A stone is thrown straight up from the edge of a roof, 875 feet above the ground, at a speed of 14 feet per second. A. Remembering that the acceleration due to gravitv is - 32 feet per second squared, how high is the stone 4 seconds later? B. At what time does the stone hit the ground? C. What is the velocity of the stone when it hits the ground?

A small object begins a free-fall from a height of 25.0 m. After 1.40 s, a second small object is launched vertically upward from the ground with an initial velocity of 37.0 m/s.

Height from which first object falls, s₁ = 25.0 m Time elapsed, t = 1.40 s Initial velocity of second object, u₂ = 37.0 m/s

For the first object that undergoes free-fall;

The vertical displacement after time t, s₁ = u₁t + 1/2 gt²  -------> (1)

Where u₁ = Initial velocity of the object, g = acceleration due to gravity = 9.81 m/s²

For the second object,

The vertical displacement after time t, s₂ = u₂t - 1/2 gt² ------> (2)

Substitute the given values in the above equations and solve for t

Using equation (1),s₁ = u₁t + 1/2 gt² = 0 + 1/2 x 9.81 x (1.40)² = 12.99 m

Thus, the first object falls a distance of 12.99 m in 1.40 seconds.Now, using equation (2),s₂ = u₂t - 1/2 gt²

Solve the above equation for t

Substitute the values u₂ = 37.0 m/s t = Time at which the two objects meet g = 9.81 m/s²∴ t = s₂/g = (u₂t - s₁)/g

On substituting the given values we get, t = (37.0 x 1.40 - 12.99) / 9.81= 3.59 s

Now, the height at which the two objects will first meet is given by the equation, s = s₁ + u₁t Where u₁ = 0 m/s (as it is in free-fall)

Substituting the values we get, s = 25.0 + 0 x 3.59= 25.0 m

Therefore, the height at which the two objects will first meet is 25.0 m.

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1)Magnitude of the linear acceleration of the hoop= 9.8 m/s²2)the magnitude of the linear acceleration of the sphere is 0. 3)The magnitude of the angular acceleration of the disk pulley α = 0.4 m/s². 4)The magnitude of the angular acceleration of the sphere= 0.23 m/s². 5)The tension in the string between the sphere and disk pulleyT1 = 40.38 N. 6)The tension in the string between the hoop and disk pulleyT = 50.68 N.7)The hoop takes time to fall 1.57 m= 0.56 s. 8)the magnitude of the velocity of the green hoop v² = 6.2 m/s. 9)The magnitude of the final angular speed of the orange sphere is 29.5 rad/s.

1) Magnitude of the linear acceleration of the hoop:The tension in the string between the hoop and disk pulley is T. Let a be the linear acceleration of the hoop, and R be the radius of the hoop. There is only one force acting on the hoop, which is the force due to tension, which acts in the forward direction. Hence,mh * a = TThus, a = T / mh. The tension is given by,T = mg - T1Here,m is the mass of the hoop, g is the acceleration due to gravity, and T1 is the tension in the string between the sphere and disk pulley. Hence,a = (mg - T1) / mhGiven that,mh = 2.6 kgm = 9.8 m/s²g = 9.8 m/s²T1 = Tension in the string between the sphere and disk pulley = 0 (Since the sphere rolls without slipping)a = (2.6 × 9.8 - 0) / 2.6 = 9.8 m/s²

2) Magnitude of the linear acceleration of the sphere:Since the sphere rolls without slipping, the acceleration of the sphere is the same as the linear acceleration of its center of mass. Let a1 be the linear acceleration of the sphere, and R1 be the radius of the sphere. Let T1 be the tension in the string between the sphere and disk pulley. Hence,mh * a1 = T1Thus, a1 = T1 / mhGiven that,T1 = 0a1 = 0Thus, the magnitude of the linear acceleration of the sphere is 0.

3) Magnitude of the angular acceleration of the disk pulley:Let I be the moment of inertia of the disk pulley, α be its angular acceleration, and R be its radius. The disk pulley is rolling without slipping. Hence, a frictional force f is acting on it, which acts opposite to the direction of motion of the pulley. Hence,ma = fThus,ma = μmgHere,μ is the coefficient of friction between the pulley and the surface it is rolling on. Thus,α = a / R = μg / RThus,α = 0.4 m/s².

4) Magnitude of the angular acceleration of the sphere:Let I1 be the moment of inertia of the sphere, α1 be its angular acceleration, and R1 be its radius. Since the sphere is rolling without slipping, we can assume that its point of contact with the ground is momentarily at rest. Hence, the frictional force f1 is acting on it, which acts opposite to the direction of motion of the sphere. Hence,ma1 = f1Thus,ma1 = μmgHere,μ is the coefficient of friction between the sphere and the surface it is rolling on. Thus,α1 = a1 / R1 = μg / R1Thus,α1 = 0.23 m/s².

5) Tension in the string between the sphere and disk pulley:Let T1 be the tension in the string between the sphere and disk pulley, and a1 be the linear acceleration of the sphere. The net force acting on the sphere is,m1a1 = T1 - m1gHere,m1 is the mass of the sphere, and g is the acceleration due to gravity. Since the sphere is rolling without slipping, its angular acceleration is,α1 = a1 / R1Hence,α1 = 0.23 m/s²The moment of inertia of the sphere is,I1 = (2/5) m1 R1²Hence,T1 = m1 (g - a1)T1 = 4.1 (9.8 - 0)T1 = 40.38 N.

6) Tension in the string between the hoop and disk pulley:Let T be the tension in the string between the hoop and disk pulley, and a be the linear acceleration of the hoop. The net force acting on the hoop is,mh a = T - mh gHere,mh is the mass of the hoop, and g is the acceleration due to gravity. Hence,T = mh (g + a)T = 2.6 (9.8 + 9.8)T = 50.68 N.

7) Time taken by the hoop to fall a distance of 1.57 m:Let h be the distance fallen by the hoop, and t be the time taken to fall this distance. Hence,1/2 mgh = mh g h t/2 = sqrt (2h/g)t = sqrt (2 × 1.57 / 9.8)t = 0.56 s.

8) Magnitude of the velocity of the hoop after it has dropped 1.57 m:Let v be the velocity of the hoop after it has dropped 1.57 m. The final velocity of the hoop is given by,v² - u² = 2ghHere,u is the initial velocity of the hoop, which is 0. Hence,v² = 2ghv² = 2 × 9.8 × 1.57v = 6.2 m/s.

9) Magnitude of the final angular speed of the sphere:Let ω be the final angular speed of the sphere, v1 be its final linear velocity, and R1 be its radius. Since the sphere rolls without slipping,ω = v1 / R1Hence,ω = v / R1Here,v is the linear velocity of the hoop just before it hits the sphere. Hence,v = 6.2 m/sAlso,R1 = 0.21 mω = v / R1ω = 29.5 rad/sThus, the magnitude of the final angular speed of the orange sphere is 29.5 rad/s.

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The dark region between the two rainbows is due to the critical angle at which light rays are reflected away from the observer's eye, and this angle depends on the size of the rain droplets.

When two rainbows form, there is a dark region in-between them because of the critical angle. This critical angle is the minimum angle of incidence beyond which total internal reflection of a light ray occurs from the water droplets in the atmosphere. Because of this angle, the light that reflects from the rain droplets moves away from the observer's eye, so a dark region is formed between the two rainbows.

The light that enters the drop slows down and bends, and the angle of bending is dependent on the color of the light. Red light is bent the least, while violet is bent the most, causing the separation of the colors in a rainbow. The angle of incidence can vary based on the size of the rain droplets, which is why two rainbows can form with different angles of incidence producing the different colors.

Thus, the dark region between the two rainbows is due to the critical angle at which light rays are reflected away from the observer's eye, and this angle depends on the size of the rain droplets.

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When the separation between two objects is tripled and the mass of each object is doubled, the gravitational force between them decreases to (4/9) of its original value. In this case, the force decreases from 100 N to approximately 44.44 N.

The gravitational force between two objects is given by the equation:

Fgrav = G * (M₁ * M₂) / r²,

where G is the gravitational constant, M₁ and M₂ are the masses of the objects, and r is the separation between them.

In this scenario, we have Fgrav = 100 N. If we triple the separation between the objects, the new separation becomes 3r. Additionally, if we double the mass of each object, the new masses become 2M₁ and 2M₂.

Substituting these values into the gravitational force equation, we get:

Fgrav' = G * ((2M₁) * (2M₂)) / (3r)²

      = (4 * G * (M₁ * M₂)) / (9 * r²)

      = (4/9) * Fgrav.

Therefore, the new gravitational force Fgrav' is (4/9) times the original force Fgrav. Substituting the given value Fgrav = 100 N, we find:

Fgrav' = (4/9) * 100 N

      = 44.44 N (rounded to two decimal places).

Hence, the new gravitational force is approximately 44.44 N.

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(a) The angular velocity of the sphere at t = 3.00 s is 45 rad/s.

(b) The angular velocity of the sphere changes direction at t = 0.857 s

(c) The sphere is in rotational equilibrium at t = 0.43 s.

(d) The net torque on the sphere at t = 0.643 s is 4.45 N m.

(e) The rotational kinetic energy of the sphere at t = 0.214 s is 0.273 J.

Radius of sphere, r = 68.0 cm = 0.68 m

Mass of the sphere, m = 1.6 kg

The angular position of sphere, θ(t) = 7t³ − 9t² + 1

(a)

We can differentiate it to obtain its angular velocity:

ω(t) = dθ/dtω(t) = 21t² - 18t

The angular velocity of the sphere at t = 3.00 s is:

ω(3.00) = 21(3.00)² - 18(3.00)

ω(3.00) = 45 rad/s

Therefore, the angular velocity of the sphere at t = 3.00 s is 45 rad/s.

(b)

The angular velocity of the sphere changes direction when:

ω(t) = 0

Therefore,

21t² - 18t = 0

t(21t - 18) = 0

t = 18/21 = 0.857 s

Thus, the angular velocity of the sphere changes direction at t = 0.857 s.

(c)

The sphere is in rotational equilibrium when its angular acceleration is zero:

α(t) = dω/dt

α(t) = 42t - 18 = 0

Thus, t = 0.43 s.

Hence, the sphere is in rotational equilibrium at t = 0.43 s.

(d)

Net torque on the sphere, Τ = Iα

Here, I is the moment of inertia of the sphere, which is given by:

I = (2/5)mr²

I = (2/5)(1.6)(0.68)²

I = 0.397 J s²/rad

The angular acceleration of the sphere at t = 0.643 s is:

α(t) = 42t - 18

α(0.643) = 42(0.643) - 18

α(0.643) = 11.21 rad/s²

The net torque at t = 0.643 s is:

Τ(t) = Iα

Τ(0.643) = (0.397)(11.21)

Τ(0.643) = 4.45 N m

Therefore, the net torque on the sphere at t = 0.643 s is 4.45 N m.

(e)

The rotational kinetic energy of the sphere, Krot = (1/2)Iω²

The angular velocity of the sphere at t = 0.214 s is:

ω(t) = 21t² - 18t

ω(0.214) = 21(0.214)² - 18(0.214)

ω(0.214) = 1.17 rad/s

The rotational kinetic energy at t = 0.214 s is:

Krot = (1/2)Iω²

Krot = (1/2)(0.397)(1.17)²

Krot = 0.273 J

Therefore, the rotational kinetic energy of the sphere at t = 0.214 s is 0.273 J.

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The net force exerted by the conducting plane on the charge, Net force = -q² / [2ε(h+z)²].

Induced charge density on the conducting plane is, Induced charge density = -q / (2πh) where q is the charge and h is the distance of charge q from the grounded conducting plane.

a. The net force exerted on the point charge by the grounded conducting plane:

Given that a point charge q is located at a distance z above a grounded conducting plane, we want to find the net force exerted by the conducting plane on the charge.

We define h as the distance of charge q from the grounded conducting plane. The net force exerted on the point charge by the grounded conducting plane is given by the equation:

F = -q² / [2ε(h+z)²]

where ε represents the permittivity of free space. The negative sign in the expression indicates that the net force exerted by the conducting plane is opposite to the direction of the charge q.

b. The induced charge density on the conducting plane:

The induced charge density can be calculated by,

Induced charge density = -q / (2πh)

This formula provides the charge density induced on the conducting plane as a result of the presence of the point charge q, where q is the charge and h is the distance of charge q from the grounded conducting plane.

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The ball rises to a height of 18.5 meters when it reaches a 25-degree incline.

When the 41 kg metal ball reaches a 25 degree incline, the height it goes to can be calculated. Here's how you can calculate the height of the ball:

First, we will calculate the potential energy of the ball by utilizing the formula: potential energy = mass * gravity * height

PE = mgh

Where m = 41 kg, g = 9.81 m/s² (the acceleration due to gravity), and h is the height in meters.

Since the ball is rolling at 19 m/s on a level surface, its kinetic energy will be:

kinetic energy = 0.5 * mass * velocity²

KE = 0.5 * m * v²

KE = 0.5 * 41 * 19²

KE = 7383.5 J

Now, we will equate the potential energy to the kinetic energy since the energy is conserved:

PE = KE => mgh = 7383.5Jh = 7383.5 / (41 * 9.81)h = 18.5 m

Therefore, the ball rises to a height of 18.5 meters when it reaches a 25-degree incline.

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A copper wire of length 10 ft, with a cross sectional area of 1.0 mm², and a Young’s modulus 10x10¹⁰ N/m² has a weight load hung on it. If its increase in length is 1/8 of inch, the value of the weight is approximately:

d. 800 kg.

To calculate the approximate value of the weight hung on the copper wire, we can use Hooke's Law, which states that the elongation of a material is directly proportional to the applied force.

Hooke's Law formula: F = k * ΔL

Where:

F = Force (weight)

k = Spring constant (Young's modulus)

ΔL = Change in length

Given:

Length of wire (L) = 10 ft = 120 inches

Cross-sectional area (A) = 1.0 mm² = 1.0 × 10⁻⁶ m²

Young's modulus (Y) = 10 × 10¹⁰ N/m²

Change in length (ΔL) = 1/8 inch = 1/8 × 1/12 = 1/96 feet

To find the spring constant (k), we can use the formula:

k = (Y * A) / L

k = (10 × 10¹⁰ N/m²) * (1.0 × 10⁻⁶ m²) / (120 inches)

Now, let's calculate the value of k:

k = (10 × 10¹⁰ N/m²) * (1.0 × 10⁻⁶ m²) / (120 inches)

= 8.33 × 10⁻⁶ N/inch

Now, we can substitute the values into Hooke's Law formula to find the approximate weight:

F = (8.33 × 10⁻⁶ N/inch) * (1/96 feet)

F = 8.33 × 10⁻⁶ N/inch * 96 inches/1 foot

= 8.33 × 10⁻⁶ N/inch * 96

= 0.799 N

To convert the force from Newtons to kilograms, we can divide it by the acceleration due to gravity (g ≈ 9.8 m/s²):

Weight (W) = F / g

W = 0.799 N / 9.8 m/s²

W ≈ 800 kg

Approximately, the value of the weight is 800 kg.

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Elasticity is of significant importance in the everyday lives of businesses and consumers as it helps them understand and respond to changes in prices and demand for goods or services. By considering elasticity, businesses can make informed decisions regarding pricing strategies, production levels, and resource allocation. Consumers, on the other hand, can assess the impact of price changes on their purchasing decisions and adjust their consumption patterns accordingly.

Elasticity, specifically price elasticity of demand, measures the responsiveness of consumer demand to changes in price. It indicates the percentage change in quantity demanded resulting from a one percent change in price. Understanding price elasticity allows businesses to determine how sensitive consumers are to changes in price and adjust their pricing strategies accordingly.

For example, let's consider the market for gasoline. Gasoline is a highly price-sensitive good, meaning that changes in its price have a significant impact on consumer demand. If the price of gasoline increases, consumers may reduce their consumption and seek alternatives such as carpooling or using public transportation. In this scenario, businesses need to consider the price elasticity of gasoline to predict and respond to changes in consumer behavior. They might lower prices to stimulate demand or introduce more fuel-efficient options to cater to price-conscious consumers.

In conclusion, elasticity matters because it provides valuable insights into the dynamics of supply and demand, enabling businesses and consumers to make informed decisions in response to price changes. By understanding elasticity, businesses can adapt their strategies to maintain competitiveness, while consumers can optimize their purchasing choices based on price sensitivity.

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The charge on the plates decreases by a factor of two, and the capacitance decreases by a factor of 2. So, the correct answer is option A.

When the separation between the plates of a parallel plate capacitor is doubled, the capacitance is reduced to half its original value. (Note that only the distance between the plates, not the area, affects capacitance in a parallel plate capacitor.)

The capacitance, C, of a parallel plate capacitor with plate area A and distance d between the plates is given by:

C = ε₀A/d ... [1]

Where ε₀ is the permittivity of free space.

The charge, Q, on a capacitor is given by:

Q = CV ... [2]

Where V is the potential difference across the capacitor.

If the separation distance between the plates is doubled, the capacitance of the capacitor is reduced to half of its original value, as per Equation [1]. If the capacitance of the capacitor reduces to half of its original value while the potential difference V across the capacitor remains constant, the charge Q on the capacitor also decreases to half of its initial value, as per Equation [2].

The charge on the plates decreases by a factor of two, and the capacitance decreases by a factor of 2.

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A parallel-plate capacitor with capacitance Co stores charge of magnitude Qoon plates of area A separated by distance do. (a)Cnew / Co = 200 / Qo(b)AV new / AVo = 200 / Qo(c)Cnew / Co = 2.(d)AV new / AVo = Qo / Qo = 1. (e)Cnew / Co = do / (2do) = 1/2. (f)AV new / AVo = Qnew / Qo = Qo / Qo = 1

(a) The ratio of the new capacitance (Cnew) to the original capacitance (Co) is equal to the ratio of the new charge (Qnew) to the original charge (Qo):

Cnew / Co = Qnew / Qo

Since the charge is doubled to 200, the ratio becomes:

Cnew / Co = 200 / Qo

(b) The ratio of the new potential difference (AV new) to the original potential difference (AVo) is equal to the ratio of the new charge (Qnew) to the original charge (Qo):

AV new / AVo = Qnew / Qo

Since the charge is doubled to 200, the ratio becomes:

AV new / AVo = 200 / Qo

(c) The ratio of the new capacitance (Cnew) to the original capacitance (Co) is equal to the ratio of the new plate area (2A) to the original plate area (A):

Cnew / Co = (2A) / A

Cnew / Co = 2

(d) The ratio of the new capacitance (Cnew) to the original capacitance (Co) is equal to the ratio of the new plate area (2A) to the original plate area (A), and the ratio of the new potential difference (AV new) to the original potential difference (AVo):

Cnew / Co = (2A) / A = 2

AV new / AVo = Qnew / Qo

Since the charge is given as Qo, the ratio becomes:

AV new / AVo = Qo / Qo = 1

(e) The ratio of the new capacitance (Cnew) to the original capacitance (Co) is equal to the ratio of the new distance between the plates (2do) to the original distance between the plates (do):

Cnew / Co = do / (2do) = 1/2

(f) The ratio of the new potential difference (AV new) to the original potential difference (AVo) is equal to the ratio of the new charge (Qnew) to the original charge (Qo):

AV new / AVo = Qnew / Qo = Qo / Qo = 1

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A series RLC circuit has an impedance of 1209 and a resistance of 642.  The average power delivered to the circuit is 12 W (Option E)

Given;

Impedance, Z = 1209 Ω

Resistance, R = 642 Ω

Voltage, Vrms = 90 volts

We are to calculate the average power delivered to the circuit.

P = Vrms2 / R *cos(Φ) ---(1)

Where Φ = angle of phase difference between the current and voltage

Since it is not given whether the circuit is capacitive or inductive or purely resistive, we will have to calculate the value of Φ to determine the nature of the circuit.

Cos(Φ) = R/Z = 642/1209 = 0.531<0.08

Thus, the circuit is inductive (since cos(Φ) is positive and < 1)

We can determine the value of angle Φ using the following equation;

Cos(Φ) = R/ZΦ = cos-1(R/Z)Φ = cos-1(642/1209)Φ = 0.08 rad

Average power delivered to the circuit;

P = Vrms2 / R *cos(Φ)

Substituting the values of Vrms, R and cos(Φ)P = (90)2 / 642 *0.531P = 12.6 W ≈ 12 W

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Part A: The intensity of the wave when it passed a point only 1.5 km from the source is 4.9×1011 J/(m2⋅s).

Part B: The energy passes through at a rate of 3.4×1012 J/s.

The intensity of an earthquake wave passing through the Earth is measured to be 2.0×106 J/(m2⋅s) at a distance of 48 km from the source. We need to find out the following:

Part A: What was its intensity when it passed a point only 1.5 km from the source?

Part B: At what rate did energy pass through an area of 7.0 m2 at 1.5 km?

Part A

The intensity I of the wave is inversely proportional to the square of the distance r from the source. The equation is given by

I1/I2 = (r2/r1)²

Where I1 is the intensity at distance r1, I2 is the intensity at distance r2.

Let's plug in the values

I1 = 2.0×106 J/(m2⋅s), r1 = 48 km = 48000 m, r2 = 1.5 km = 1500 m

I2 = (r1/r2)² × I1

I2 = (48000/1500)² × 2.0×106 J/(m2⋅s)

I2 = 4.9×1011 J/(m2⋅s)

Part B

The rate at which energy is transmitted through a surface area is called the intensity. Intensity is the energy per unit area per unit time. The equation for the intensity is given by

I = P/A

Where P is the power transmitted and A is the area.

Let's plug in the values

I = 4.9×1011 J/(m2⋅s), A = 7.0 m2I = P/PI = A × P/PI = (7.0 m²) × P/tP/t = I/A

Area A = 7.0 m², distance r = 1.5 km = 1500 m

The rate at which energy passes through an area of 7.0 m² at 1.5 km is given by

P/t = (4.9×1011 J/(m²⋅s)) × (7.0 m²)

P/t = 3.4×1012 J/s

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The angular size of the footprints from the new, much closer distance of 50.0 km above the surface of the Moon is 4 × 10¹⁰.

Given data:

Orbital radius/viewing distance, d = 50.0 km = 50.0 × 10³ m

To convert the orbital radius/viewing distance from kilometers to meters, we use the conversion factor:

1 km = 1 × 10³ m

Thus, d = 50.0 × 10³ m

The formula for calculating the angular size of footprints is given below:

θ = d / D

Where,

θ = Angular size of footprints.

d = Distance of telescope from the footprints.

D = Length of the footprints.

The Lunar Reconnaissance Orbiter satellite orbits 50 km above the surface of the Moon. So, the distance of the telescope from the footprints is d = 50.0 × 10³ m.

From Part II, the length of the footprints is D = 1.25 × 10⁻³ m.

Using the above formula, we can calculate the angular size of footprints as:

θ = d / D

θ = (50.0 × 10³) / (1.25 × 10⁻³)

θ = (50.0 × 10³) × (10³ / 1.25)

θ = (50.0 × 10³) × (8 × 10²)

θ = 4 × 10¹⁰

Therefore, the angular size of footprints from this new, much closer distance is 4 × 10¹⁰.

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To determine the volume of a sample of strontium with a given mass, we can use the formula:

Volume = Mass / Density

Given:

Density of strontium (d) = 2.60 g/cm^3

Mass of the sample = 47.2 pounds

Before we proceed, let's convert the mass from pounds to grams, as the density is given in grams per cubic centimeter (g/cm^3).

1 pound is approximately equal to 453.592 grams.

Mass of the sample in grams = 47.2 pounds * 453.592 grams/pound

Now, we can calculate the volume using the formula:

Volume = Mass / Density

Volume = (47.2 * 453.592) / 2.60

By performing the calculations, we can determine the volume of the strontium sample in cubic centimeters.

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Answer:

The correct option is (c) 0.08 A.

To find the current through Ry in the following circuit, we will apply Kirchhoff's Rules.

Kirchhoff's Rules are the basic rules used to analyze a circuit.

There are two rules:

Kirchhoff’s First Law (KCL) and Kirchhoff’s Second Law (KVL).

Kirchhoff’s First Law (KCL) states that the total current entering a junction is equal to the total current leaving the junction.

Kirchhoff’s Second Law (KVL) states that the total voltage around a closed circuit is zero.

For Junction A, the current entering the junction is equal to the current leaving the junction:

For junction B, the current entering the junction is equal to the current leaving the junction:

From the above two equations, we get:

This is equation 1.

We apply Kirchhoff's Second Law to the outer loop as shown below:

This is equation 2

Putting the values of equations 1 and 2, we get:

The current through Ry is:

Ry = R2 || R3

=> Ry = 209*30/(209+30)

=> Ry = 25.14Ω

Iy = 0.0795 A ≈ 0.08

Therefore, the correct option is (c) 0.08 A.

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The inductance of the precision laboratory resistor, when shortened to half its length and with its 500 turns counter-wound, is approximately 7.36 millihenries (mH).

To calculate the inductance of the precision laboratory resistor, we can use the formula for the inductance of a solenoid:

L = (μ₀ * N² * A) / l

Where:

L is the inductance,

μ₀ is the permeability of free space (4π × 10^-7 H/m),

N is the number of turns,

A is the cross-sectional area of the solenoid, and

l is the length of the solenoid.

Given that the original coil has a diameter of 1.55 cm, the radius (r) is half of that, which is 0.775 cm or 0.00775 m. The cross-sectional area (A) of the coil is then:

A = π * r² = π * (0.00775 m)²

The length of the original coil is 3.75 cm or 0.0375 m, and the number of turns (N) is 500.

Substituting these values into the inductance formula:

L = (4π × 10^-7 H/m) * (500²) * (π * (0.00775 m)²) / (0.0375 m)

Simplifying the expression gives:

L = (4π × 10^-7 H/m) * (500²) * (π * 0.00775²) / 0.0375

L ≈ 7.36 × 10^-4 H

Converting to millihenries:

L ≈ 7.36 mH

Therefore, the inductance of the precision laboratory resistor, when shortened to half its length and with its 500 turns counter-wound, is approximately 7.36 millihenries (mH).

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The magnitude is 5000 V/m. The work done by the electric field on the electron is -5 x 10^2 eV or -8 x 10^-17 J. The change in potential energy is -8 x 10^-17 J.The kinetic energy of the electron when it reaches the positive plate will be 8 x 10^-17 J.

a) The magnitude of the electric field between the two plates can be determined using the formula:

E = V / d

where E is the electric field, V is the voltage difference, and d is the distance between the plates.

Given that V = 500 V and d = 10 cm = 0.1 m, we can calculate the electric field:

E = 500 V / 0.1 m = 5000 V/m

b) The work done by the electric field on the electron as it moves from the negative plate to the positive plate can be calculated using the formula:

Work = q * V

where Work is the work done, q is the charge of the electron, and V is the voltage difference.

The charge of an electron is approximately -1.6 x 10^-19 C (coulombs). The voltage difference is given as V = 500 V.

Work = (-1.6 x 10^-19 C) * (500 V) = -8 x 10^-17 J

To express the answer in electron volts (eV), we can convert from joules to electron volts using the conversion factor:

1 eV = 1.6 x 10^-19 J

Work = (-8 x 10^-17 J) / (1.6 x 10^-19 J/eV) = -5 x 10^2 eV

c) The change in potential energy of the electron as it moves from the negative plate to the positive plate is equal to the work done by the electric field. From part (b), we found that the work done is -8 x 10^-17 J.

d) The change in potential energy of the electron is equal to the change in kinetic energy. Therefore, when the electron reaches the positive plate, its kinetic energy will be equal to the magnitude of the change in potential energy.

Since the change in potential energy is -8 x 10^-17 J, the kinetic energy of the electron when it reaches the positive plate will be 8 x 10^-17 J.

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A diverging lens with focal length |f|= 20.0 cm produces an image with a magnification of +0.680. What are the object and image distances? (Include the sign of the value in your answers.) object distance -3.125 cm  image distance  2.125 cm.

To find the object and image distances for a diverging lens, we can use the lens formula:

1/f = 1/do - 1/di

where f is the focal length, do is the object distance, and di is the image distance.

Given:

Focal length (f) = |20.0 cm|

Magnification (m) = +0.680

Since the lens is diverging, the focal length is negative.

We can start by rearranging the lens formula to solve for the image distance:

1/di = 1/f - 1/do

Substituting the given values:

1/di = 1/(-20.0 cm) - 1/do

Simplifying:

1/di = -1/20.0 cm - 1/do

Next, we can substitute the magnification formula into the equation:

m = -di/do

Substituting the given magnification:

0.680 = -di/do

Now we have two equations:

1/di = -1/20.0 cm - 1/do

0.680 = -di/do

We can solve these equations simultaneously to find the object and image distances.

From equation (1):

1/di = -1/20.0 cm - 1/do

Multiplying through by do*di:

do*di = -do - 20.0 cm * di

From equation (2):

0.680 = -di/do

Rearranging:

di = -0.680 * do

Substituting the expression for di in equation (1):

do*(-0.680 * do) = -do - 20.0 cm * (-0.680 * do)

Simplifying:

-0.680 * do² = -do + 20.0 cm * do²

Rearranging and combining like terms:

0.680 * do² - do² = do

Simplifying further:

-0.320 * do² = do

Dividing through by do:

-0.320 * do = 1

Solving for do:

do = 1 / -0.320

do ≈ -3.125 cm

Substituting the value of do into the expression for di:

di = -0.680 * (-3.125 cm)

di ≈ 2.125 cm

Therefore, the object distance is approximately -3.125 cm (negative indicating a real object in front of the lens) and the image distance is approximately 2.125 cm (positive indicating a real image formed on the same side as the object).

object distance.

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a) The correct answer is C. A series of bright and dark fringes with the central bright fringe being wider and brighter than the other bright fringes.

b) The correct answer is A. The two waves reach their maximum value at the same time and their minimum value at the same time.

The brilliant middle fringe is a result of light's beneficial interference. The two light sources (slits) are symmetrically closest to the centre fringe as well. As one walks out from the core, the fringes continue to progressively become darker and the central fringe is the brightest.

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The magnetic field at the center of a single circular loop of wire or radius 10 cm carrying a current of 2.5 A is             3.14 × 10-5 T.

Magnetic field at the center of a single circular loop of wire or radius 10 cm carrying a current of 2.5 A can be calculated using the formula;

B=μ0I/2R

where B is the magnetic field, I is the current flowing, R is the radius of the loop and μ0 is the permeability of free space.The given values are;I = 2.5 AR = 10 cm = 0.1 mμ0 = 4π × 10-7 T m/A.

Substitute the values into the formula; B = μ0I/2R = (4π × 10-7 T m/A) × (2.5 A)/2(0.1 m)= 3.14 × 10-5 T

Therefore, the magnetic field at the center of a single circular loop of wire or radius 10 cm carrying a current of 2.5 A is 3.14 × 10-5 T.

Answer: 3.14×10^−5T.

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a. The resistance of 100 meters of #18 AWG Copper wire at 20°C is 0.2098 Ω

b. To calculate the resistance of a wire, the cross-sectional area of the wire is required.

c. The area required to calculate the resistance is 2.155 × [tex]10^{-10}[/tex] m². The resistance of copper wire at 100°C is 92.2 Ω.

a. The resistance of 100 meters of #18 AWG Copper wire at 20°C can be determined using the formula;

R = ρL/A

A = πr²ρ

where;

R = resistance

ρ = resistivity

L = length of the wire

A = area of cross-section

r = radius of the wire

Substituting the given values;

Length of wire L = 100 meters

Area of cross-section A = ?

Diameter of wire d = 0.0403 inches or 1.02462 mm

Cross-sectional area A = πd²/4 = π(1.02462 mm)²/4 = 0.8231 mm²

Resistivity ρ = 1.724 x [tex]10^{-8}[/tex] Ω-m (at 20°C for copper)

Thus;

R = ρL/A = 1.724 x [tex]10^{-8}[/tex] Ω-m x 100 meters / 0.8231 mm²R = 0.2098 Ω

a. The resistance of 100 meters of #18 AWG Copper wire at 20°C is 0.2098 Ω

b. To calculate the resistance of a wire, the cross-sectional area of the wire is required.

c. To find the resistance of 600 meters of solid Copper wire with a diameter of 5 mm, we need to know the cross-sectional area of the wire. The formula for the cross-sectional area is;

A = πr²A = π(5/2)²A = 19.63 mm²

The resistivity of copper is 1.724 × [tex]10^{-8}[/tex] Ωm. Using the formula;

R = ρL/A

where;

L = 600 mA = 19.63 mm²

ρ = 1.724 × [tex]10^{-8}[/tex] Ωm

R = 0.16 ΩP.

To find the area required to calculate the resistance, the cross-sectional area of the wire is required. If the resistance of copper wire is 80 ohms at 20°C, we can use the above formula for resistivity.

ρ = RA/L

where;

R = 80 Ω

A = ?

L = 1 m

ρ = 1.724 × [tex]10^{-8}[/tex] Ωm

A = ρL/R = 1.724 × [tex]10^{-8}[/tex] × 1/80A = 2.155 × [tex]10^{-10}[/tex] m²

The resistance of copper wire at 100°C can be determined using the formula;

Rt = R0 [1 + α(T[tex]_{t}[/tex] - T[tex]_{0}[/tex])]

where;

R0 = resistance at 20°C = 80 Ω

T0 = temperature at 20°C = 293 K (20 + 273)

Tt = temperature at 100°C = 373 K (100 + 273)

α = temperature coefficient of copper = 0.00393/°C

Rt = 80 [1 + 0.00393(373 - 293)]R[tex]_{t}[/tex] = 92.2 Ω

Answer:

Therefore area required to calculate the resistance is 2.155 × [tex]10^{-10}[/tex] m². The resistance of copper wire at 100°C is 92.2 Ω.

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The person must point the boat in the direction measured from north at an angle of approximately 59.1 degrees to the west (clockwise direction) so that the boat goes straight across the river traveling due north. To determine the direction in which the person must point the boat, we need to consider the vector components of the boat's velocity and the river's velocity.

Let's define the x-axis as pointing east and the y-axis as pointing north. The river's velocity is given as 2.00 m/s in the positive x-direction (west to east). The person wants the boat to travel due north, which means the boat's velocity in the y-direction should be 3.47 m/s.

To achieve this, the boat's x-component of velocity must be equal and opposite to the river's velocity. In other words, the x-component of the boat's velocity should be -2.00 m/s.

Now, we can use vector components to find the direction in which the person must point the boat. The boat's velocity vector can be represented as the sum of its x-component and y-component:

[tex]V_{boat[/tex] =[tex]V_x[/tex]î +[tex]V_y[/tex]ĵ

Given that [tex]V_x[/tex] = -2.00 m/s and [tex]V_y[/tex] = 3.47 m/s, the boat's velocity vector can be written as:

[tex]V_{boat[/tex]= (-2.00 î) + (3.47 ĵ)

To find the direction of the boat's velocity, we can calculate the angle it makes with the positive y-axis (north). The angle θ is given by:

θ =[tex]tan^(-1)(V_y/V_x)[/tex]

θ = [tex]tan^(-1[/tex])(3.47/-2.00)

Using a calculator, we find θ ≈ -59.1 degrees.

Therefore, the person must point the boat in the direction measured from north at an angle of approximately 59.1 degrees to the west (clockwise direction) so that the boat goes straight across the river traveling due north.

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When two people pull a rock in different directions with forces of 450 N and 300 N, vector addition shows that the resultant force is 375 N directed south-southeast.

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Answer: Reflecting telescopes focus light with a series of mirrors, while refracting telescopes use lenses.

Explanation:

Therefore, the change in electric potential energy is $1.62 \times 10^{10} J$, which is approximately $1.6 \times 10^{10} J$.Hence, the correct option is $1.6 \times 10^{10} J$.

Electric potential energy is calculated using the formula :$E_{p}=k \frac{q_{1} q_{2}}{r}$where,$k$ is Coulomb's constant, $9 \times 10^9 Nm^2/C^2$$q_1$ is the magnitude of charge 1$q_2$ is the magnitude of charge 2$r$ is the distance between the chargesFrom the above formula,$E_{p} \propto \frac{1}{r}$ which implies that when the distance between the two charges decreases, the electric potential energy will increase.

The change in electric potential energy, $\Delta E_{p}$ can be calculated using the formula,$\Delta E_{p} = E_{p final} - E_{p initial}$Given,$q_{1} = 1C$$q_{2} = 0.2C$$r_{initial} = 1m$$r_{final} = 0.1m$Let's find the initial electric potential energy:$E_{p initial} = k \frac{q_{1} q_{2}}{r_{initial}}$$E_{p initial} = 9 \times 10^9 \frac{(1)(0.2)}{1}$$E_{p initial} = 1.8 \times 10^9 J$Now,

let's find the final electric potential energy:$E_{p final} = k \frac{q_{1} q_{2}}{r_{final}}$$E_{p final} = 9 \times 10^9 \frac{(1)(0.2)}{0.1}$$E_{p final} = 1.8 \times 10^{10} J$The change in electric potential energy is $\Delta E_{p} = E_{p final} - E_{p initial}$$\Delta E_{p} = (1.8 \times 10^{10}) - (1.8 \times 10^9)$$\Delta E_{p} = 1.62 \times 10^{10} J$

Therefore, the change in electric potential energy is $1.62 \times 10^{10} J$, which is approximately $1.6 \times 10^{10} J$.Hence, the correct option is $1.6 \times 10^{10} J$.

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The curve-fit function (1-²) ¹/n V₂ R 2, max is commonly used to approximate the behavior of turbulent flow in smooth circular tubes. The values of n vary depending on the Reynolds number (Re) of the flow. Near Re-4x10³, n is approximately 6; near Re-1.1x105, n is around 7; and near 3.2x10^6, n is approximately 10. This function helps to describe the relationship between velocity (V), radius (R), and the maximum radius (R 2, max) in turbulent flow conditions.

The given curve-fit function (1-²) ¹/n V₂ R 2, max represents a relationship observed in turbulent flow within smooth circular tubes. The function involves three variables: velocity (V), radius (R), and the maximum radius (R 2, max).

The term (1-²) ¹/n represents the ratio of the difference between the maximum radius (R 2, max) and the radius (R) to the maximum radius raised to the power of 1/n. This term accounts for the influence of the radius on the behavior of the turbulent flow.

The values of n vary depending on the Reynolds number (Re) of the flow. Near Re-4x10³, the value of n is approximately 6, indicating a certain relationship between the variables in this range. Near Re-1.1x105, the value of n is approximately 7, and near 3.2x10^6, the value of n is approximately 10. These different values of n reflect the changing behavior of turbulent flow at different Reynolds numbers.

Overall, the given curve-fit function helps approximate the relationship between velocity, radius, and the maximum radius in turbulent flow conditions, with different values of n accounting for the varying behavior at different Reynolds numbers.

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You pull downward with a force of 31 N on a rope that passes over a disk-shaped pulley of mass of 1.5 kg and a radius of 0.075 m . Therefore, the tension in the rope on both sides of the pulley is:T1 = 25.155 N and T2 = 15.345 N

When a 31N force is applied to a rope that passes over a disk-shaped pulley of mass of 1.5 kg and a radius of 0.075 m, the tension in the rope on both sides of the pulley is as follows:

T1 = (m1g + T2)/(1)T2 = (m2g - T1)/(2)

Where,m1=1.5 kgm2=0.77 kg T1 = tension in the rope on the side with the mass m1, T2 = tension in the rope on the side with the mass m2g = acceleration due to gravity = 9.81 m/s²

T1:T1 = (m1g + T2)/(1)T1 = (1.5 kg × 9.81 m/s² + T2)/(1)

Substitute the given value for T2:31 N = (1.5 kg × 9.81 m/s² + T2)/(1)T2 = (31 N - 1.5 kg × 9.81 m/s²)T2 = 15.345 N

Therefore, T1 = (1.5 kg × 9.81 m/s² + 15.345 N)/(1)T1 = 25.155 N

Therefore, the tension in the rope on both sides of the pulley is:T1 = 25.155 N and T2 = 15.345 N

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47 degrees is equal to 0.8203 radians.

To convert degrees to radians, we can use the following conversion formula:

radians = (degrees * π) / 180

Where:

degrees is the measurement in degrees

π (pi) is a mathematical constant approximately equal to 3.14159

To convert 47 degrees into radians, we will use the following formula;

Radian = (Degree × π) / 180 Where π = 3.14 radians

47 degrees is given, so we can substitute it into the formula:

Radian = (Degree × π) / 180

Radian = (47 × 3.14) / 180

Radian = 0.8203 radians

Therefore, 47 degrees is equal to 0.8203 radians.

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