4) a) Show that a small change dn in the index of refraction of the material of a lens produces a small change in the focal length df given by: df -dn f n-1 b) Use this result to find the focal length of a thin lens for blue light for which n = 1.53, if the focal length for red light, for which n = 1.47 is 20 cm.

The magnitude of the electric potential difference between the plates is 37.63 V.

Given:

Distance between the plates, d = 9 cm

Surface charge density, ρ = 3.7 nC/m² = 3.7 × 10⁻⁹ C/m²

Calculation:

For two parallel plate conductors, the electric field is given as:

E = ρ/ε₀

where ρ is the charge density

           ε₀ is the permeability of free space

Applying values in the above equation we get:

E = ( 3.7 × 10⁻⁹ C/m²)/(8.85 × 10⁻¹² C²/m)

  = 418.08 V/m

Now, the potential difference between the plates separated by a distance 'd' is given as:

ΔV = Ed

where E is the electric field

           d is the distance of separation between the plates

Applying values in the above equation we get:

ΔV = (418.08 V/m) × (0.09 m)

     = 37.63 V

Therefore, the magnitude of the electric potential difference between the plates is 37.63 V.

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A rubber ball and a stone of the same size are examples which will have more inertia and is therefore denoted as option A.

This is referred to as the property exhibited by a body in which it has the tendency to remain at rest or in uniform motion.This property is dependent on the mass of the substance as we can deduce that the greater the mass, the greater the inertia and vice versa.

The size of a rubber ball and stone will have different masses in which that of the stone will be greater. This is as a result of the difference in the nature of the substances which are used to make both items mentioned above.

This is therefore the reason why  a rubber ball and a stone of the same size as having more inertia(mass) where chosen as the most appropriate choice in this scenario.

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SOLUTION:

Acceleration = Force/mass

= 100N / 2 kgs

= 50 m/s

.

.

________________________________

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The speed of the ambulance is 4.61 m/s.

Doppler's effect formula is used, which is

[tex]f' = f (\frac{v + vo}{v + vs})[/tex]

Where,

f' = observer frequency of sound

f = actual frequency of sound waves

v = speed of sound waves

vo = velocity of observer

vs = velocity of source

Here, observer is the cyclist and source is the ambulance.

Given,

f' = 1590 Hz

f = 1600 Hz

v = 343 m/s

vo = 2.44 m/s

Now put the given values in the above formula,

[tex]f' = f (\frac{v + vo}{v + vs})[/tex]

[tex]1590 = 1600 (\frac{343 + 2.44}{343 + vs})[/tex]

[tex]1590 = 1600( \frac{345.44}{343 + vs})[/tex]

[tex]1590 = \frac{552,704}{343+vs}[/tex]

[tex]343+vs = \frac{552,704}{1590\\}[/tex]

[tex]343 + vs = 347.61\\[/tex]

[tex]vs = 347.61 - 343[/tex]

[tex]vs = 4.61 m/s[/tex]

Hence, the speed of the ambulance is 4.61 m/s.

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The frequency heard by an observer in the moving car as he approaches the police car is 558.6 Hz.

The frequency heard by the observer is calculated by applying the principle of Doppler effect.

f₀ = f'[(v/v - vs)]

where;

f₀ = 500[(343 / 343 - 36)]

f₀ = 500(343/307)

f₀ = 558.6 Hz

Thus, the frequency heard by an observer in the moving car as he approaches the police car is 558.6 Hz.

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Since , e"(h) = M >0

hence ,  At p3 3ε/M  function e(h) has a minimum of function.

In mathematics, point at which the value of a function is less than or equal to the value at any nearby point (local minimum) or at any point (absolute minimum) or the minimum value of a function is the lowest point of a vertex

e(h) = ∈ /h +( [tex]h^{2}[/tex]/6) M

e'(h) = - ∈ /[tex]h^{2}[/tex] + (2h/6) M

putting , e'(h)  = 0

- ∈ /[tex]h^{2}[/tex] + (2h/6) M  = 0

h = [tex](3 /M)^{1/3}[/tex] *  (∈^(1/3))

e"(h) = 2∈ / [tex]h^{3}[/tex] + (2M)/6                          equation 1

substituting value of h in equation 1, we get

e"(h) = 2∈M / 3∈ + M/3

e"(h) = M >0

hence ,  At p3 3ε/M  function e(h) has a minimum

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If a truck in front of your car begins to turn left, but its right turn signals are flashing, it is most likely that the truck driver is making a mistake or he is not conscious of his driving

Traffic light can simply be defined as a set of automatically operated coloured lights such as red, amber, and green, for controlling traffic at road junctions, pedestrian crossings and roundabouts.

Below are the meaning of some traffic light colours:

So therefore, if a truck in front of your car begins to turn left, but its right turn signals are flashing, it is most likely that the truck driver is making a mistake

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The center of gravity is the point in an object where there is as much weight on one side as on the other side. This is also called the center of balance.

what do you mean by center of balance and center of gravity ?

A location where an object is balanced in relation to applied forces is known as the center of balance (COB).

The average location of an object's weight is known as its center of gravity. Any object's travel through space may be completely explained in terms of how its center of gravity moves from one location to another and, if it is free to spin, how it rotates around that center of gravity.

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(A) The speed of the pendulum when it reaches the bottom is 3.83 m/s.

(B) The speed of the  pendulum when it reaches the bottom after losing 18% of its energy is 3.47 m/s.

(C) The height reached by the pendulum after losing another 7% of its energy is 0.57 m.

(D) When the pendulum stops swinging, it has used all its energy to overcome frictional force of air.

Apply the principle of conservation of energy;

K.E = P.E

¹/₂mv² = mgh

v² = 2gh

v = √2gh

v = √(2 x 9.8 x 0.75)

v = 3.83 m/s

K.E = (100 -  18)P.E

¹/₂mv² = 0.82mgh

V = √(0.82 x 2gh)

v = √(0.82 x 2 x 9.8 x 0.75)

v = 3.47 m/s

K.E = 0.5(1)(3.47)² = 6.02 J

When it losses 7% = 6.02 - (0.07 x 6.02) = 5.598 J

Height reached:

mgh = 5.598

h = 5.598/mg

h = 5.598/(1 x 9.8)

h = 0.57 m

When the pendulum stops swinging, it has used all its energy to overcome frictional force of air.

Thus, the speed of the pendulum when it reaches the bottom is 3.83 m/s.

The speed of the  pendulum when it reaches the bottom after losing 18% of its energy is 3.47 m/s.

The height reached by the pendulum after losing another 7% of its energy is 0.57 m.

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The pattern of organization Alicia chose for her speech outline is referred to as topical.

This type of organization involves the arrangement of subtopics within a large or broad topic.

Alicia talked about speech pathology and the different subtopics include challenges, therapy etc which depicts topical pattern.

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Answer:

See below

Explanation:

The force applied and the length of the arm

The magnitude of an applied torque depend on force vector, 'θ' is the angle between r and F, and the distance vector.

What is torque?

The force that can cause an object to rotate along an axis is measured as torque. Similar to how force accelerates an item in linear kinematics, torque accelerates an object in an angular direction. A vector quantity is a torque.

Torque is defined as Γ=r×F=rFsin(θ). In other words, torque is the cross product of the force vector, where 'θ' is the angle between r and F, and the distance vector (the distance between the pivot point and the place where force is applied).

The magnitude of an applied torque depend on force vector, 'θ' is the angle between r and F, and the distance vector.

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Increasing a car's speed from 0 mph to 30 mph requires more work than that of 50 mph to 60 mph.

Work is a measure of energy expended in moving an object. It is calculated by multiplying the force by the distance.

According to this question, work is required to move a car from one position to another.

The distance moved from 0 - 30mph is more than that moved from 50 - 60mph, therefore, increasing a car's speed from 0 mph to 30 mph requires more work than that of 50 mph to 60 mph.

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In the pinball game, momentum is transferred from the plunger to the ball.

What is momentum?

Momentum is the quantity of motion of a moving body, measured as a product of its mass and velocity.

The formula of the momentum is: p = m*v

Here a pinball game machine is given.

We know that,

Hence,

In the pinball game, momentum is transferred from the plunger to the ball.

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Answer:

1. 571.43m/s

2. 142.9m and 342.9m

Explanation:

1.Take the difference in time.

1.2-0.7=0.7 seconds

Take the distance between them and divide with differnce in time.

400÷0.7=571.43 seconds.

2.Take the time of the two men and divide by two.

0.5÷2= 0.25 secs

1.2÷2= 0.6 secs

multiply each with the velocity.

0.25×571.43=142.9m

0.6×571.43=342.9m

The x-component of the given vector is determined as 10.53 m.

The x-component of a vector is defined as the value of the vector or component of the vector in the x direction.

For a vector of 14.4 m long and pointing in a 133 degree direction.

The x-component of the vector is calculated as follows;

Vx = V cosθ

where;

θ = 133⁰ - 90⁰ = 43⁰

Vx = 14.4 x cos(43)

Vx = 10.53 m

Thus, the x-component of the given vector is determined as 10.53 m.

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Hi there!

First, let's find the period of the pendulum. This can be found by solving for the amount of time it takes for the pendulum to make ONE complete swing.

[tex]T = \frac{\text{Total time}}{\text{Number of complete swings} }\\\\T = \frac{145}{110} = 1.318 s[/tex]

Now, let's use the equation for the period of a simple pendulum:
[tex]T = 2\pi \sqrt{\frac{L}{g}}[/tex]

T = Period (1.318 s)
L = length of string (0.55 m)

g = acceleration due to gravity on planet (? m/s²)

Let's solve for 'g' doing some quick rearranging of the equation:
[tex]T^2 = 4 \pi^2 (\frac{L}{g})\\\\g = \frac{4\pi^2 L}{T^2}\\\\[/tex]

Solving for 'g' by plugging in values:
[tex]g = \frac{4\pi^2 (0.55)}{(1.318)^2}\\ \\= \boxed{12.496 \frac{m}{s^2}}[/tex]

The applied voltage is 1.41* 10⁴ kV.

Calculation of applied voltage-

The capacitance of parallel plates in the absence of a dielectric is:

C = ∈₀A/d

where, "d" = the spacing between the plates = 0.01 * 10⁻³ m

"A" = the area of the plates = 4 m²

"∈₀" = the constant = 8.85419*10⁻¹² C²/Nm²

So, C₀ = [tex]\frac{8.85419*10^{-12}*4 }{0.0100*10^{-3} }[/tex]

          = 3541.67 *10⁻⁹ F

The dielectric constant, κ= 3.4, is defined as:

κ = C/C₀

The effective capacitance (with the dielectric) is denoted by "C", whereas the original capacitance is denoted by "C₀". (without the dielectric).

As a result, the new capacitance is:

C = κC₀

However, capacitance is connected to a voltage by:

C = Q/V

utilizing the increased capacitance and accounting for "V":

κC₀ = Q/V

V = Q/κC₀ = [tex]\frac{0.170}{3.4 * 3541.67 *10^{-9} }[/tex] = 1.41* 10⁴ kV

Therefore the applied voltage is 1.41* 10⁴ kV.

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In order to use the efficiency of a car engine in an optimal way, a maximum engine speed of 2,500 rpm for shifting is recommended for gas engines.

The maximum thermal efficiency of modern gasoline engines is over 50%, but legal cars on the road are only about 20% to 35% when used to power cars.

Unfortunately, even with regular car maintenance such as tuning and oil changes, the efficiency of today's gasoline engines is about 30-35%. In other words, about 65 cents per dollar spent on gasoline is wasted.

EPA's 2020 Automotive Trend Report preliminary data shows that the average fuel economy of 2020 model light trucks has increased to 25.7 miles per gallon (MPG). If this preliminary data holds, it will mark the highest average fuel economy ever for a new light commercial vehicle.

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The clothing seems white because the entire light is reflected.

Absorption:

This occurs when light is transformed into another type of energy. This happens when light vanishes while travelling through another medium.

reflection:

It is referred to as light reflection when a light ray hits a polished, smooth surface and bounces back. The incident light rays hit the surface and are reflected off of it. The beam that returns is referred to as the reflected ray.

according to the question we know that a shirt appears a color because it reflects that color of light. A shirt appears black when it absorbs all wavelengths of light. In case  of white shirt no wavelength is absorbed by the shirt due to which the shirt appears white and in simple words the entire light is reflected back.

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Answer:

Explanation:

Formula

E = I * R                                      Substitute the givens into the formula

Givens

E = 120 volts

I = ?

R = 4800 0hms

Solution

Ohms law is a 3 letter equation that relates volts ohms (resistance) * current (in amperes)

E = i * R

120 V = I * 4800 ohms               Divide both sides by 4800  

120 V/4800 = I * 4800/4800     Divide

0.025 amps = I

Answer

I = 0.025 amps

The temperature rises of in the block if the current flows for 10 minutes is, 34.78K.

To find the answer, we need to know about the power of the electric heater.

                           [tex]P=40 W\\m=1.5kg\\S=460\\t=10minutes=10*60s=600s[/tex]

                             [tex]P*t= mS[/tex]ΔT

                           ΔT [tex]=\frac{P*t}{mS} =\frac{40*600}{1.5*460} =34.78K[/tex]

Thus, we can conclude that, the temperature rises in the block if the current flows for 10 minute is 34.78K.

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If the current flows for 10 minutes, the temperature in the block rises by 34.78 K.

We must understand the electric heater's power in order to determine the solution.

              [tex]P=40Watts\\m=1.5kg\\s=460\\t=10 min=600s[/tex]

                    [tex]Pt=ms[/tex]ΔT

We must determine the increase in temperature, T.

             ΔT[tex]=\frac{Pt}{ms} =34.78 K[/tex]

Thus, we can infer that if the current flows for 10 minutes at 34.78K, the temperature in the block will increase.

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Answer: 37 Celsius and 310.15 K

Explanation:

C/5 = (F-32)/9 = (98.6-32)/9 = 66.6/9 = 7.4

C = 7.4 * 5 = 37 so the temperature is 37 Celsius

K = C + 273.15 = 37 + 273.15 = 310.15 K

The answer is 5.618928 kcal

The work done going up the stairway is the amount of gravitational potential energy that you gain.

GPE = m * g * h

= 80 * 9.8 * 30

=23,520J

converting to kcal

= 23520 * (0.2389 cal / 1J) * (1kcal / 1000cal)

= 5.618928 kcal

A calorie is a unit of energy. In nutrition, calories refer to the energy people get from the foods and drinks they eat, and the energy they use in physical activity. calories are indexed in the dietary records on all food packaging. Many weight loss programs center round reducing the consumption of calories.

There are many factors that go into determining your calorie needs. Your age, weight, height, and activity level all play a role. according to nutritional guidelines in the united states, adults 21 years old and older must consume anywhere between 1,600 and 3,000 calories per day.

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The amount of excess load that can be imposed on the wing of an airplane depends upon the speed of the airplane.

What is meant by wing loading?

Wing loading in aerodynamics is the ratio of the loaded weight of the aircraft and the area of each wing. It is given as:

W_L = W (weight of the aircraft) / S (gross wing area of the airplane)

The more speed an aircraft gains, the more lift is produced by each unit of wing area.

Thus, with a high value of wing loading the aircraft lands and takes-off at higher speeds with low maneuvering control. Generally, Airliners have high wing loading while smaller airplanes have lower wing loading.

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They are useful in providing larger frictional force.

The force produced by two surfaces coming into contact and sliding against one another is referred to as frictional force.

The following variables impact the frictional force: Surface roughness and the amount of force pressing them together have the biggest impact on these forces.

Football shoes have spikes or studs because they offer more frictional force when running on grass than regular shoes do.

The studs make it easier for players to move faster and change directions rapidly without slipping while also preventing them from slipping on the grass.

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The direction of the force vector and acceleration are identical (downwards) during the motion. The simulation also indicates the same relation.

Newton's 2nd regulation, F = ma is a vector equation. It says that the net force (a vector) acting on a mass m (a range of) causes an acceleration (a vector) of the object within the same path as the net force.

A simple mathematical relationship exists between the mass of an item (m), the net force at the item (f) and its acceleration (a). The acceleration of an object is without delay proportional to the net force and indirectly proportional to the item's mass (a = f/m).

For example, displacement, velocity, and acceleration are vector quantities, while speed (the magnitude of velocity), time, and mass are scalars. To qualify as a vector, a quantity having magnitude and direction must also obey certain rules of combination.

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21 cm is  the wavelength of the 21-cm-line if it was coming from a distance of 8,000,000,000 (8*10^9) light-years

Only when relative velocity changes do wavelength alter. Thus, the wavelength will not change. A waveform signal that is carried in space or down a wire has a wavelength, which is the separation between two identical places (adjacent crests) in consecutive cycles.

This length is typically defined in wireless systems in meters (m), centimeters (cm), or millimeters (mm). The relationship between frequency and wavelength is inversely proportional. The wavelength of the wave with the highest frequency is the shortest. Half the wavelength corresponds to twice the frequency. The wavelength in the red spectrum is the longest.



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Answer:

1.66m/s

Explanation:

here

initial velocity(u): 0m/s

final velocity (v):60km= 60×1000/60×60

:16.6m/s

time taken(t): 10s

now

acceleration:v-u/t

or a: 16.6-0/10

:16.6/10

acceleration: 1.66 m/s

In physics, acceleration is the rate at which the velocity of an object changes in relation to time. The sum of all forces exerted on an object results in its acceleration. The acceleration is  1.66 m/s².

The rate at which velocity, either in terms of speed or direction or both, varies over time is known as acceleration. A circle's motion accelerates despite its constant speed because its direction is continually shifting.

Here

initial velocity(u) = 0m/s

final velocity (v) = 60km

v = 60 × 1000 / 60 × 60

v = 16.6 m/s

Time taken(t) =  10s

Now.

acceleration =  v-u / t

a = 16.6-0/10

a = 16.6/10

a = 1.66 m/s²

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The mass of Jupitar is obtained from the calculations as 5.8 * 10^-14 Kg.

There are nine planets in the solar system and the sun lies at the enter of our solar system. This is the heliocentric model of the solar system.

Given that;

T^2 = GMr^3/4π

T = period

G = gravitational constant

r = radius

M = mass of Jupitar

Now;

1 day = 86400 seconds

1.77 days = 1.77 days *  86400 seconds/1 day

= 152928 seconds

Making M the subject of the formula;

M =4πT^2/Gr^3

M = 4 * 3.142 * (152928)^2/6.67 × 10^-11 * (422 × 10^9)^3

M = 2.9 * 10^11/5.0 * 10^24

M = 5.8 * 10^-14 Kg

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The particle that was given off in the reaction beta rays is β–. Option C

A nuclear reaction will involve the change in the nucleus of an atom. It often leads to the production of new nucleus. As we can see the equation, there was a transformation from nitrogen nucleus to oxygen atom.

Thus, the particle that was given off in the reaction beta rays is β–. The fact that the mass number remained the same but the atomic number increased by one unit from left to right implies that a beta decay has occurred.

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