SI), although other units such as kilometres per hour (km/h) and miles per hour (mph) are also commonly used. It can be calculated as the ratio of distance traveled by an object to the time taken to cover that distance.
For example, if a car travels 100 meters in 10 seconds, its speed can be calculated as 100 meters divided by 10 seconds, which equals 10 meters per second. Speed can also be constant, if the object is moving at a constant rate without changing its speed, or variable, if the object's speed changes over time.
(a) To determine how far travelled before stopping, use the kinematic equation:
[tex]v^2 = u^2 + 2as[/tex]
where,
v = final velocity (0 m/s, since you come to a stop)
u = initial velocity (12 m/s)
a = acceleration (-3.5 m/s^2, since decelerating)
s = distance traveled (unknown)
Rearranging the equation to solve for s, we get:
[tex]s = (v^2 - u^2) / (2a)[/tex]
Substituting in the known values, we get:
[tex]s = (0 - (12 m/s)^2) / (2*(-3.5 m/s^2)) = 20.57 m[/tex]
Therefore, travelled 20.57 meters before coming to a stop.
(b) To determine speed, travelled half the distance, use the kinematic equation:
[tex]s = ut + 0.5at^2[/tex]
where,
t = time elapsed (unknown, but use the fact that have travelled half the distance)
s = distance traveled (half of 20.57 m = 10.285 m)
u = initial velocity (12 m/s)
a = acceleration (-3.5 m/s^2)
Substituting the known values,
[tex]10.285 m = (12 m/s)t + 0.5(-3.5 m/s^2)t^2[/tex]
Simplifying and rearranging, a quadratic equation is obtained:
[tex]1.75t^2 - 12t + 10.285 = 0[/tex]
Solving t using the quadratic formula,
t = 2.52 s (rounded to two decimal places)
Now, to find your speed at this point, we can use the kinematic equation:
[tex]v = u + at[/tex]
Substituting the known values,
[tex]v = 12 m/s + (-3.5 m/s^2)(2.52 s) = 2.6 m/s[/tex]
Therefore, when travelled half the distance, the speed is 2.6 m/s, which is less than 6 m/s.
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Find the marginal cost functions for each of the following average cost functions. a) AC= 1.5Q + 4 + 460 b) AC = 160 +5+3Q+2Q²
(a) The marginal cost function is MC = 3Q + 464.
(b) The marginal cost function is MC = 160 + 5 + 6Q + 6Q^2.
What is the marginal cost function?The marginal cost (MC) of a product is the change in the total cost of producing an additional unit of the product.
(a) AC = 1.5Q + 4 + 460
The total cost (TC) of producing Q units is given by:
TC = AC * Q
TC = (1.5Q + 4 + 460) * Q
TC = 1.5Q^2 + 4Q + 460Q
TC = 1.5Q^2 + 464Q + 460
The marginal cost function is given by the derivative of the total cost function with respect to Q:
MC = dTC/dQ
MC = d/dQ (1.5Q^2 + 464Q + 460)
MC = 3Q + 464
b) AC = 160 + 5 + 3Q + 2Q²
The total cost of producing Q units is given by:
TC = AC * Q
TC = (160 + 5 + 3Q + 2Q^2) * Q
TC = 160Q + 5Q + 3Q^2 + 2Q^3
The marginal cost function is given by the derivative of the total cost function with respect to Q:
MC = dTC/dQ
MC = d/dQ (160Q + 5Q + 3Q^2 + 2Q^3)
MC = 160 + 5 + 6Q + 6Q^2.
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Which of the following actions will increase the electric field strength at the position of the dot? - Make the rod longer without changing the charge. - Make the rod shorter without changing the charge. - Make the rod fatter without changing the charge.- Make the rod narrower without changing the charge. - Remove charge from the rod
E: "Removing charge from the rod" will increase the electric field strength at the position of the dot.
The electric field strength at the position of the dot depends on the charge and the distance from the charge. Therefore, any change that affects the charge or the distance will also affect the electric field strength.
A and B - Changing the length of the rod without changing the charge does not affect the charge density, and thus does not affect the electric field strength at the position of the dot. Therefore, options A and B are not correct.C and D - Changing the cross-sectional area of the rod without changing the charge does affect the charge density, and therefore can affect the electric field strength at the position of the dot. However, it is not clear how changing the rod's shape will affect the charge density, and so it is not possible to determine whether option C or D will increase or decrease the electric field strength. Therefore, options C and D are not correct.E - Removing charge from the rod will decrease the charge, and therefore decrease the electric field strength at the position of the dot. Therefore, option E is correct.You can learn more about electric field strength at
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a parallel-plate vacuum capacitor is connected to a battery and charged until the stored electric energy is u. the battery is removed, and then a dielectric material with dielectric constant k is inserted into the capacitor, filling the space between the plates. finally, the capacitor is fully discharged through a resistor (which is connected across the capacitor terminals).
All of the electrical energy held in the capacitor is released as heat when it is fully discharged through the resistor.
What purpose does a capacitor serve?Two conductors that are separated from one another and positioned close to one another make up a capacitor, an electrical energy storage device.A straightforward illustration of one such storage device is a parallel-plate capacitor.
The following formula can be used to determine how much energy was lost inside the resistor:
= ½ × ×
A parallel-plate capacitor's capacitance is determined by:
= × 0 × /,
Since the capacitor is charged until the stored electric energy is , we can write:
½ × × =
Solving for gives:
= 2 × /
Substituting this expression for into the formula for, we get:
½ × × ² × ( − 1)
(B) The potential difference across the capacitor will rise as long as the charging battery is still attached and the dielectric is inserted because of the higher capacitance. The following formula can be used to compute the new potential difference:
The capacitor's newly accumulated electrical energy is:
½ × × × ²
= ½ × × ² × ( − 1)
= ½ × × ² × ( − 1)
= ½ × × (² − ²)
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Question:
A parallel-plate vacuum capacitor is connected to a battery and charged until the stored electric energy is . The battery is removed, and then a dielectric material with dielectric constant is inserted into the capacitor, filling the space between the plates. Finally, the capacitor is fully discharged through a resistor (a resistor is connected across the capacitor terminals).
A) Find , the energy dissipated in the resistor.
Express your answer in terms of and other given quantities.
B)Consider the same situation as in the previous part, except that the charging battery remains connected while the dielectric is inserted. (Part B figure) The battery is then disconnected and the capacitor discharged. For this situation, what is , the energy dissipated in the resistor?
Express your answer in terms of and other given quantities.
Introduce CT scan and explain its working mechanism.
What two processes are part of electromagnetism?
A. The attraction of electric charges for each other
B. The production of a magnetic field by an electric current
C. The production of an electric current by a moving magnet
D. The production of a magnetic field by permanent magnets
Answer:
C. The production of an electric current by a moving magnet
B. The production of a magnetic field by an electric current
What is electromagnetism?
The " real definition " of electromagnetism is the following:
Electromagnetism is the physical interaction among electric charges, magnetic moments, and the electromagnetic field.
There are different " types " of electromagnetic fields
StaticSlowly changingWavesThe last thing that YOU should know is that electromagnetic waves are generally known as light and obey the laws of optics. Which means in similar terms they just obey the laws of optics.
Thus, the answers to your question is, C & B
A GPS satellite orbits at an altitude of 20,200 km above the surface of the earth. What is the speed of the satellite? Recall that RE= 6.37 x 10^6 m
The speed of the GPS satellite can be calculated using the following formula:
How to calculate GPS satellite ?v = √(GM/r)
where G is the gravitational constant, M is the mass of the Earth, r is the distance between the center of the Earth and the satellite, and v is the speed of the satellite.The distance between the satellite and the center of the Earth can be found by adding the radius of the Earth to the altitude of the satellite:r = RE + h = 6.37 x 10^6 m + 20,200,000 m = 2.06 x 10^7 m
The mass of the Earth is M = 5.97 x 10^24 kg, and the gravitational constant is G = 6.67 x 10^-11 Nm^2/kg^2. Substituting these values into the formula, we get:v = √(GM/r) = √((6.67 x 10^-11 Nm^2/kg^2)(5.97 x 10^24 kg)/(2.06 x 10^7 m)) ≈ 3,870 m/s
Therefore, the speed of the GPS satellite is approximately 3,870 m/s.
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Calculate whether the kinetic energy of the particle increase, decrease or remain the same if the particle`s velocity changes (a) from -3 m/s to -2 m/s and (b) from -2 m/s to 2 m/s if the particle moves along x axis.? (c) In each situation, is the work done on the particle positive, negative or zero?
Answer:
(a) The kinetic energy of the particle will decrease if the particle's velocity changes from -3 m/s to -2 m/s.
(b) The kinetic energy of the particle will increase if the particle's velocity changes from -2 m/s to 2 m/s.
(c) The work done on the particle in both situations is positive. This is because work is the energy transferred to or from a system and the kinetic energy of the particle increases in both cases, indicating energy was transferred to the particle.
The area labeled B is four times the area labeled A. Express b in terms of a.
The x y-coordinate plane is given. A curve and shaded region are graphed.
The curve y = ex enters the window in the second quadrant, goes up and right, crosses the y-axis becoming more steep, then exits the window in the first quadrant.
The shaded region A is below the curve and above the x-axis between x = 0 and x = a.
b in terms of a that satisfies the given condition is, [tex]b = \ln(3e^a - 2)[/tex].
The equation of the curve is y = e^x.
The shaded region A is the area under the curve between x = 0 and x = a, so its area is given by,
[tex]A = \int_0^a e^x dx = e^a - 1[/tex]
The area labeled B is four times the area labeled A, so its area is given by,
B = 4A = 4(e^a - 1)
To express b in terms of a, find the value of b that satisfies,
[tex]\int_0^a e^x dx = 3(e^a - 1)[/tex]
Using the formula for the integral of e^x, we get:
[tex]e^b - e^a = 3(e^a - 1)[/tex]
Solving for b, we get:
[tex]b = \ln(3e^a - 2)[/tex]
So the area labeled B is [tex]4(e^a - 1)[/tex], and the value of b that satisfies the given condition is [tex]b = \ln(3e^a - 2)[/tex].
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A metallic surface is illuminated by light. There is another metal plate a few cm away, which is held at an electric potential of V the plate is connected to a circuit which measures the current of electrons. When light of wavelength 520 nm is incident on the metal surface, electrons are emitted from the metal surface and absorbed by the second plate, which creates a current. When the potential of the second plate V is increased to Vstop =0.3 V, the current stops. 3.1 Photon energyWhat is the energy of each photon in electron volts?
The energy of each photon in electron volts is, 2.07.
Stopping potential is the minimum negative voltage applied to the anode to stop the photocurrent. The maximum kinetic energy of the electrons equal the stopping voltage, when measured in electron volt.
Stopping potential = Energy of light wave falling - energy of each photon
Energy of light wave falling = [tex]\dfrac{hc}{\lambda}[/tex]
Energy of light wave falling = [tex]\dfrac{hc}{\lambda}[/tex]
Energy in electron volts for a wave of wavelength 520 nm is, 2.37 eV.
Now,
Stopping potential = Energy of light wave falling - energy of each photon
0.3V = 2.37 - energy of each photon
Energy of each photon = 2.07
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A hot rod can accelerate from 0 to 60 km/h in 7.5 s. (a) What is its average acceleration, in m/s2, during this time? (b) How far will it travel during the 7.5 s, assuming its acceleration is constant? (c) From rest, how much time would it require to go a distance of 0.40 km if its acceleration could be maintained at the value in (a)?
Average acceleration is a) 2.22 [tex]m/s^2[/tex] b) hot rod travels 78.4 m c) time is 20 seconds
(a) To find the average acceleration, we need to use the formula: average acceleration = change in velocity / time.
First, we convert 60 km/h to m/s:
60 km/h x 1000 m/km x 1 h/3600 s = 16.67 m/s
So the change in velocity is:
16.67 m/s - 0 m/s = 16.67 m/s
And the time is:
7.5 s
Therefore, the average acceleration is:
average acceleration = (16.67 m/s) / (7.5 s) = 2.22 m/s^2
(b) To find the distance traveled, we can use the formula: distance = initial velocity x time + 0.5 x acceleration x time^2.
The initial velocity is 0 m/s, so the formula becomes:
distance = 0.5 x acceleration x time^2
Plugging in the values, we get:
distance = 0.5 x 2.22 m/s^2 x (7.5 s)^2 = 78.4 m
Therefore, the hot rod will travel 78.4 meters during the 7.5 seconds.
(c) We can use the formula: distance = 0.5 x acceleration x time^2, and solve for time.
Plugging in the values, we get:
0.40 km x 1000 m/km = 400 m
acceleration = 2.22 m/s^2
So the formula becomes:
400 m = 0.5 x 2.22 m/s^2 x time^2
Solving for time, we get:
time = sqrt(400 m / (0.5 x 2.22 m/s^2)) = 20 s
Therefore, it would take 20 seconds for the hot rod to travel 0.40 km with constant acceleration of 2.22 m/s^2.
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show that for every k, there is a set of k positive integers such that the difference of any two of these has an even number of prime factors, counted with multiplicity. (hint: you don't need to know any number theory.)
It is demonstrated that for each k, there exists a set of k positive integers whose difference has an even number of prime components, counted with multiplicity.
Let's construct the set of positive integers using the following algorithm:
Start with the first k prime numbers: 2, 3, 5, 7, 11, ..., p_k.
Define the i-th integer in the set as the product of the first i prime numbers: a_i = 235*...*p_i.
Claim: For any two integers in the set, their difference has an even number of prime factors, counted with multiplicity.
Proof:
Let a_i and a_j be two integers in the set such that i < j. We want to show that the difference a_j - a_i has an even number of prime factors, counted with multiplicity.
Notice that a_j - a_i can be written as:
a_j - a_i = 235*...p_i(235*...*p_j/2 - 1)
The first term on the right-hand side has i prime factors, and the second term has an odd number of prime factors. To see why the second term has an odd number of prime factors, notice that all the prime factors of 235*...p_j are also prime factors of 235...*p_i, so they cancel out in the difference.
a_j - a_i has i prime factors plus an odd number of prime factors, which is an even number of prime factors. This completes the proof.
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in a frictionless setup, two block of different masses are laying on two incline planes with different slopes and joined together by a pulley (see figure). find the acceleration of each block and the tension in the rope. also write the criteria (or equation) needed to be satisfied for block 2 to be moving down its incline plane.
Answer:
Explanation:
The problem you're describing involves two blocks connected by a rope over a pulley, with each block resting on a separate incline plane. The acceleration of each block and the tension in the rope can be determined using the principles of dynamics and conservation of energy.
Let's call the masses of the two blocks m1 and m2, the incline angles of the two planes θ1 and θ2, and the acceleration of each block a1 and a2.
The net force on each block is given by the sum of the gravitational force, the normal force, and the tension in the rope.
For block 1:
m1 * g * sin(θ1) - m1 * a1 = T
For block 2:
m2 * g * sin(θ2) - m2 * a2 = -T
where g is the acceleration due to gravity and T is the tension in the rope.
From these equations, we can see that the tension in the rope is equal in magnitude but opposite in direction for each block. The acceleration of each block can be found by rearranging the equations and solving for a1 and a2.
The condition for block 2 to be moving down its incline plane is that the net force on the block must be in the direction of motion, which means that the tension in the rope must be less than the gravitational force acting on the block. This can be expressed as:
T < m2 * g * sin(θ2)
Here is a definition of each variable in the problem:
m1: mass of block 1
m2: mass of block 2
θ1: incline angle of the first plane
θ2: incline angle of the second plane
a1: acceleration of block 1
a2: acceleration of block 2
g: acceleration due to gravity
T: tension in the rope
I hope this helps! Let me know if you have any other questions or if you need further clarification.
a cooler (mass 3.6 kg) is initially at rest on a horizontal floor. it is then pushed in a straight line for 2 m by a small child who exerts a horizontal force with magnitude 30 n.
A.) Use the Work-Energy Theorem to determine the final speed of the cooler if there is no friction between the cooler and the floor. B.) Use the Work-Energy Theorem to determine the final speed of the cooler if the coefficient of kinetic friction between the cooler and the floor is 0.30.
(a) The final speed of the cooler is 5.77 m/s.
(b) In the presence of friction, the final speed of the cooler is 3.43 m/s.
What is the final speed of the cooler?
The final speed of the cooler is calculated by applying work energy theorem as shown below;
Change in kinetic energy of the cooler = work done on the cooler
¹/₂mv² = fd
v² = (2fd) / m
v = √(2fd) / m
where;
f is the applied forced is the displacement of the coolerm is the mass of the coolerThe final speed of the cooler is calculated as;
v = √(2 x 30 x 2) / 3.6
v = 5.77 m/s
At the given coefficient of friction, the final speed is calculated as;
v = √(2Ffd) / m
where;
Ff is force of frictionFf = μW
where;
μ is coefficient of frictionW is the normal forcev = √(2μWd) / m
v = √(2 x 0.3 x 3.6 x 9.8 x 2) / 3.6
v = 3.43 m/s
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use your function to predict the force on the force sensor when the ramp angle is 90 degrees. to make your prediction, assume a cart mass of 2.25 kg.
When the angle is 90 degrees up, the only force acting in a downward direction is gravitational force, with magnitude mg, where m is mass and g is gravitational acceleration.
m = 2.25 kg, g = 9.8 /s²
F = mg = 2.25 × 9.8
F = 22.05 N.
An item falling freely in a vacuum is said to accelerate gravitationally in physics. This is the constant acceleration brought on just by the gravitational pull.
By putting an object in a vacuum chamber and monitoring the object's speed as a function of time as it accelerates, one can calculate the gravitational acceleration. This is the approach that Galileo popularized.
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A student wants to launch a toy dart toward a target that hangs from a light string: As time t = 0,the dart is launched with an initial speed v at an angle 0 above the horizontal ground: At the instant the dart is launched, the string is cut such that the target begins to fall straight down. The positive horizontal direction is considered to be to the right, and the positive vertical direction is considered to be up.
A student wants to launch a toy dart As time t = 0, release a ball that has been resting near the Earth's surface. Allow a cart to leave rest and move down a 40°40° inclination with regard to the ground.
The graphs which could represent the vertical component of the velocity as a function of time for the dart and the target immediately after the dart is launched and the target begins to fall is Dart: above x-axis, decreasing; Target: below x-axis, decreasing.
Velocity is the directional speed of a moving object as an indicator of the rate at which its position changes as perceived from a given frame of reference and measured by a particular standard of time. The idea of speed is important in kinematics, the part of classical mechanics that explains the motion of things.
Velocity is a physical vector quantity that requires both magnitude and direction to be determined. Velocity is a scalar absolute value of speed, a consistently derived unit whose quantity is measured in SI (metric system) in meters per second.
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The above question is incomplete, the complete question is-
A student wants to launch a toy dart toward a target that hangs from a light string. At time t=0, the dart is launched with an initial speed v0 at an angle θ0 above the horizontal ground. At the instant the dart is launched, the string is cut such that the target begins to fall straight down. The positive horizontal direction is considered to be to the right, and the positive vertical direction is considered to be up.
Which of the following graphs could represent the vertical component of the velocity as a function of time for the dart and the target immediately after the dart is launched and the target begins to fall?
Help!!
A horse does 860 J of work in 420 seconds while pulling a wagon. What is the power output of the horse? Round your answer to the nearest whole number.
The power output of the horse is [???] W.
What will happen to an acceleration if the mass of an object is tripled, but the force is kept the same?
If the mass of an object is tripled while the force acting upon it is kept constant, the acceleration of the object will be reduced to one-third of its original value.
This can be explained using Newton's second law of motion, which states that the acceleration of an object is directly proportional to the force applied to it and inversely proportional to its mass.
In this scenario, the force is held constant, but the mass is increased by a factor of three. As a result, the acceleration will decrease by a factor of three, since the same force is now acting on a much heavier object. Therefore, the object will experience a lower acceleration than before, making it harder to move or change its speed or direction.
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