A 2-m telescope can collect a given amount of light in 1 hour. Under the same observing conditions, how much time would be required for a 6-m telescope to perform the same task? A 12-m telescope?

Answer:

(a) Surface charge density is the charge per unit area.

[tex]\sigma = 1.46 \times 10^{-7}~{\rm Q/m^2}[/tex]

(b) [tex]\vec{E} = (+\^z)~8.34\times 10^3~{\rm N/C}[/tex]

(c) [tex]\vec{E} = (-\^z)~8.34\times 10^3~{\rm N/C}[/tex]

Explanation:

(a) Surface charge density is the charge per unit area. The area of the square plate can be calculated by its side length.

[tex]A = l^2 = (0.4)^2 = 0.16 ~{\rm m^2}[/tex]

Half of the total charge is distributed on one side and the other is distributed on the other side.

Therefore, surface charge density on each face of the plate is

[tex]\sigma = Q/A = \frac{2.35 \times 10^{-8}}{0.16} = 1.46 \times 10^{-7}~{\rm Q/m^2}[/tex]

(b) To find the electric field just above the plate, Gauss' Law can be used. Normally, Gauss' Law can only be used in infinite sheet (considering the flat surfaces), but just above the surface can be considered that the distance from the surface is much much smaller than the length of the plate (x << l).

In order to apply Gauss' Law, we have to draw an imaginary cylinder with radius r. The cylinder has to stay perpendicular to the plane.

[tex]\int\vec{E}d\vec{a} = \frac{Q_{enc}}{\epsilon_0}\\E2\pi r^2 = \frac{\pi r^2 \sigma}{\epsilon_0}\\E = \frac{\sigma}{2\epsilon_0}\\E = \frac{1.46 \times 10^{-7}}{2\times 8.8\times 10^{-12}} = 8.34\times 10^3~{\rm N/C}[/tex]

The direction of the electric field is in the upwards direction.

(c) The magnitude of the electric field is the same as that of upper side. Only the direction is reversed, downward direction.

Final answer:

The amplitude of the oscillator is 1.989 m.

Explanation:

The amplitude of an oscillator can be determined using the initial displacement and initial velocity of the system. In this case, the initial displacement is given as 1.00 m and the initial velocity as 1.72 m/s. The amplitude, also known as the maximum displacement, is equal to the square root of the sum of the squares of the initial displacement and initial velocity.

Using the formula:

X = √(x₀² + v₀²)

Where X is the amplitude, x₀ is the initial displacement, and v₀ is the initial velocity.

Substituting the given values into the formula:

X = √(1.00² + 1.72²)

= √(1 + 2.9584)

= √3.9584

= 1.989 m

The amplitude of the oscillator is 1.989 m.

Answer:

76.3 m

Explanation:

We are given that

Initial speed of the ball,u=(15+A)m/s

Height of cliff,h=(25.9+B) m

We have to find the distance from the base of the cliff the ball will land in the water below.

A=4 and B=54

Distance=[tex]u\sqrt{\frac{2h}{g}}[/tex]

Using the formula and substitute the values

[tex]D=(15+4)\sqrt{\frac{2(25+54)}{9.8}}[/tex]

Because [tex]g=9.8m/s^2[/tex]

[tex]D=19\sqrt{\frac{158}{9.8}}[/tex]

D=76.3

Hence, the distance from the base of the cliff the ball will land in the water below=76.3 m

Answer:

[tex]a=-3449.67\frac{m}{s^2}[/tex]

Explanation:

The car is under an uniforly accelerated motion. So, we use the kinematic equations. We calculate the acceleration from the following equation:

[tex]v_f^2=v_0^2+2ax[/tex]

We convert the initial speed to [tex]\frac{m}{s}[/tex]

[tex]39\frac{km}{h}*\frac{1000m}{1km}*\frac{1h}{3600s}=10.83\frac{m}{s}[/tex]

The car stops, so its final speed is zero. Solving for a:

[tex]a=\frac{v_0^2}{2x}\\a=-\frac{(10.83\frac{m}{s})^2}{2(1.7*10^{-2}m)}\\a=-3449.67\frac{m}{s^2}[/tex]

To calculate the acceleration of the car, use the formula a = (vf - vi) / t. Convert the initial velocity and distance to the appropriate units before substituting them into the formula.

To calculate the acceleration of the car, we can use the formula:

a = (vf - vi) / t

Where a is the acceleration, vf is the final velocity, vi is the initial velocity, and t is the time taken. In this case, the initial velocity is 39 km/h, which is converted to m/s by vi = 39 km/h * (1000 m/1 km) * (1 h/3600 s). The final velocity is 0 m/s since the car stops. The time taken can be found by t = d / vi, where d is the distance and vi is the initial velocity. The distance is given as 1.7 cm, which is converted to m by d = 1.7 cm * (1 m/100 cm). Substituting these values into the formula, we get the acceleration of the car.

So the acceleration of the car is approximately -8.45 m/s^2.

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Answer:

E= 11.3 N/C.

Explanation:

The electric filed at any point on the z-axis is given by the formula

[tex]E= \frac{\sigma}{2\epsilon_0}[/tex]

here, sigma is the charge density and ε_o is the permitivity of free space.

therefore,

[tex]E= \frac{0.2\times10^{-9}}{2\times8.85\times10^{-12}}[/tex]

solving it we get

E= 11.3 N/C.

Hence, the required Electric field is E= 11.3 N/C.

Final answer:

The speed of the putty just before it strikes the ceiling is approximately 0.342 m/s, and it takes approximately 0.97 seconds to reach the ceiling.

Explanation:

To calculate the speed of the putty just before it strikes the ceiling, we can use the equation for vertical motion:

v = u + gt

Where:

v is the final velocity

u is the initial velocity

g is the acceleration due to gravity (-9.8 m/s^2)

t is the time

In this case, the object is thrown upwards, so the final velocity when it reaches the ceiling will be zero. The initial velocity is 9.50 m/s and the acceleration due to gravity is -9.8 m/s^2. Substitute these values into the equation and solve for t:

0 = 9.50 - 9.8t

t = 0.97 seconds

Therefore, the speed of the putty just before it strikes the ceiling is 9.50 - 9.8(0.97) = -0.342 m/s. Since speed cannot be negative, the actual speed is 0.342 m/s.

To calculate the time it takes for the putty to reach the ceiling, we can use the same equation and solve for t:

0 = 9.50 - 9.8t

t = 0.97 seconds

So, it takes the putty approximately 0.97 seconds to reach the ceiling.

Final answer:

The speed of the putty just before it strikes the ceiling is 9.50 m/s, the same as its initial speed due to conservation of energy. The time it takes for the putty to reach the ceiling can be calculated using kinematic equations.

Explanation:

The problem presented concerns the motion of a glob of putty thrown upward and involves calculations based on the principles of kinematics in physics.

Answer to Part (a): What is the speed of the putty just before it strikes the ceiling?

The speed of the putty just before it strikes the ceiling can be found by using the kinematic equation for velocity under constant acceleration (gravity). Since the putty is thrown upwards, it will slow down under the influence of gravity until it reaches its maximum height. At this point, it will start to fall back down, accelerating under gravity until it hits the ceiling. Assuming no energy loss, the speed of the putty just before it strikes the ceiling will be the same as its initial speed when it leaves the hand, which is 9.50 m/s.

Answer to Part (b): How much time from when it leaves your hand does it take the putty to reach the ceiling?

The time it takes for the putty to reach the ceiling can be calculated using the kinematic equation s = ut + (1/2)at², where s is the displacement, u is the initial velocity, a is the acceleration due to gravity (approximately -9.81 m/s² when upwards is considered positive), and t is the time. Solving for t gives us the time taken for the putty to reach the height of 3.60 m.

The speedometer of a car measures speed, which is a scalar quantity indicating how fast the car is moving without regard to direction. Velocity, on the other hand, includes both speed and direction, which the speedometer does not display. The odometer measures total distance traveled, not displacement, and dividing distance by time gives average speed, not velocity.

The speedometer of a car measures speed, not velocity. Speed is a scalar quantity, which means it only describes how fast an object is moving regardless of its direction. On the other hand, velocity is a vector quantity that describes both the speed and the direction of an object's movement. For example, if a car is moving at 60 miles per hour (mph), the speedometer shows this speed, but it does not indicate whether the car is traveling north, south, east, or west – that would be necessary information to determine the car's velocity.

A car's odometer, in contrast, measures the total distance traveled by the car. This distance is a scalar quantity as well, which means it does not account for the direction of travel, only the cumulative distance covered. When you divide the total distance traveled, as shown on the odometer, by the total time taken for the trip, you are calculating the average speed of the car, not the magnitude of average velocity. These two quantities – average speed and the magnitude of average velocity – are the same when the car moves in a straight line without changing its direction.

Answer:

elastic force and weight are related to the acceleration of the System.

Explanation:

The relationship between these two forces can be found with Newton's second law.

        [tex]F_{e}[/tex] - W = m a

        K x - m g = m a

We see that elastic force and weight are related to the acceleration of the System.

If a harmonic movement is desired, an extra force that increases the elastic force is applied, but to begin the movement this force is eliminated, in general , if the relationship between this external and elastic force is desired, the only requirement is that it be small for harmonic movement to occur

The relationship between the applied force of a hanging mass and the spring force of the spring is governed by Hooke's law, where the force is proportional to displacement. At equilibrium, the spring force balances the gravitational force, and any displacement results in harmonic oscillation.

The relationship between the applied force of a hanging mass on a spring and the spring force is described by Hooke's law. When a mass is attached to a vertical spring, it stretches until the spring force equals the force of gravity on the mass. This point is called the equilibrium position. The spring force at this point is k times the displacement from the spring's unstretched length, where k is the spring constant. If the mass is displaced from this equilibrium position, a restoring force acts to return the mass to equilibrium, leading to harmonic oscillation.

In the context of springs, two scenarios arise. When the spring is relaxed, there is no force on the block, indicating an equilibrium situation. Upon stretching or compressing the spring, Hooke's law predicts the force on the block as being proportional to the displacement from its equilibrium position, x, with force F being -Kx. Here, -K represents the negative spring constant, signifying a restoring force opposite to the direction of displacement.

For a massive continuous spring with negligible gravitational effect compared to the spring force (KL » mg), when the top of the spring is driven up and down, the position of the bottom can be described as a function of time. This shows that both vertical and horizontal spring-mass systems follow similar dynamics and obey the laws governing harmonic motion.

The volumetric rate or flow rate of a fluid is defined as the amount of the volume of a fluid circulating on a surface per unit of time. In this case we have units given initially: Gallons and minutes. For the first part we will convert the minutes to seconds, and we will obtain the flow rate under that measure. For the second case we will convert the gallons to cubic meters and obtain the desired value. Recall the following conversion rates,

[tex]1 min = 60s[/tex]

[tex]1 U.S Gal = 0.00378541178 m^3[/tex]

If the flow rate is defined as the volume by time, the flow rate with the given values is

[tex]Q = \frac{V}{t}[/tex]

[tex]Q = \frac{50Gal}{8min}[/tex]

[tex]Q = 6.25 Gal/min[/tex]

PART A ) Converting to Gal/seconds, we have,

[tex]Q = 6.25 \frac{Gal}{min}(\frac{1min}{60s})[/tex]

[tex]Q = 0.10416Gal/s[/tex]

PART B) Converting Gal/seconds to [tex]m^3/s[/tex]

[tex]Q = 0.104116\frac{Gal}{s} (\frac{0.00378541178 m^3}{1 Gal})[/tex]

[tex]Q = 3.941*10^{-4}m^3/s[/tex]

The rate of filling the gasoline tank is 0.104 gallons per second or approximately 0.000383 cubic meters per second.

To calculate the rate at which the gasoline tank is being filled, we need to first convert the given quantities into the relevant units. Given that the gasoline tank is 50.0 gallons and it takes 8.00 minutes to fill it, the flow rate is 50/480 (since 8 min = 480 s) = 0.104 gallons/second. Since 1 US gallon = 231 cubic inches, this gives us a flow rate of 0.104 x 231 = 23.5 cubic inches per second.

To convert this to the rate in cubic meters per second, we use the fact that 1 inch = 0.254 cm and 1 cubic meter = 1,000,000 cubic cm. Therefore, 23.5 cubic inches = 23.5 x 0.254^3 cubic meters = approximately 0.000383 cubic meters per second (m^3/s).

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Answer:

The magnitude of the electric field be 171.76 N/C so that the electron misses the plate.

Explanation:

As data is incomplete here, so by seeing the complete question from the search the data is

vx_0=1.1 x 10^6

ax=0 As acceleration is zero in the horizontal axis so

Equation of motion in horizontal direction is given as

[tex]s_x=v_x_0 t[/tex]

[tex]t=\frac{s_x}{v_x}\\t=\frac{2 \times 10^{-2}}{1.1 \times 6}\\t=1.82 \times 10^{-8} s[/tex]

Now for the vertical distance

vy_o=0

than the equation of motion becomes

[tex]s_y=v_y_0 t+\frac{1}{2} at^2\\s_y=\frac{1}{2} at^2\\0.5 \times 10^{-2}=\frac{1}{2} a(1.82 \times 10^{-8})^2\\a=3.02 \times 10^{13} m/s^2[/tex]

Now using this acceleration the value of electric field is calculated as

[tex]E=\frac{F}{q}\\E=\frac{ma}{q}\\E=\frac{m_ea}{q_e}\\[/tex]

Here a is calculated above, m is the mass of electron while q is the charge of electron, substituting values in the equation

[tex]E=\frac{9.1\times 10^{-31} \times 3.02 \times 10^{13} }{1.6 \times 10^{-19}}\\E=171.76 N/C[/tex]

So the magnitude of the electric field be 171.76 N/C so that the electron misses the plate.

If the electron misses the upper plate, the magnitude of the electric field is equal to 171.88 Newton per coulomb.

Given the following data:

Scientific data:

To calculate the magnitude of the electric field:

First of all, we would determine the time taken by this electron to travel through the plates.

Mathematically, time is given by this formula:

[tex]Time = \frac{distance}{speed} \\ \\ Time = \frac{0.02}{1.10 \times 10^6} \\ \\ Time = 1.82 \times 10^{-8}\;m/s[/tex]

Next, we would find the acceleration of the electron in the vertical direction by using this formula:

[tex]a=\frac{2y}{t^2} \\ \\ a=\frac{2 \times 0.005}{(1.82 \times 10^{-8})^2}\\ \\ a=\frac{0.01}{3.31 \times 10^{-16}}\\ \\ a=3.02 \times 10^{13}\;m/s^2[/tex]

Mathematically,  the electric field is given by this formula:

[tex]E=\frac{ma}{q}[/tex]

Where:

Substituting the given parameters into the formula, we have;

[tex]E=\frac{9.1 \times 10^{-31} \times 3.02 \times 10^{13}}{1.6 \times 10^{-19}}\\ \\ E=\frac{2.75 \times 10^{-17}}{1.6 \times 10^{-19}}[/tex]

Electric field, E = 171.88 N/C.

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Complete Question:

An electron is projected with an initial speed v0 = [tex]1.6 \times 10^6[/tex] m/s into the uniform field between the parallel plates. The distance between the plates is 1 cm and the length of the plates is 2 cm. Assume that the field between the plates is uniform and directed vertically downward, and that the field outside the plates is zero. The electron enters the field at a point midway between the plates. E = N/C

(a) If the electron just misses the upper plate as it emerges from the field, find the magnitude of the electric field.

Answer:

Final velocity of the elevator will be 4.453 m/sec

Explanation:

Let mass is m

Acceleration due to gravity is g m/sec^2

Distance s = 2.2 m

As the elevator is moving upward so net force on elevator

[tex]F=mg+ma[/tex]

So according to question

[tex]1.46mg=mg+ma[/tex]

0.46 mg = ma

a = 0.46 g

a = 0.46×9.8 = 4.508 [tex]m/sec^2[/tex]

Initial velocity of elevator is 0 m/sec

From third equation of motion

[tex]v_f^2=v_i^2+2as[/tex]

[tex]v_f^2=0^2+2\times 4.508\times 2.2[/tex]

[tex]v_f=4.453m/sec[/tex]

So final velocity of the elevator will be 4.453 m/sec

Answer:

a. They both reach at the same time.

Explanation:

On a frictionless incline, the only force that moves the person downwards is the x-component of the persons weight. (x-direction is the direction along the incline.)

[tex]F = mg\sin(\theta)[/tex]

Here, θ is the angle of the incline above horizontal.

This force is equal to 'ma' according to Newton's Second Law.

Comparing the weights of the two persons gives

[tex]F_1 = 85g\sin(\theta) = 85a_1\\F_2 = 75g\sin(\theta) = 75a_1\\a_1 = g\sin(\theta)\\a_2 = g\sin(\theta)[/tex]

Since the accelerations of both persons are the same, they reach the bottom at the same time.

The crucial point here is that the acceleration on a frictionless incline is independent from the mass of the object. If there were friction on the surface, then the person with smaller mass would reach the bottom first.

Answer:

True

Explanation:

The total mass of all the planets is much less than the mass of the Sun is a True statement.

Sun contributes to most of the mass of the solar system. Sun contributes to 99.8% mass of the solar system. Only 0.2% of the mass of the solar system is given by the planets. Hence, the above statement is absolutely correct.

Answer:

[tex]|R|=4.373km[/tex]

Explanation:

Given data

For first plate let it be p₁

[tex]p_{z1}=1162 m\\ p_{x1}=10.1km\\\alpha _{1}=34.2^{o}[/tex]

For second plate let it be p₂

[tex]p_{z2}=4162 m\\p_{x2}=9.5km\\\alpha _{2}=21.5^{o}[/tex]

To find

Distance R between them

Solution

To find distance between two plates first we need to find p₁ and p₂

Finding p₁

According to vector algebra

[tex]p_{1}=p_{x1}i+p_{y1}j+p_{z1}k\\ as\\tan\alpha =tan(34.2^{o} )=(p_{y1}/p_{x1})\\p_{y1}=10.1tan(34.2^{o} )\\p_{y1}=6.864km[/tex]

So we get

[tex]p_{1}=-10.1i-6.864j+1.162k[/tex]

Now to find p₂

[tex]p_{2}=p_{x2}i+p_{y2}j+p_{z2}k\\ as\\tan\alpha =tan(21.5^{o} )=(p_{y2}/p_{x2})\\p_{y2}=9.5tan(21.5^{o} )\\p_{y2}=3.74km[/tex]

So we get

[tex]p_{2}=-9.5i-3.74j+4.162k[/tex]

Now for distance R

According to vector algebra the position vector R between p₁ and p₂

[tex]R=p_{1}-p_{2}\\ R=(p_{x1}-p_{x2})i+(p_{y1}-p_{y2})j+(p_{z1}-p_{z2})k\\R=(-10.1-(-9.5))i+(-6.864-(-3.74))j+(1.162-4.162)k\\R=-0.6i-3.124j-3k\\|R|=\sqrt{(0.6)^{2}+(3.124)^{2}+(3)^{2} }\\ |R|=4.373km[/tex]

To find the distance between two planes, calculate the horizontal distances using the given angles and distances. Use the Pythagorean theorem to find the distance by applying the formula. The distance between the two planes is approximately 14.1 km.

To find the distance between the two planes, we can use the Pythagorean theorem. First, we calculate the horizontal distances using the given angles and distances. For the first plane, the horizontal distance is 10.1 km * cos(34.2°), and for the second plane, it is 9.5 km * cos(21.5°). Next, we use the Pythagorean theorem to find the distance between the two planes: d = √((9.5 km * cos(21.5°))^2 + (10.1 km * cos(34.2°))^2).

Calculating the values, we get d ≈ 14.1 km.

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Answer:

0.00461031264 m/s

Explanation:

G = Gravitational constant = 6.67 × 10⁻¹¹ m³/kgs²

M = Mass of the Earth =  6 × 10²⁴ kg

r = Distance between Earth and Sun = [tex]1.5\times 10^{11}\ m[/tex]

t = Time taken = 3 days

Acceleration is given by

[tex]a=\dfrac{GM}{r^2}\\\Rightarrow a=\dfrac{6.67\times 10^{-11}\times 6\times 10^{24}}{(1.5\times 10^{11})^2}\\\Rightarrow a=1.77867\times 10^{-8}\ m/s^2[/tex]

Velocity of the star

[tex]v=u+at\\\Rightarrow v=0+1.77867\times 10^{-8}\times 3\times 24\times 60\times 60\\\Rightarrow v=0.00461031264\ m/s[/tex]

The Sun's speed will be 0.00461031264 m/s

Answer:

Explanation:

Given

Force applied [tex]F=80\ N[/tex]

Door is [tex]d=3\ m[/tex] wide

for gate to revolve Properly Pivot Point must be at center i.e. 1.5 from either end

Torque applied is [tex]T=force\times distance[/tex]

Maximum torque

[tex]T_{max}=F\times \frac{d}{2}[/tex]

[tex]T_{max}=80\times \frac{3}{2}[/tex]

[tex]T_{max}=120\ N-m[/tex]

Final answer:

The pivot point of a revolving door is at its central axis, and the maximum torque exerted by the businesswoman on the revolving door is 120 Nm, calculated by multiplying the force (80 N) by the perpendicular distance from the pivot point to the point of application (1.5 m).

Explanation:

The question is asking about the concept of torque, which is a measure of the turning force on an object. In the scenario described, a businesswoman is applying a force of 80 N to a revolving door. The maximum torque can be calculated by multiplying the force applied by the perpendicular distance from the pivot point to the point of application. The pivot point of a revolving door is its central axis, which is the center of the door.

To calculate the maximum torque, we use the formula:

Torque = Force * Distance.

Here, Torque = 80 N * 1.5 m (since the force is applied at the edge of the 3 m wide door, the distance from the pivot point is half the width).

Thus, Torque = 120 Nm

This is the maximum torque exerted by the woman on the door relative to its central.

Answer:

725.2‬ N

Explanation:

Since it is not stated the scale, the person or both accelerated or experience weightlessness, the net force acting on the bathroom scale is the weight of the person acting downward as the person stands on the scale .

                       Weight = mass of a body × acceleration due to gravity

                                    = 74 kg × 9.8 m/s²

                                    = 725.2‬ N    

Answer:

a,b and c are true.

Explanation:

Following are true statements

a. Electric field lines and Equipotential surfaces are always mutually perpendicular is  a true statement.

b. When all charges are at rest, the surface of a conductor is always an equipotential surface.

c. An equipotential surface is a three-dimensional surface on which the electric potential is the same at every point.

Following are False statements

d. The potential energy of a test charge increases as it moves along an equipotential surface.

e. The potential energy of a test charge decreases as it moves along an equipotential surface.

Reason: A t any point in an equipotential surface, the potential is same throughout. There is no increase or decrease in potential energy as the test charge moves in an equipotential environment.

Statements a, b, and c are correct: Electric field lines and equipotential surfaces are always mutually perpendicular; when all charges are at rest, the surface of a conductor is always an equipotential surface; and an equipotential surface is a three-dimensional surface on which the electric potential is the same at every point. Statements d and e are not correct because the potential energy of a test charge does not change as it moves along an equipotential surface.

The following statements are true about electrical fields and potential:

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To find the spring constant (k) of the spring, the potential energy of the falling safe is equated with the elastic potential energy of the spring at maximum compression. The calculated spring constant is approximately 20641 N/m.

To determine the spring constant of the spring, we can use the conservation of energy principle. Since the safe falls from a height onto the spring and compresses it, we can equate the potential energy of the safe at the height to the elastic potential energy stored in the spring at maximum compression.

The potential energy (PE) of the safe when it is above the spring is given by PE = mgh, where m is the mass (1100 kg), g is the acceleration due to gravity (9.81 m/s2), and h is the height (2.4 m). The elastic potential energy (EPE) stored in the spring when compressed is given by [tex]EPE = (1/2)kx^2[/tex], where k is the spring constant we need to find, and x is the compression distance (0.52 m).

Equating these two energies, [tex]mgh = (1/2)kx^2[/tex], we solve for the spring constant (k). Plugging in the values gives us:

[tex]1100 kg * 9.81 m/s^2 * 2.4 m = (1/2)k * (0.52 m)^2[/tex]

Solving for k, we get:

[tex]k = (1100 kg * 9.81 m/s^2 * 2.4 m) / (0.5 * (0.52 m)^2)[/tex]

After calculating, we find that the spring constant k is approximately 20641 N/m.

The spring constant k of the spring is 232614 N/m.

Use energy conservation: lost gravitational energy equals stored elastic potential energy in spring.

Gravitational Potential Energy (GPE) lost by the safe:

Height dropped by safe: [tex]\( h = 2.4 \, \text{m} + 0.52 \, \text{m} = 2.92 \, \text{m} \)[/tex]

Mass of safe (m): 1100 kg

Acceleration due to gravity [tex](\( g \)): \( 9.81 \, \text{m/s}^2 \)[/tex]

GPE lost = [tex]\( mgh = 1100 \times 9.81 \times 2.92 \)[/tex]

Elastic Potential Energy (EPE) stored in the spring:

Compression of spring [tex](\( x \)): 0.52 m (52 cm)[/tex]

EPE stored = [tex]\( \frac{1}{2} k x^2 \)[/tex]

Set lost gravitational potential energy equal to spring's stored elastic energy.

[tex]\[ mgh = \frac{1}{2} k x^2 \][/tex]

Rearrange the formula to solve for k:

[tex]\[ k = \frac{2mgh}{x^2} \][/tex]

Substitute the values:

[tex]\[ k = \frac{2 \times 1100 \times 9.81 \times 2.92}{(0.52)^2} \][/tex]

[tex]\[ k = \frac{2 \times 1100 \times 9.81 \times 2.92}{0.2704} \][/tex]

[tex]\[ k = \frac{62898.672}{0.2704} \][/tex]

[tex]\[ k \approx 232613.6 \, \text{N/m} \][/tex]

Answer:

The general equation of movement in fluids is obtained from the application, at fluid volumes, of the principle of conservation of the amount of linear movement. This principle establishes that the variation over time of the amount of linear movement of a fluid volume is equal to that resulting from all forces (of volume and surface) acting on it. Expressed in This equation is called the Navier-Stokes equation.

The equation is shown in the attached file

Explanation:

The derivative of velocity with respect to time determines the change in the velocity of a particle of the fluid as it moves in space. It also includes convective acceleration, expressed by a nonlinear term that comes from convective inertia forces). With this equation, Stokes studied the motion of an infinite incompressible viscous fluid at rest at infinity, and in which a solid sphere of radius r makes a rectilinear and uniform translational motion of velocity v. It assumes that there are no external forces and that the movement of the fluid relative to a reference system on the sphere is stationary. Stokes' approach consists in neglecting the nonlinear term (associated with inertial forces due to convective acceleration).

Answer:

 λ₂ = 1.8 m

Explanation:

given,

wavelength of the string 1 = 0.90 m

frequency of the string 1 = 600 Hz

wavelength of string 2 = ?

frequency of the string 2 = 300 Hz

we now,

[tex]f\ \alpha\ \dfrac{1}{\lambda}[/tex]

now,

[tex]\dfrac{f_1}{f_2}=\dfrac{\lambda_2}{\lambda_1}[/tex]

[tex]\dfrac{600}{300}=\dfrac{\lambda_2}{0.9}[/tex]

λ₂ = 2 x 0.9

 λ₂ = 1.8 m

Hence, the wavelength of the second string is equal to  λ₂ = 1.8 m

To solve this problem we will apply the concepts related to power as a function of the change of energy with respect to time. But we will consider the energy in the body equivalent to kinetic energy. The change in said energy will be the difference between the two velocity data given by half of the mass. We will first convert the given units into an international system like this

Initial Velocity,

[tex]V_i = 60km/h (\frac{1000m}{1km})(\frac{1h}{3600s})[/tex]

[tex]V_i = 16.6667m/s[/tex]

Final Velocity,

[tex]V_f = 100km/h (\frac{1000m}{1km})(\frac{1h}{3600s})[/tex]

[tex]V_f = 27.7778m/s[/tex]

Now Power is defined as the change of Energy over the time,

[tex]P = \frac{E}{t}[/tex]

But Energy is equal to Kinetic Energy,

[tex]P = \frac{\frac{1}{2} m\Delta v^2}{t}[/tex]

[tex]P = \frac{\frac{1}{2} m(v_f^2-v_i^2)}{t}[/tex]

Replacing,

[tex]P = \frac{\frac{1}{2} (900)(27.7778^2-16.6667^2)}{4}[/tex]

[tex]P = 56kW[/tex]

Therefore the correct answer is A.

Final answer:

The additional power needed to achieve the acceleration is 62 kW.So,option (d) 62 kW is correct.

Explanation:

To calculate the additional power needed to achieve acceleration, we can use the formula for power: Power = Force x Velocity. We know the mass of the car (900 kg), the initial velocity (60 km/h), and the final velocity (100 km/h). We can convert the velocities to m/s and calculate the force required to accelerate the car. Then, we can multiply the force by the change in velocity to find the additional power needed.

In this case, the additional power needed is approximately 62 kW. Therefore, option (d) 62 kW is the correct answer.

Answer:

494.262996942 N

Upward

494.262996942 N

Downward

Explanation:

W = Weight of the firefighter = 712 N

g = Acceleration due to gravity = 9.81 m/s²

Mass is given by

[tex]m=\dfrac{W}{g}\\\Rightarrow m=\dfrac{712}{9.81}\\\Rightarrow m=72.5790010194\ kg[/tex]

Force is given by

[tex]T-W=-ma\\\Rightarrow T=W-ma\\\Rightarrow T=712-72.5790010194\times 3\\\Rightarrow T=494.262996942\ N[/tex]

The force on the firefighter is 494.262996942 N

directed upward

On the pole the force will be the same 494.262996942 N

But the direction will be downward

Final answer:

The magnitude of the vertical force on the firefighter from the pole is determined by the net force required for the given downward acceleration. The direction of the force the firefighter applies to the pole is downward while the reaction force from the pole is upward, with the magnitude being the same for both forces.

Explanation:

The question relates to the physics concept called Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force acting on the object and inversely proportional to the object's mass. The law is usually expressed by the equation F = ma, where F is the net force, m is the mass, and a is the acceleration.

To find the magnitude of the vertical force on the firefighter from the pole, we need to take into account both the downward force due to gravity (which is the firefighter's weight) and the additional force needed to produce the acceleration. If the firefighter weighs 712 N and has an acceleration of 3.00 m/s2 downward, we can calculate the force exerted on the pole using F = ma. The mass (m) of the firefighter can be obtained by dividing the weight (W) by the acceleration due to gravity (g), m = W/g. From F = ma, we get F = (W/g)a. Inserting the given values, we have F = (712 N/9.8 m/s2) × 3.00 m/s2. The resulting force is less than the weight of the firefighter because the net force is reduced by the downward acceleration. The direction of the force exerted by the firefighter on the pole is downward since the firefighter is sliding down.

The reaction force exerted on the firefighter by the pole, according to Newton's third law, is equal in magnitude but opposite in direction to the force exerted by the firefighter on the pole. So, the magnitude would be the same, and the direction would be upward.

Answer:

a) The acceleration of the rocket is 249 m/s².

b) The acceleration of the rocket is 25 times the acceleration of a free-falling body (25 g),

c) The distance traveled in 0.900 s was 101 m.

d) The figures are not consistent. The acceleration of the rocket was 20 g.

Explanation:

Hi there!

a) To calculate the acceleration of the rocket let's use the equation of velocity of the rocket:

v = v0 + a · t

Where:

v = velocity of the rocket.

v0 = initial velocity.

a = acceleration.

t = time.

We know that at t = 0.900 s, v = 224 m/s. The initial velocity, v0, is zero because the rocket starts from rest.

v = v0 + a · t

Solving for a:

(v - v0) / t = a

224 m/s / 0.900 s = a

a = 249 m/s²

The acceleration of the rocket is 249 m/s²

b) The acceleration of gravity is ≅ 10 m/s². The ratio of the acceleration of the rocket to the acceleration of gravity will be:

249 m/s² / 10 m/s² = 25

So, the acceleration of the rocket is 25 times the acceleration of gravity or 25 g.

c) The equation of traveled distance is the following:

x = x0 + v0 · t + 1/2 · a · t²

Where:

x = position of the rocket at time t.

x0 = initial position.

v0 = initial velocity.

t = time.

a = acceleration.

Since x0 and v0 are equal to zero, then, the equation of position gets reduced to:

x = 1/2 · a · t²

x = 1/2 · 249 m/s² · (0.900 s)²

x = 101 m

The distance traveled in 0.900 s was 101 m.

d) Now, using the equation of velocity, let's calculate the acceleration. We know that at 1.40 s the velocity of the rocket is zero and that the initial velocity is 283 m/s.

v = v0 + a · t

0 m/s = 283 m/s + a · 1.40 s

-283 m/s / 1.40 s = a

a = -202 m/s²

The figures are not consistent because 40 g is equal to an acceleration of 400 m/s² and the magnitude of the acceleration of the rocket was ≅20 g.

Final answer:

When a charge is accelerated through a potential difference, its speed can be calculated using the equation v = √(2qV/m). In the given example, an electron is accelerated through a potential difference of 4V, resulting in a speed of approximately 290 mV.

Explanation:

When a charge is accelerated through a potential difference, it gains kinetic energy. The relationship between the potential difference and the speed of the charge can be calculated using the equation:

v = √(2qV/m)

Where v is the final speed of the charge, q is the charge of the particle, V is the potential difference, and m is the mass of the particle.

In the given example, an electron with a charge of -1.60 × 10-19 C and a mass of 9.11 × 10-31 kg is accelerated to a speed of 10 × 104 m/s through a potential difference. To calculate the potential difference, we can rearrange the equation to:

V = m(v²)/(2q)

Substituting the values:

V = (9.11 × 10-31 kg)(10 × 104 m/s)²/ (2)(-1.60 × 10-19 C)

V = 2.91 × 10-2 V

So, the potential difference is approximately 29 mV.

Final answer:

The magnitude of the average velocity equals the average speed when the direction of motion doesn't change and the speed is constant. These conditions are met in straightforward travel without directional change or speed variations. For round trips or trips involving direction changes, the average speed may differ from the magnitude of the average velocity.

Explanation:

The conditions under which the magnitude of the average velocity equals the average speed occur when the motion does not involve a change in direction. The average speed is calculated by dividing the total distance traveled by the elapsed time, while the magnitude of the average velocity is the total displacement divided by the elapsed time. For these two quantities to be the same, the direction of travel must remain constant, meaning there is no reversal or change in direction.

When you take a road trip and do not change direction, and your speed is consistent, then your average speed is equal to the magnitude of the average velocity. However, if you ended up back at your starting point, despite having moved, your displacement would be zero, and hence so would your average velocity, even though your average speed is greater than zero.

If you're calculating the ratio of the total distance as shown on the car's odometer to the time of the trip, you're calculating average speed. The speedometer of a car measures instantaneous speed, not velocity, because it does not provide information about direction.

Answer:

Explanation:

It is given that the sphere is insulated from ground and a large charge is placed on the sphere. The charge on the hollow sphere will always remain on the outer surface of the sphere and there will be no charge on the inner surface of the sphere.

If a person touches the inner surface of the sphere then he will not be harmed as there is no charge on the inner surface of the sphere.

If a person carries the charge of the opposite sign of the same magnitude then the sphere and person get neutralized upon touching the sphere.

If a person does not touches the sphere then the charge on the outer surface will be zero and there will be a positive charge on the inner surface of the sphere                        

A person inside a charged, hollow, metallic sphere, known as a Faraday cage, would not be harmed upon touching the interior, regardless of their own charge, due to the neutralization of the electric field inside the conductor.

A person placed inside a large, hollow, metallic sphere that is insulated from the ground and charged will not be harmed upon touching the inside of the sphere. The phenomenon that explains this is known as the Faraday cage effect. According to this principle, an external static electric field will cause the charges within a conductor to rearrange themselves in such a way that the electric field inside the conductor cancels out.

Now, if the person inside the sphere also has a charge of the opposite sign to that of the sphere, an interesting interaction occurs. However, due to the conductor's nature, the electric field inside the metallic sphere remains null. The charges within the conductor would still redistribute to neutralize the electric field within the conductor. Therefore, if a person inside touched the inner surface of the sphere, they would not be directly harmed by the electric field, as it is neutralized within the conductor. The charges from the person would likely redistribute on the sphere's inner surface to maintain electrostatic equilibrium without causing harm.

To find the volume of the snorkeler's lungs at a depth of 49m, use Boyle's Law by calculating the volume at the new pressure using the initial volume and pressure.

The volume of the snorkeler's lungs at a depth of 49 m can be calculated using Boyle's Law. Boyle's Law states that pressure and volume are inversely proportional when temperature is constant. To find the volume at depth, you can use P1V1 = P2V2, where P1V1 is the initial condition and P2V2 is the final condition.

Using the given data:

By rearranging the formula:

V2 = (P1 * V1) / P2 = (1.0 atm * 4.3 L) / 5.9 atm = 0.72 L

Therefore, at a depth of 49 m, the volume of the snorkeler's lungs would be approximately 0.72 liters.

Answer:

The magnitude of electric field at the center of each ring is 129.96 N/C

Explanation:

As per the question:

The diameter of the ring , d = 10 cm = 0.1 m

Radius, [tex]r = \frac{d}{2} = \frac{0.1}{2} = 0.05\ m[/tex]

Separation between the rings, d = 20.0 cm = 0.20 m

Charge on a ring, q = +20 nC = [tex]20\times 10^{- 9}\ C[/tex]

Now,

The electric field at the center of either ring is given by:

[tex]E = \frac{1}{4\pi \epsilon_{o}}\frac{qd}{(d^{2} + r^{2})^{\frac{3}{2}}}[/tex]

where

[tex]\frac{1}{4\pi \epsilon_{o}} = 9\times 10^{9}[/tex]

Thus

[tex]E = 9\times 10^{9}\times \frac{20\times 10^{- 9}\times 0.20}{(0.20^{2} + 0.05^{2})^{\frac{3}{2}}}[/tex]

E = 129.96 N/C