The power in watts (P) in a circuit is equal to the voltage (E) multiplied by the current (I). This rule written as a formula would read I = P × E. Once you solve an equation, you can prove your answer by substituting the values into the original equation and verify equality.

Solving the equation f(x) = 0 will give us the equilibrium points, the differential equation governing z(t) is \(\frac{dz}{dt} = \frac{dx}{dt} = f(x)\), the system trajectories is negative semidefinite: \(\frac{dV(z)}{dt} \leq 0\) for all z and The solution to the linear differential equation can be written as:

\(z(t) = Ce^(^A^t^)\)

a) To find the equilibrium points of the given scalar nonlinear system, we need to set the derivative of the system with respect to the variable (in this case, x) equal to zero. Let's denote the system as dx/dt = f(x).

So, we have:

\(\frac{dx}{dt} = f(x) = 0\)

Solving the equation f(x) = 0 will give us the equilibrium points.

b) For the positive equilibrium point \(xe^+\), let \(z = x - xe^+\). To find the differential equation governing z(t), we substitute this value of z into the original system equation:

\(\frac{dz}{dt} = \frac{dx}{dt} = f(x)\)

Substituting \(z = x - xe^+\) into f(x), we get:

\(\frac{dz}{dt} = f(x) = f(z + xe^+)\)

c) To show that \(V(z) = (1/2)z^2\) is a Lyapunov function, we need to demonstrate the following properties:

V(z) is positive definite: V(z) > 0 for all z ≠ 0, and V(0) = 0.

V(z) is radially unbounded: V(z) approaches infinity as z approaches infinity.

The derivative of V(z) along the system trajectories is negative semidefinite: \(\frac{dV(z)}{dt} \leq 0\) for all z.

d) For z ≈ 0, we can consider the linearized form of the system around the equilibrium point z = 0. This involves finding the linear differential equation governing z(t), which can be obtained by linearizing the nonlinear system equation around z = 0.

The linearized equation will have the form:

\(\frac{dz}{dt} = Az\)

Where A is the Jacobian matrix evaluated at the equilibrium point z = 0.

The solution to the linear differential equation can be written as:

\(z(t) = Ce^(^A^t^)\)

Where C is a constant determined by the initial conditions.

To argue that the equilibrium point \(ze = 0\) is asymptotically stable, we need to show that all trajectories starting sufficiently close to z = 0 converge to the equilibrium point as t → ∞. This can be inferred by examining the eigenvalues of the matrix A. If all eigenvalues have negative real parts, then the equilibrium point is asymptotically stable.

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Answer: Negative

Explanation: If electric field lines point radially toward a charge, that charge must be Negative

Answer:

Light is faster in air than in glass so the speed of light decreases when it passes through glass.

The distance from the observer to the rocket is increasing at a rate of 1000 mph.

To find the rate at which the distance from the observer to the rocket is increasing, we can use the concept of relative velocity. Since the rocket is traveling straight up at a constant velocity of 750 mph and the observer is stationary, the only motion that affects the distance between them is the vertical motion of the rocket.

At the moment when the rocket is 2 miles high, we can consider a right-angled triangle formed by the observer, the rocket, and the height of the rocket. The distance from the observer to the rocket forms the hypotenuse of this triangle, and the height of the rocket is one of the perpendicular sides. We can use the Pythagorean theorem to relate these quantities:

(distance from observer to rocket)² = (height of rocket)² + (horizontal distance)²

Let's call the distance from the observer to the rocket D, the height of the rocket H, and the horizontal distance X. We are given that D = 1.5 miles and H = 2 miles. We need to find the rate at which D is changing when H is 2 miles.

Differentiating both sides of the equation with respect to time, we get:

2 * D * dD/dt = 2 * H * dH/dt + 2 * X * dX/dt

Since the observer is stationary, the horizontal distance X remains constant. Also, the rocket is traveling straight up, so dX/dt = 0. Therefore, the equation simplifies to:

D * dD/dt = H * dH/dt

Plugging in the known values, we have:

1.5 * dD/dt = 2 * dH/dt

Now, solving for dD/dt, we find:

dD/dt = (2 * dH/dt) / 1.5

Since the rocket is traveling at a constant velocity of 750 mph, the rate at which the height of the rocket is changing, dH/dt, is also 750 mph. Substituting this value, we get:

dD/dt = (2 * 750 mph) / 1.5

dD/dt = 1500 mph / 1.5

dD/dt = 1000 mph

Therefore, the distance from the observer to the rocket is increasing at a rate of 1000 mph.

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The acceleration of the object with an applied force of 100N and a mass of 50kg is 2m/s².

Acceleration is defined as the change in velocity per unit time. Velocity is defined as the speed of the object in a particular direction. The acceleration is a vector quantity. The unit of acceleration is m/s².

According to Newton's second law, Force is directly proportional to the acceleration of the object. F = m×a, where F is the force of the object, m is the mass of the object and a is the acceleration of the object.

From the given,

Force, F = 100N

mass of the object, m=50kg

acceleration of the object, a=?

F = m×a

a =F/m

  = 100/50

  =2m/s².

Thus, the acceleration of the object is 2m/s².

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The charge on each plate of a capacitor has a magnitude of  6.7 × 10⁻⁹ C when the electric field between the plates is 29,000 V/m. If the plate separation is 3.5 × 10⁻⁴ m, 2.31 × 10⁻¹⁶ F is the capacitance of the capacitor

The capacitance of a capacitor can be calculated based on the charge on each plate, the electric field between the plates, and the plate separation. In this case, the charge on each plate is given as 6.7 × 10⁻⁹ C, the electric field is 29,000 V/m, and the plate separation is 3.5 × 10⁻⁴ m. The capacitance (C) of a capacitor is given by the formula:

\(C = Q / V\)

where Q is the charge on each plate and V is the voltage or electric field between the plates. In this case, the charge on each plate is 6.7 × 10⁻⁹ C and the electric field is 29,000 V/m. Therefore, we can calculate the capacitance as follows:

C = (6.7 × 10⁻⁹ C) / (29,000 V/m)

Simplifying the expression, we can divide the charge by the electric field:

C = 2.31 × 10⁻¹⁶ F

Therefore, the capacitance of the capacitor is 2.31 × 10⁻¹⁶ Farads.

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Answer:

a) 588.235 ohm ( all the 3 resistors are in parallel )

b) 10 A

c) Current flowing -

in 20 ohm resistor = 6.25 A

in 100 ohm resistor = 1.25 A

in 50 ohm resistor = 2.5 A

d) Voltage drop across all the resistor = 125 V

(As all the resistors are in parallel all the resistors will have the same voltage drop )

e) Power dissipated -

in 20 ohm resistor = 781.25 W

in 100 ohm resistor = 156.25 W

in 50 ohm resistor = 312.5 W

(Power = Current × Potential Difference)

The distance travelled by the plane is found to be 1600 m.

The plane is flying at a speed of 150m/s and it is accelerating uniformly at a rate of 2m/s² the airplane is flying for 10 seconds.

According to the equation of motion,

S = ut+1.2at²

Where, t is the time required by the body during the motion, u is the initial velocity, a is the net acceleration and S is the distance travelled by the body.

Putting values in the above relation to find the distance travelled by the plane,

S = 150(10)+0.5(2)(100)

S = 1600m

So, the plane would have travelled a distance of 1600 m.

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Answer:

34º

Explanation:

The computation of the angle of refraction in the glass for this ray is shown below:

Given that

the angle of incidence of another red ray is 65º

And, n_1 = 1

n_2 = 1.62

According to shell law

n_1 sin \(\theta\)_1 = n_2 sin \(\theta\)_r

sin \(\theta\)_r = 1 sin 65÷ 1.62

= 0.559

\(\theta\)_r  = sin^-1 (0.559)

= 34º

Answer:

Fr = 26.83 [N]

Explanation:

To solve this problem we must use the Pythagorean theorem, since the forces are vector quantities, that is, they have magnitude and density. Therefore the Pythagorean theorem is suitable for the solution of this problem.

\(F_{r}=\sqrt{(12)^{2}+(24)^{2} } \\F_{r}=26.83[N]\)

Answer:They have 10

Explanation:

Answer:

A is true ,b is true

Explanation:

Answer:

The magnitude of the surface charge density on either plate is approximately 5.41 × 10⁻⁸ C/(m²)

Explanation:

The magnitude of the charge on each of the conducting plates = Equal

The orientation of the charges on each plate = Opposite charges

The magnitude of the point charge placed between the plates = +3.60 nC

The force acting on the point charge, q = 22.0 μN

Therefore, we have;

F = E × q

E = F/q = 22.0 μN/+3.60nC = 6,111.1111 V/m

We have;

\(E = \dfrac{\rho }{\epsilon_0}\)

Where;

ρ = The surface charge density

ε₀ = Absolute permittivity ≈ 8.85 × 10⁻¹² C²/(N·m²)

Therefore, we have;

ρ = E × ε₀ = 6,111.1111 V/m × 8.85 × 10⁻¹² C²/(N·m²) = 5.40833332 × 10⁻⁸ C/(m²)

∴ The magnitude of the surface charge density on either plate, ρ ≈ 5.41 × 10⁻⁸ C/(m²).

A jet engine emits sound uniformly in all directions, radiating an acoustic power of 3.07×105 W. The intensity of the sound at a distance of 54.5 m from the engine is calculated below: The sound power is given as P = 3.07 × 10⁵ W Distance, r = 54.5 m

I = 3.07 × 10⁵ / (4π × (54.5)²)I = 0.0584 W/m²Therefore, the intensity of the sound at a distance of 54.5 m from the engine is 0.0584 W/m².The sound intensity level β is given by β = 10 log(I/I₀), where I₀ is the reference intensity of 1 × 10⁻¹² W/m².

Substituting the values,β = 10 log(0.0584/1 × 10⁻¹²)β = 10 log(5.84 × 10¹⁰)β = 10(10.766)β = 107.66 dB Accordingly, the intensity of the sound at a distance of 54.5 m from the engine is 0.0584 W/m² and the corresponding sound intensity level β is 107.66 dB (decibels).

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Answer:

the motion of the sled decrease

Explanation:

the mass increase

The acceleration of the sled will is inversely proportional to the mass on it. Therefore, the motion of the sled would slow if Joslyn's friend Lisa also wanted to ride.

Acceleration in mechanics can be described as the rate of change of velocity of an object with respect to time. The acceleration of an object is a vector quantities with magnitude and direction.

The S.I. unit for acceleration can be represented as m/s² the net balance of all external forces acting is equal to the magnitude of the net resulting force.

The force (F) acting on an object is equal to the product of the mass (m) of the object and acceleration (a) of an object.

F = ma

And, a = F/m

As shown in the above equation, the mass of the object is inversely proportional to the acceleration of the object, therefore, the acceleration of the sled decrease as the mass on it increases.

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The principle or law that states "an induced emf in a circuit loop produces a current whose magnetic field opposes further change of magnetic flux" is credited to Lenz. The correct option is b.

The principle described in the question is known as Lenz's law, named after the Russian physicist Heinrich Lenz. Lenz's law is a fundamental law in electromagnetism that relates to electromagnetic induction.

According to Lenz's law, when there is a change in magnetic flux through a circuit loop, an induced electromotive force (emf) is produced in the loop. This induced emf generates a current that creates a magnetic field that opposes the change in the magnetic flux.

In simpler terms, the induced current in a circuit loop always acts in a way that opposes the cause that produces it.

Lenz's law is based on the conservation of energy and is a consequence of Faraday's law of electromagnetic induction.

It provides a fundamental understanding of the relationship between magnetic fields, electric currents, and changes in magnetic flux, and it has important applications in various areas of electromagnetism, including the design and operation of electric generators, transformers, and other electromagnetic devices. The correct option is b.

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the principle or law that says" an induced emf in a circuit loop produces a current whose magnetic field opposes further change of magnetic flux" is credited to

a. Faraday.

b. Lenz.

c. Ampere.

d. Volta.

The equation Jerry should use for the total resistance in a series circuit is R = R₁ + R₂ + R₃.

option A.

In a series circuit, all components are connected end-to-end, forming a single path for current flow.

A series circuit can contain any combination of resistors, capacitors, and inductors.

In a series circuit, the current flowing in each circuit component is the same  and the voltage drop in each circuit component is different.

Based on this, the total formula for the total resistance in a series circuit is given as;

R = R₁ + R₂ + R₃

where;

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Answer:

W = 600 N

Explanation:

The mechanical advantage of any machine is the ratio of the load or weight to the effort applied. Hence, the general formula for mechanical advantage is as follows:

M.A = W/P

where,

M.A =Mechanical Advantage of Hammer = 10

W = Maximum force that could be generated = ?

P = Force created by me = 60 N

Therefore,

10 = W/60 N

W = 10*60 N

W = 600 N

The overall charge of an object that has 82 protons, 82 neutrons, and 109 electrons is -27. Details about charge can be found below.

The overall charge of an atom can be calculated by subtracting the number of electrons from the number of protons.

The proton number represents the number of positively charged atom while the electron number represents the number of negatively charged atom.

According to this question, a charged object has 82 protons, 82 neutrons, and 109 electrons. The overall charge of the atom is calculated as follows:

Overall charge = 82 - 109

Charge = -27

Therefore, the overall charge of an object that has 82 protons, 82 neutrons, and 109 electrons is -27.

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To calculate the efficiency of a cyclical heat engine, we can use the formula: Efficiency = (Work Output / Heat Input) * 100%

Given that the efficiency of the cyclical heat engine is 21.5% and the heat input is 5.2 × 10^9 J, we can calculate the work output.

Efficiency = (Work Output / Heat Input) * 100%

0.215 = (Work Output / (5.2 × 10^9 J)) * 100%

Rearranging the equation to solve for the work output:

Work Output = Efficiency * Heat Input / 100%

Work Output = (21.5% * (5.2 × 10^9 J)) / 100%

Calculating the work output:

Work Output = (0.215 * 5.2 × 10^9 J) / 100%

Work Output ≈ 1.118 × 10^9 J

Therefore, the work output of the cyclical heat engine is approximately 1.118 × 10^9 J.

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Answer:

The Enlightenment.

Explanation:

The Enlightenment took place during the eighteen century and it typically involved a period of philosophical, cultural and intellectual movement in which ideas on nature, reason, humanity and God were emphasized on greatly over science and superstition through meetings that were held at coffeeshops, scientific academies, literary salons, masonic lodges, etc. Some of the influential philosophers of the enlightenment were Adam Smith, John Locke, Jean Jacques Rousseau, David Hume, Baruch Spinoza, Montesquieu, Hugo Grotius, Denis Diderot, Gottfried Wilhelm, Cesare Beccaria, Immanuel Kant etc. The enlightenment was also known as the age of reason or the age of enlightenment.

Hence, the enlightenment was a period in European history when many educated people stressed the importance of learning and reasoning; education was considered the key to understanding and solving society's problems.

The magnitude of the magnetic field at the location of the charge due to the current-carrying wire is given as B = μI/2πr.

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials.

A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field.

The magnitude of the magnetic field at the location of the charge due to the current-carrying wire is calculated as follows;

B = μI/2πr

where;

Thus, the magnitude of the magnetic field at the location of the charge due to the current-carrying wire is given as B = μI/2πr.

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Answer: CO

I hope this helps

Answer:

co

Explanation:

Answer:

V (Joules / Coulomb)       definition of voltage

E = Joules          energy expressed in Joules

V Q = energy expressed in Joules and Coulombs

E = 12 Joules / Coulomb * 5 Coulomb = 60 Joules

Answer:

D) Kira will have more endurance because she spends more time moving heavy objects.

Explanation:

Because Kira works in a warehouse, she most likely spends a significant amount of time lifting and carrying heavy items every day she works. By doing this, Kira builds up muscle endurance, which makes lifting heavy items easier over longer periods for her, versus Marta, who only does it occasionally, and may struggle lifting heavy objects over a consistent amount of time.

Kira is pretty strong. She works in a warehouse, so she moves a lot of heavy things. Her sister Marta can lift more weight than Kira can but only does it occasionally, likely to have more endurance is D. Kira will have more endurance because she spends more time moving heavy objects is correct option.

A warehouse is a large commercial building used for the storage of goods, materials, merchandise, and products. Warehouses are typically equipped with various storage systems, such as racks, shelves, and bins, to organize and store items efficiently. They serve as distribution centers for businesses, where goods are received, sorted, stored, and eventually shipped out to retailers or directly to customers. Warehouses are crucial components of supply chain management, providing a space to hold inventory and manage the flow of products throughout the distribution process.

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Answer:

47 mW

Explanation:

The average value of the Poynting vector, S = 0.939 W/m² = Intensity of wave, I

S = I S

Also, I = P/A where P = Et, P = power of electromagnetic wave, E = energy of electromagnetic wave in time t and t = time = 1 min = 60 s and A = area = lb since the electromagnetic waves falls on area equal to that of a rectangle.

So, S = Et/A

E = SA/t

= Slb/t

= 0.939 W/m² × 1.5 m × 2.0 m/60 s

= 2.817 W/60 s

= 0.047 W

= 47 mW

So, 47 mW of electromagnetic energy falls on the area in 1.0 minute.

Explanation:

damming of water which changes water flow, in turn could/does cause road/ property flooding

directly being related to water usage, if we have droughts(lack of rain) there we have less water to use for the power.

Environmental consequences

Limited Reservoirs

At F the object moving to the left, if the positive x-direction corresponds to its rightward movement.

Δx = \(X _{2}\) - \(X_{1}\)

where,

Δx = Change in position

\(X _{2}\) = Final position

\(X_{1}\) = Initial position

If Δx is positive, the object is moving forwards ( Right ).

If Δx is negative, the object is moving backwards ( Left ).

If the curve is moving from top to bottom Δx will be negative and if the curve is moving from bottom to top Δx will be positive.

So, in this given curve, at the points B, C and D, the curve is moving from bottom to top meaning that the object is moving to the right and at the point F, the curve is moving from top to bottom meaning that the object is moving to the left.

Therefore, At F the object moving to the left, if the positive x-direction corresponds to its rightward movement.

The given question contains no distance - time graph. I considered the attached image as the graph.

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Answer:

volume

Explanation:

it depends on the volume of the items

hope this helps

Answer:mass

Explanation:

gravity depends directly upon the masses of the two objects, and inversely on the square of the distance between them. This means that the force of gravity increases with mass, but decreases with increasing distance between objects.

The direction of acceleration due to gravity is always towards earth, going downwards.

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