Light of wavelength 588.0 nm illuminates a slit of width 0.74 mm. (a) At what distance from the slit should a screen be placed if the first minimum in the diffraction pattern is to be 0.93 mm from the central maximum(b) Calculate the width of the central maximum.

Answer:

Check the explanation

Explanation:

Stationary states might as well be illustrated in a simpler form of the Schrödinger equation, It utilizes the theory of energy conservation (Kinetic Energy + Potential Energy = Total Energy) to acquire information about an electron’s behavior that is been bound to a nucleus.

Kindly check the attached image below to get the step by step explanation to the above question.

Answer:

The object will continue spinning as it has been.

Explanation:

When an object is spinning in a closed system with no unbalance

forces and no external force or torques are applied to it, it will have no change in angular momentum.

According to the law of conservation of angular momentum which states that when no external torque acts on an object, no change of angular momentum will occur

If the change in angular momentum is zero, then the angular momentum is constant; therefore, the object will continue spinning as it has been.

The same scenario is applicable to the rotation and the spinning of the earth.

Answer: The object will continue spinning as it has been.

Explanation: Unbalanced forces refers a force that changes the position, speed or direction of the object to which it is applied. While torque refers to otational or twisting effect of a force; a moment of force, defined for measurement purposes as an equivalent straight line force multiplied by the distance from the axis of rotation.

Any object that spins in place with no unbalanced forces or torques acting upon it will continue spinning as it has been. An example can be seen in the rotation of planets.

Answer:

[tex]18.8\times 10^{-3} T[/tex]

Explanation:

We are given that

Number of turns,N=500

Radius,r=0.04 m

Length of solenoid,L=40 cm=[tex]\frac{40}{100}=0.4 m[/tex]

1 m=100 cm

Current,I=12 A

We have to find the magnitude of magnetic field near the center of the solenoid.

Number of turns per unit length,n=[tex]\frac{500}{0.4}=1250[/tex]

Magnetic field near the center of the solenoid,B=[tex]\mu_0 nI[/tex]

Where [tex]\mu_=0=4\pi\times 10^{-7}Tm/A[/tex]

[tex]B=4\pi\times 10^{-7}\times 1250\times 12=18.8\times 10^{-3} T[/tex]

[tex]B=18.8\times 10^{-3} T[/tex]

Answer:

D

Explanation:

An inverse resistor giving power to the system

A capacitor hooked up to a battery acts as another battery aiding the first battery as time progresses.

As time progresses, a capacitor hooked up to a battery begins to act as another battery aiding the first battery. The capacitor acts as a temporary storehouse of energy and can drive its collected charge through a second circuit.

Answer:

[tex]-16.6 rad/s^2[/tex]

Explanation:

The torque exerted on a rigid body is related to the angular acceleration by the equation

[tex]\tau = I \alpha[/tex] (1)

where

[tex]\tau[/tex] is the torque

I is the moment  of inertia of the body

[tex]\alpha[/tex] is the angular acceleration

Here we have a solid sphere: the moment of inertia of a sphere rotating about is centre is

[tex]I=\frac{2}{5}MR^2[/tex]

where

M = 240 g = 0.240 kg is the mass of the sphere

[tex]R=\frac{2.50}{2}=1.25 cm = 0.0125 m[/tex] is the radius of the sphere

Substituting,

[tex]I=\frac{2}{5}(0.240)(0.0125)^2=1.5\cdot 10^{-5} kg m^2[/tex]

The torque exerted on the sphere is

[tex]\tau = Fr[/tex]

where

F = -0.0200 N is the force of friction

r = 0.0125 m is the radius of the sphere

So

[tex]\tau=(-0.0200)(0.0125)=-2.5\cdot 10^{-4} Nm[/tex]

Substituting into (1), we find the angular acceleration:

[tex]\alpha = \frac{\tau}{I}=\frac{-2.5\cdot 10^{-4}}{1.5\cdot 10^{-5}}=-16.6 rad/s^2[/tex]

Answer:

a) The shear stress is 0.012

b) The shear stress is 0.0082

c) The total friction drag is 0.329 lbf

Explanation:

Given by the problem:

Length y plate = 2 ft

Width y plate = 10 ft

p = density = 1.938 slug/ft³

v = kinematic viscosity = 1.217x10⁻⁵ft²/s

Absolute viscosity = 2.359x10⁻⁵lbfs/ft²

a) The Reynold number is equal to:

[tex]Re=\frac{1*3}{1.217x10^{-5} } =246507, laminar[/tex]

The boundary layer thickness is equal to:

[tex]\delta=\frac{4.91*1}{Re^{0.5} } =\frac{4.91*1}{246507^{0.5} } =0.0098[/tex] ft

The shear stress is equal to:

[tex]\tau=0.332(\frac{2.359x10^{-5}*3 }{1} )(246507)^{0.5} =0.012[/tex]

b) If the railing edge is 2 ft, the Reynold number is:

[tex]Re=\frac{2*3}{1.215x10^{-5} } =493015.6,laminar[/tex]

The boundary layer is equal to:

[tex]\delta=\frac{4.91*2}{493015.6^{0.5} } =0.000019ft[/tex]

The sear stress is equal to:

[tex]\tau=0.332(\frac{2.359x10^{-5}*3 }{2} )(493015.6^{0.5} )=0.0082[/tex]

c) The drag coefficient is equal to:

[tex]C=\frac{1.328}{\sqrt{Re} } =\frac{1.328}{\sqrt{493015.6} } ==0.0019[/tex]

The friction drag is equal to:

[tex]F=Cp\frac{v^{2} }{2} wL=0.0019*1.938*(\frac{3^{2} }{2} )(10*2)=0.329lbf[/tex]

Final answer:

To balance the rod with the attached objects, you need to place a support at a certain distance from the rod's center. The support should be placed at (2/3) times the length of the rod to the right of the rod's center.

Explanation:

To balance the rod with the attached objects, you need to place a support at a certain distance from the rod's center. Let's call this distance 'x'. The torque on the rod is balanced when the clockwise torque is equal to the counterclockwise torque.

The torque on the left end of the rod is given by the force on the left object times the distance from the left end of the rod to the support, which is (m * x). The torque on the right end of the rod is given by the force on the right object times the distance from the right end of the rod to the support, which is ((2m) * (L - x)). Since the torque must be balanced, we set the two torques equal to each other:

(m * x) = ((2m) * (L - x))

Simplifying this equation gives us:

x = (2/3) * L

Therefore, you should place the support at a distance of (2/3) times the length of the rod to the right of the rod's center.

Answer: 4.821m/s

Explanation:

initial momentum = final momentum 

(80 x 4.1) + (85 x 5.5) = (80 + 85)u

328 + 467.5 = 165u  =  795.5 / 165 = 4.821m/s

Answer:

Explanation:

find the solution below

Final answer:

Calculating voltage, current at maximum power, and fill factor for a PV cell involves applying the Shockley diode equation under illumination. These calculations help in identifying the Maximum Power Point (MPP), crucial for optimizing the cell's performance. Detailed calculations were not provided, but the importance of plotting the I-V characteristics to validate the MPP was emphasized.

Explanation:

The task requires calculating the voltage and current at maximum power and the fill factor for a photovoltaic (PV) cell, given its light current density and reverse saturation current density, at an ambient temperature of 32°C. This involves understanding the behavior of PV cells under varying conditions and applying theoretical models to determine the operating characteristics that yield maximum power output. The fill factor (FF), the maximum power point (MPP), and the open-circuit voltage (Voc) are crucial aspects in assessing the performance and efficiency of PV cells. To derive an accurate answer, one would typically employ the Shockley diode equation adjusted for light conditions and solve for the power at different voltages to find the maximum. However, the calculation details are beyond this response scope due to the required complexity and dependency on specific model parameters not provided. Generally, these calculations involve detailed semiconductor physics and numerical methods to solve nonlinear equations reflective of the cell's I-V characteristic in illuminated conditions. Plotting current and power as functions of voltage from V=0 to Voc, helps validate the maximum power point approximation. This plot demonstrates the balance between voltage and current that a PV cell must achieve to optimize power output, usually found using methods like the perturb and observe algorithm in practical MPPT (Maximum Power Point Tracking) systems.

Answer:

Yes it would

Explanation:

The submerged portion of the pen would be affected by water buoyancy force which equals to the water weight that it displaces. By Newton's 3rd law, the pen would then make a reaction force on the water itself, which in turn affect the whole reading on the balance.

Yes, it would: The submerged part of the pen would be influenced by water buoyancy force which equals the water weight that it replaces.

The submerged part of the pen would be influenced by water buoyancy force which equals the water weight that it replaces. According to Newton's 3rd law, the pen would then make a reaction force on the water itself, which in turn impacts the whole reading on the balance.

Newton's Third Law refers to There are two forces resulting from this interaction. A force on the chair and also a force on your body. These two forces are called action and reaction forces and also are the subject of Newton's third law of motion.

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Answer:

When the midpoint of the magnet is in the plane of the loop, the induced current in the loop viewed from above is essentially zero.

Explanation:

At some point far away from the plane of the loop of the wire, there's no flux and hence, no induced emf, and no current in the loop. As the magnet descends and gets close to the entrance of the coil, some of the magnetic field from the magnet threads the top few imaginary disks of the coil. The time changing flux induces some EMF and subsequently some induced current.

As the magnet continues to fall, and enters the coil, more of its magnetic field is threading imaginary disks in the coil, so as it moves, the time rate of change of total flux increases, so the EMF goes up. Note that the field lines above and below the magnet's midpoint point in the same direction from the the North pole to the south.

As the bar moves through the plane of the coil, the North end first, flux is added by the motion of the magnet and flux is removed by the motion of South end.

At some point, the bar reaches the middle of the coil. At this point, the amount of flux added to the top half of the coil by a small motion of the magnet is equal to the amount of flux removed from the bottom half. Therefore, at this point the EMF is zero. And hence, there is no induced current observed in the wire at this point.

Hope this Helps!!!

When viewed from the above, there is no induced current observed in the wire at this point due to zero EMF.

The given problem is based on the concept of electric flux and induced emf. The electric flux is dependent on induced emf, which is produced to current. Since,  point far away from the plane of the loop of the wire, there's no flux and hence, no induced emf, and no current in the loop.

Note: - The field lines above and below the magnet's midpoint point in the same direction from the the North pole to the south.

Thus, we conclude that when viewed from the above, there is no induced current observed in the wire at this point due to zero EMF.

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Answer:

F'=(8/3)F

Explanation:

to find the change in the force you take into account that the electric force is given by:

[tex]F=k\frac{q_1q_2}{(18.0u)^2}=k\frac{q_1q_2}{324.0}u^2[/tex]

However, if q1'=1/3*q, q2'=2*q2 and the distance is halved, that is 18/=9.o unit:

[tex]F'=k\frac{q_1'q_2'}{(9u)^2}=k\frac{(1/3)q_1(2)q_2}{81.0u^2}=\frac{2}{3}k\frac{q_1q_2}{81.0u^2}[/tex]

if you multiply this result by 4 and divide by 4 you get:

[tex]F'=\frac{8}{3}k\frac{q_1q_2}{324.0u^2}=\frac{8}{3}F[/tex]

hence, the new force is 8/3 of the previous force F.

Answer:

The third ball emerges from the right side.

Explanation:

This is a conservation of Momentum problem

In an explosion or collision, the momentum is always conserved.

Momentum before explosion = Momentum after explosion

Since the rigid pipe was initially at rest,

Momentum before explosion = 0 kgm/s

- Taking the right end as the positive direction for the velocity of the balls

- And calling the speed of the 6 g ball after explosion v

- This means the velocity of the 4 g ball has to be -2v

- Mass of the third ball = m

- Let the velocuty of the third ball be V

Momentum after collision = (6)(v) + (4)(-2v) + (m)(V)

Momentum before explosion = Momentum after explosion

0 = (6)(v) + (4)(-2v) + (m)(V)

6v - 8v + mV = 0

mV - 2v = 0

mV = 2v

V = (2/m) v

Note that since we have established that the sign on m and v at both positive, the sign of the velocity of the third ball is also positive.

Hence, the velocity of the third ball according to our convention is to the right.

Hope this Helps!!!

Answer:

the third ball emerge from right end .

Explanation:

initially the three balls are at rest so u = 0  

Given m1 = 6g

v1 = v(i) (Right end indicates positive X axis so it is represented as (i))

m2 = 4g

v2 = 2(v)(-i) (left end indicates negative X axis so it is represented as (-i))

m3 = m

v3 = v'

Applying conservation of momentum

m1(u1) + m2(u2) + m3(u3) = m1(v1) + m2(v2) + m3(v3)

u1 = u2 = u3 = u = 0 ( at rest)

0 = (6×10⁻³)(v)(i) + (4×10⁻³)(2v)(-i) + m(v')

m(v') = (2×10⁻³)(v)(i)

v' = (2×10⁻³)(v)(i)/(m)

So positive (i) indicates right end

So the third ball emerge from right end .

Answer:

Using equation 2dsinФ=n*λ

given d=2.41*10^-6m

λ=512*10^-12m

θ=52.64 degrees

Answer:

a

The angle between the center and the third side bright fringe  is

       [tex]\theta = 39.60^o[/tex]

b

The third side bright fringe move away from the pattern's center

c

The distance by which it moves away is [tex]\Delta z=0.3906 m[/tex]

Explanation:

From the question the

           The wavelength is [tex]\lambda = 512nm[/tex]

In the first question we a asked to obtain the angle between the center and the third side bright fringe

  since we are considering the third side of the bright fringe the wavelength of light on  the three sides would be  evaluated as

                 [tex]\lambda_{3} = 3 * 512nm[/tex]

The slit separation is given as  [tex]d = 2.41 \mu m[/tex]

    The angle between the center and the third side bright fringe is

              [tex]\theta = sin^{-1} (\frac{\lambda_3}{d} )[/tex]

              [tex]\theta = sin^{-1} (\frac{3 *512*10^{-9}}{2.24*10^{-6}} )[/tex]

                [tex]= sin^{-1} (0.6374)[/tex]

                [tex]\theta = 39.60^o[/tex]

When the frequency of the light is reduced the wavelength is increased

                i.e [tex]f = \frac{c}{\lambda}[/tex]

and this increase would cause the third side bright to move away from the pattern's center

        Now from the question frequency is reduce to 94.5% this mean that the  wavelength would also increase by the same as mathematically represented below

                  [tex]\lambda_{new} = \frac{512 *10^{-9}}{0.945}[/tex]

                          [tex]= 0.542 \mu m[/tex]

The angle between the center and the third side bright fringe is for new wavelength

                   [tex]\theta = sin^{-1} (\frac{3 *512*10^{-9}}{2.41*10^{-6}} )[/tex]

                     [tex]= 42.46^o[/tex]

 The distance traveled away from the pattern's center is mathematically represented as

                [tex]z = A tan \theta[/tex]

 Where A is the separation between the slits and the screen

              [tex]\Delta z = 4.45(tan 42.46 - tan39.60 )[/tex]

                    [tex]\Delta z=0.3906 m[/tex]

Answer:

[tex]\frac{dz}{dt} = 65 mi/h[/tex]

Explanation:

let distance between two  cars  is = z  mi

we have to find  =[tex]\frac{dz}{dt}[/tex]

One travels south at  = 60 mi/h  = [tex]\frac{dx}{dt}[/tex]   (given)

the other travels west at =25 mi/h.= [tex]\frac{dy}{dt}\\[/tex]  (given)

since both car have constant speed

at t = 3 hrs

x = 3× 60  = 180 mi/h

y = 3 × 25  = 75 mi/h

from the figure (i)  we get

[tex]z = \sqrt{( x^2+ y^2)}[/tex] ...............(i)

put x and y values

we get

[tex]z = \sqrt{(180)^2 + 75^2}[/tex]

[tex]z = \sqrt{32400 + 5625} \\z = \sqrt{38025} \\z = 195 mi/h[/tex]

differentiate the equation (i) w r to t

[tex]z^2 = x^2 +y^2[/tex]

[tex]2z\frac{dz}{dt} = 2x\frac{dx}{dt}+ 2y\frac{dy}{dt}\\[/tex]

put each values

[tex]2 \times195\frac{dz}{dt} = 2 \times 180\frac{dx}{dt}+2 \times75\frac{dy}{dt}\\[/tex]

[tex]2 \times195\frac{dz}{dt} = 2 \times 180\times 60}+2 \times75\times25\\\frac{dz}{dt} = \frac{{2 \times 180\times 60+2 \times75\times25}}{ 2 \times195}\\\frac{dz}{dt} = 65 mi/h[/tex]

Answer: Both cars have equal kinetic energy

Explanation:

Answer:

Explanation:

Diffraction is the term used to describe the bending of a wave around an obstacle. It is one of the general properties of waves.

1. Diffraction of sound is the bending of sound waves around an obstacle which propagates from source to a listener. Two of the daily phenomena that exhibit diffraction of sound are:

i. The voices of people talking outside a building can be heard by those inside.

ii. The sound from the horn of a car can be heard by people at certain distances away.

When sound waves are produced, the surrounding air molecules are required for its transmission. This is because sound wave is a mechanical wave which requires material medium for its propagation. When a source produces a sound, the sound waves bend around obstacles on its path to reach listeners.

2. Light waves are electromagnetic waves which can undergo diffraction. Diffraction of light is the bending of the rays of light around an obstacle. Two of the daily phenomena that exhibit diffraction of light are:

i. The shadow of objects which has the umbra and penumbra regions.

ii. The apparent color of the sky.

A ray of light is the path taken by light, and the combination of two or more rays is called a beam. A ray or beam of light travels in a straight line, so any obstacle on its path would subject the light to bending around it during propagation. These are major applications in pin-hole cameras, shadows, rings of light around the sun etc.

Some significant differences between diffraction of light and that of the sound are:

i. Diffraction of light is not as common as that of sound.

ii. Sound propagates through a wider region than light waves.

iii. Sounds are longitudinal waves, while lights are transverse waves.

a) 10 kg

b) 60 N

c) 98 N

d) 0 N

Explanation:

a)

In this problem, there are two forces acting on the backpack:

- The restoring force from the spring, upward, of magnitude [tex]F=60 N[/tex]

- The weight of the backpack, downward, of magnitude [tex]mg[/tex], where

m = mass of the backpack

[tex]g=9.8 m/s^2[/tex] acceleration due to gravity

So the net force on the backpack is (taking upward as positive direction)

[tex]F_{net}=F-mg[/tex]

According to Newton's second law of motion, the net force must be equal to the product between the mass of the backpack and its acceleration, so

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

where

[tex]a=-3.8 m/s^2[/tex] is the acceleration of the backpack and the elevator, downward (so, negative)

If we solve the formula for m, we can find the mass of the backpack:

[tex]F=m(a+g)\\m=\frac{F}{a+g}=\frac{60}{-3.8+9.8}=10 kg[/tex]

b)

In this case, the elevator is moving upward, and it is slowing down at a rate of [tex]3.8 m/s^2[/tex].

Since the elevator is slowing down, it means that the direction of the acceleration is opposite to the direction of motion: and since the elevator is moving upward, this means that the direction of the acceleration is downward: so the acceleration is negative,

[tex]a=-3.8 m/s^2[/tex]

The net force acting on the backpack is still:

[tex]F_{net}=F-mg[/tex]

where

F is the restoring force in the spring, which this time is unknown (it corresponds to the reading on the scale)

Using again Newton's second law of motion,

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

Therefore in this case, the reading on the scale will be:

[tex]F=m(g+a)=(10)(9.8-3.8)=60 N[/tex]

So the reading is the same as in part a).

c)

In this case, the elevator is moving at a constant velocity.

The net force on the backpack is still:

[tex]F_{net}=F-mg[/tex]

However, since here the elevator is moving at constant velocity, and acceleration is the rate of change of velocity, this means that the acceleration of the elevator is zero:

[tex]a=0[/tex]

So Newton's second law of motion can be written as:

[tex]F_{net}=0[/tex]

So

[tex]F-mg=0[/tex]

Which means that the reading on the scale is equal to the weight of the backpack:

[tex]F=mg=(10)(9.8)=98 N[/tex]

d)

In this  case, the elevator had no brakes and the cable supporting it breaks loose.

This means that the elevator is now in free fall.

So its acceleration is simply the acceleration due to gravity (which is the acceleration of an object in free fall):

[tex]a=-9.8 m/s^2[/tex]

And the direction is downward, so it has a negative  sign.

The net force on the backpack is still

[tex]F_{net}=F-mg[/tex]

So Newton's second law can be rewritten as

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

Therefore, we can re-arrange the equation to find F, the reading on the scale, and we find:

[tex]F=m(g+a)=(10)(9.8-9.8)=0 N[/tex]

So, the reading on the scale is 0 N.

Answer:

The energy of the photon is  [tex]x = 2.86 eV[/tex]

Explanation:

From the question we are told that  

      The first orbit is [tex]n_1 = 5[/tex]

       The second orbit is [tex]n_2 = 2[/tex]

According to  Bohr model

     The energy of difference of the electron as it moves from on orbital to another is mathematically represented as

              [tex]\Delta E = k [\frac{1}{n^2 _1} + \frac{1}{n^2 _2} ][/tex]

  Where k is a constant which has a value of [tex]k = -2.179 *10^{-18} J[/tex]

       So

              [tex]\Delta E = - 2.179 * 10^{-18} [\frac{1}{5^2 _1} + \frac{1}{2^2 _2} ][/tex]

                     [tex]= 4.576 *10^{-19}J[/tex]

Now we are told from the question that

         [tex]1 eV = 1.602 * 10^{-19} J[/tex]

so      x eV  =  [tex]= 4.576 *10^{-19}J[/tex]

  Therefore

                [tex]x = \frac{4.576*10^{-19}}{1.602 *10^{-19}}[/tex]

                   [tex]x = 2.86 eV[/tex]

The energy of the photon produced by an electron in a Hydrogen atom moving from the 5th orbit to the 2nd orbit is 2.85 electron Volts as calculated via the Rydberg formula.

The energy of the photon produced by the transition of an electron in a hydrogen atom from the 5th orbit to the 2nd orbit can be calculated using Rydberg's formula.

Rydberg's formula for energy is given as E = 13.6 * (1/n1^2 - 1/n2^2), where n1 and n2 are the initial and final energy levels respectively, and E is the energy difference in eV (electron volts). Here n1 = 2 and n2 = 5.

So, substituting these values in the formula E = 13.6 * (1/2^2 - 1/5^2) = -13.6 * (-0.21) = 2.85 eV. Notice that the energy is negative which signifies a transition down to a lower energy level (which is exothermic), however, we are interested in the magnitude of the energy which is 2.85 eV.

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Answer:

Between R and S

Explanation:

Answer:

C) Between R and S

Explanation:

The angle is approximately 50°

Explanation:

We know,

h = L – L * cos θ

  = 1.55 – 1.55 (cos θ )

To determine the maximum height, we use conservation potential and kinetic energy. As 2nd ball rises to its maximum height, the increase of its potential energy is equal to the decrease of its kinetic energy. Below is the equation for determining the value of h from the 2nd ball’s initial velocity.

Potential energy, PE = mgh = m X 9.8 X h

Kinetic energy, KE = [tex]\frac{1}{2} mv^2[/tex]

Set PE equal to KE and solve for h.

h = v²/19.6  

To determine the initial kinetic energy of the second ball, we need to the velocity of the 2nd ball immediately after the collision. To do this we need to use conservation of momentum. Below is the equation for determining the value of h from the 2nd ball’s initial velocity.

For the 1st ball, horizontal momentum = 0.26 X 2.5 = 0.65

Since this ball falls straight down after the collision, its final horizontal momentum is 0.

For the 2nd ball, horizontal momentum = 0.61 X vf

0.64 * vf = 0.65

vf = 0.65/.0.64

vf = 0.02m/s

h = (0.65/0.64)²/19.6

h = 0.00002m

0.00002 = 1.45 – 1.45 * cos θ

Subtract 1.55 from both sides.

0.00002 – 1.45 = -1.45 * cos θ

θ = 50°

Answer: The riders are subjected to 11.5 revolutions per minute

Explanation: Please see the attachments below

Answer:

What is the average velocity of the car over the first 0.25m?  

⇒ 0.11 m/s

What is the average velocity of the car over the second 0.25m?  

⇒ 0.28 m/s

Explanation:

Just did this problem! :)

Answer:

⇒ 2.23

⇒ 0.90

⇒ 0.28

Explanation:

Just did the problem (⌐■_■)

Answer:

See explaination and attachment please.

Explanation:

Potential energy is defined as the energy possessed by a body by virtue of its position relative to others, stresses within itself, electric charge, and other factors.

Kinetic energy on the other hand is defined as the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity.

These understand will foster our knowledge to the best way of solving the question.

Please kindly check attachment for a step by step approach on the graph.

Final answer:

In a bungee jumping scenario, the stretching of the cord can be expressed in terms of length and height. Gravitational potential energy dominates early in the fall, while elastic potential energy is more significant later. Graphs can visually represent how potential energy changes with height during the jump.

Explanation:

a) Stretching of the cord: The stretching ?L of the cord can be expressed as ?L = d - z. The total potential energy U can be represented as U = UG + US, where UG is the gravitational potential energy and US is the elastic potential energy.

b) Largest potential energy: For large z (early in the fall), UG is largest. For small z (late in the fall), US is largest.

c) Graphs: Sketch a graph of UG(z) on the left plot, US(z) on the center plot, and U(z) = UG(z) + US(z) on the rightmost plot.

Answer:

The velocity of mass 2m is  [tex]v_B = 0.67 m/s[/tex]

Explanation:

From the question w are told that

     The mass of the billiard ball A is =m

     The initial speed  of the billiard ball A = [tex]v_1[/tex] =1 m/s

    The mass of the billiard ball B is = 2 m

    The initial speed  of the billiard ball  B = 0

Let the final speed  of the billiard ball A  = [tex]v_A[/tex]

Let The finial speed  of the billiard ball  B = [tex]v_B[/tex]

      According to the law of conservation of Energy

                 [tex]\frac{1}{2} m (v_1)^2 + \frac{1}{2} 2m (0) ^ 2 = \frac{1}{2} m (v_A)^2 + \frac{1}{2} 2m (v_B)^2[/tex]

              Substituting values  

                [tex]\frac{1}{2} m (1)^2 = \frac{1}{2} m (v_A)^2 + \frac{1}{2} 2m (v_B)^2[/tex]

Multiplying through by [tex]\frac{1}{2}m[/tex]

                [tex]1 =v_A^2 + 2 v_B ^2 ---(1)[/tex]

    According to the law of conservation of Momentum

            [tex]mv_1 + 2m(0) = mv_A + 2m v_B[/tex]

    Substituting values

            [tex]m(1) = mv_A + 2mv_B[/tex]

Multiplying through by [tex]m[/tex]

           [tex]1 = v_A + 2v_B ---(2)[/tex]

making [tex]v_A[/tex] subject of the equation 2

            [tex]v_A = 1 - 2v_B[/tex]

Substituting this into equation 1

         [tex](1 -2v_B)^2 + 2v_B^2 = 1[/tex]

         [tex]1 - 4v_B + 4v_B^2 + 2v_B^2 =1[/tex]

          [tex]6v_B^2 -4v_B +1 =1[/tex]

          [tex]6v_B^2 -4v_B =0[/tex]

Multiplying through by [tex]\frac{1}{v_B}[/tex]

          [tex]6v_B -4 = 0[/tex]

            [tex]v_B = \frac{4}{6}[/tex]

            [tex]v_B = 0.67 m/s[/tex]

The stationary billiard ball of mass 2m will have a final velocity of 1/3 m/s after the head-on collision with a ball of mass m moving at 1 m/s due to conservation of momentum.

In this problem, we are dealing with a one-dimensional collision. The principle of conservation of momentum is key in solving this problem. Momentum before the collision is the same as momentum after the collision. The initial momentum is given by mass * velocity of moving ball = m * v1, while the stationary ball has 0 momentum.

So, we can write this as m * 1 = (m + 2m) * v', where v' is the final velocity of the stationary ball after the collision. Solving for v', we get v' = m / 3m = 1/3 m/s. Hence, the stationary ball (of mass 2m) has a final velocity of 1/3 m/s after the collision.

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Answer:

Explanation:

find the solution below

Complete question:

A 45-mH ideal inductor is connected in series with a 60-Ω resistor through an ideal 15-V DC power supply and an open switch. If the switch is closed at time t = 0 s, what is the current 7.0 ms later?

Answer:

The current in the circuit 7 ms later is 0.2499 A

Explanation:

Given;

Ideal inductor, L = 45-mH

Resistor, R =  60-Ω

Ideal voltage supply, V = 15-V

Initial current at t = 0 seconds:

I₀ = V/R

I₀  = 15/60 = 0.25 A

Time constant, is given as:

T = L/R

T = (45 x 10⁻³) / (60)

T = 7.5 x 10⁻⁴ s

Change in current with respect to time, is given as;

[tex]I(t) = I_o(1-e^{-\frac{t}{T}})[/tex]

Current in the circuit after 7 ms later:

t = 7 ms = 7 x 10⁻³ s

[tex]I(t) = I_o(1-e^{-\frac{t}{T}})\\\\I =0.25(1-e^{-\frac{7*10^{-3}}{7.5*10^{-4}}})\\\\I = 0.25(0.9999)\\\\I = 0.2499 \ A[/tex]

Therefore, the current in the circuit 7 ms later is 0.2499 A

The current later after 7 ms is 0.2499 A

The current after a time (t) is given by:

[tex]I(t)=I_o(1-e^{-\frac{t}{\tau} })\\\\I_o=initial\ current\ at\ t=0,\tau=time \ constant\\\\Given\ that\ L=45mH=45*10^{-3}H,R=60\ ohm,V = 15V,t=7\ ms\\\\\tau=\frac{L}{R}=\frac{45*10^{-3}}{60} =7.5*10^{-4} \ s\\\\I_o=\frac{V}{R}=\frac{15}{60}=0.25\ A \\\\\\I(t)=I_o(1-e^{-\frac{t}{\tau} })=0.25(1-e^{-\frac{7*10^{-3}}{7.5*10^{-4}} })\\\\\\I(t)=0.2499\ A[/tex]

The current later after 7 ms is 0.2499 A

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Answer:

[tex]u(t)=1.15 \sin (8.68t)cm[/tex]

0.3619sec

Explanation:

Given that

Mass,m=148 g

Length,L=13 cm

Velocity,u'(0)=10 cm/s

We have to find the position u of the mass at any time t

We know that

[tex]\omega_0=\sqrt{\frac{g}{L}}\\\\=\sqrt{\frac{980}{13}}\\\\=8.68 rad/s[/tex]

Where [tex]g=980 cm/s^2[/tex]

[tex]u(t)=Acos8.68 t+Bsin 8.68t[/tex]

u(0)=0

Substitute the value

[tex]A=0\\u'(t)=-8.68Asin8.68t+8.68 Bcos8.86 t[/tex]

Substitute u'(0)=10

[tex]8.68B=10[/tex]

[tex]B=\frac{10}{8.68}=1.15[/tex]

Substitute the values

[tex]u(t)=1.15 \sin (8.68t)cm[/tex]

Period =T = 2π/8.68

After half period

π/8.68 it returns to equilibruim

π/8.68 = 0.3619sec

Answer:

Explanation:

We shall solve this question with the help of Ampere's circuital law.

Ampere's ,law

∫ B dl = μ₀ I , B is magnetic field at distance x from the axis within wire

we shall find magnetic field at distance x . current enclosed in the area of circle of radius x

=  I x π x²  / π R²

= I x²  /  R²

B x 2π x = μ₀  x current enclosed

B x 2π x = μ₀  x  I x²  /  R²

B =  μ₀   I x  / 2π R²

Maximum magnetic B₀ field  will be when x = R

B₀ = μ₀I   / 2π R

Given

B = B₀ / 3

μ₀   I x  / 2π R² = μ₀I   / 2π R x 3

x = R / 3

b ) The largest value of magnetic field is on the surface of wire

B₀ = μ₀I   / 2π R

At distance x outside , let magnetic field be B

Applying Ampere's circuital law

∫ B dl = μ₀ I

B x 2π x = μ₀ I

B = μ₀ I / 2π x

Given B = B₀ / 3

μ₀ I / 2π x = μ₀I   / 2π R x 3

x = 3R .

The location inside the wire where the magnetic field equals a third of its maximum value is at r/R = 1/3. There is no point outside the wire where the magnetic field reaches a third of its maximum value because it monotonically decreases.

To determine at what location inside the wire the magnetic field produced by the current is equal to a third of its largest value, we need to apply Ampère's Law. Inside a conductor carrying uniform current, the magnetic field B increases linearly with the distance r from the center of the wire due to the proportion of current enclosed. Thus, the magnetic field is given by B = μ_0 J r / 2, where J is the current density and μ_0 is the permeability of free space. Since the value at the surface (r=R) will be maximum (B_max), for B to be a third of B_max, we must have (1/3)B_max = (μ_0 J r / 2). Solving for r, we find r/R = 1/3.

For locations outside the wire, Biot-Savart Law or Ampère's Law show that the magnetic field decreases with 1/r. However, since the field is maximum at the surface, and we do not have an equation that varies outside the wire, we cannot directly calculate when the field will be a third without additional information on how the field varies with r beyond the wire's surface. Normally, the largest value of B is at the surface, and it decreases monotonically outside, never increasing again to allow for a location at which it is a third of its largest value.

The magnetic field for points inside the wire is proportional to the distance from the center, and for points outside the wire, it is inversely proportional to the distance from the center. However, the magnetic field does not reach a third of its largest value at any given point outside the cylindrical wire since it monotonically decreases.

The position of the mass after t seconds can be found using the equation of motion for a mass-spring system. We can calculate the angular frequency, determine the phase angle, and substitute the values into the equation to find the position.

The position of the mass after t seconds can be found using the equation of motion for a mass-spring system. The equation is given by:

x(t) = A * cos(ω * t + φ)

Where A is the maximum displacement, ω is the angular frequency, and φ is the phase angle. In this problem, the maximum displacement is 0.2 meters, the angular frequency can be calculated using ω = sqrt(k/m) (where k is the spring constant and m is the mass), and the phase angle can be determined using the initial conditions.

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The problem is a case of simple harmonic motion which is solved by first identifying the spring constant using Hooke's Law, then finding the equation for the motion, resulting in x(t) = 0.5 cos(sqrt(41.67) * t + π/2), which gives the position of the mass as a function of time.

The problem presented is a classical physics problem dealing with the motion of a mass attached to a spring. This is a scenario of simple harmonic motion which is characterized by an oscillating motion about a stable equilibrium.

In simple harmonic motion, the spring force is given by Hooke's law, F = -kx, where k is the spring constant and x is the displacement of the spring from its equilibrium. In this case, we are given that the force needed to stretch the spring 0.2m beyond its natural length is 50N. So we can find the spring constant k by rearranging Hooke's law: k = F/x = 50N / 0.2m = 250 N/m.

Next, we know that the equation for the position of the mass in simple harmonic motion is given by x(t) = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency and φ is the phase.

Here, the amplitude A is the initial position, which is 0. The initial velocity is given as 0.5 m/s, and ω can be found using the relation ω = sqrt(k/m) = sqrt(250 / 6) = sqrt(41.67) rad/s. The phase can be found using the fact that the initial velocity v0 is v0 = ωA sin(φ), and since A=0 and v0=0.5, φ = arcsin(v0/ωA) = arcsin((0.5 m/s) / (sqrt(41.67) rad/s * 0) = π/2.

Therefore, the equation for the motion becomes x(t) = 0.5 cos(sqrt(41.67) * t + π/2). This equation gives the position x of the mass as a function of time t.

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Answer:

A) = 0.63 m

B)This is approximately 0.21

Explanation:

Part A)

Now if spring is connected to the block then again we can use energy conservation

so we will have

[tex]\frac{1}{2}kx^2 = mg(x + x')sin\theta + \frac{1}{2}kx'^2[/tex]

so we will have

[tex]\frac{1}{2}(70)(0.5^2) = 2(9.81) (0.50 + x') sin41 + \frac{1}{2}(70)x'^2[/tex]

[tex]8.75 = 6.43 + 12.87 x' + 35 x'^2[/tex]

[tex]x' = 0.13 m[/tex]

so total distance moved upwards is

[tex]L = 0.5 + 0.13 = 0.63 m[/tex]

Part B

Now the incline has a coefficient of kinetic friction μk. If the block, attached to the spring, is observed to stop just as it reaches the spring's equilibrium position, what is the coefficient of friction μk

KE = 8.75 – 10.78 * sin 41

This is approximately 1.678 J. This is the kinetic energy at the equilibrium position. For the block to stop moving at this position, this must be equal to the work that is done by the friction force.

Ff = μ * 10.78 * cos 41

Work = 0.5 * μ * 10.71 * cos 41

μ * 10.78 * cos 41 = 8.75 – 10.78 * sin 41

μ = (10 – 8.82 * sin 41) ÷ 8.82 * cos 41

μ = 1.678/8.136

μ = 0.206

This is approximately 0.21

Answer:

A) d = 0.596m

B) μ = 0.206

Explanation:

A) The potential energy stored in a spring when it is compressed is given as;

U = (1/2)kx² - - - - - - (eq1)

Where

U is potential energy stored in spring

k is spring constant

x is compressed length of the spring

Let the height the mass moved before coming to rest be h.

Thus, the potential energy at this height is;

U = mgh - - - - - - (eq2)

Since the mass is connected to the spring, according to the principle of conservation of energy, initial potential energy of the spring is equal to the sum of the final potential energy in the spring and the potential energy of the mass. Thus, we have;

(1/2)K(x1)² = (1/2)K(x2)² + mgh

Where x1 is the initial compressed length and x2 is the final compressed length.

Now, h will be dsin41 while x2 will be d - x1

Where d is the distance the mass moves up before coming to rest

Thus, we now have;

(1/2)K(x1)² = (1/2)K(d - x1)² + mg(dsin41)

(1/2)K(x1)² = (1/2)K(d² - 2dx1 + (x1)²) + mg(dsin41)

(1/2)K(x1)² = (1/2)Kd² - Kdx1 + (1/2)K(x1)² + mg(dsin41)

(1/2)Kd² - Kdx1 + mg(dsin41) = 0

(1/2)Kd² + d[mg(sin41) - Kx1] = 0

From the question,

k = 70 N/m

m = 2.2 kg

x1 = 0.5m

g = 9.8 m/s²

Thus, plugging in these values, we now have;

(1/2)(70)d² + d[(2.2•9.8•0.6561) - (70•0.5)] = 0

35d² - 20.8545d = 0

35d² = 20.8545d

Divide both sides by d to get;

35d = 20.8545

d = 20.8545/35

d = 0.596m

B) Here we are looking for the coefficient of friction.

First of all, let's find the kinetic energy at the equilibrium position;

The potential energy of the spring;

P.E = (1/2)K(x1)² = (1/2)(70)(0.5)² = 8.75J

The energy that will cause the block to decelerate = mg(x1)sin41 = 2.2 x 9.8 x 0.5 x 0.6561 = 7.073

So,

Net KE = 8.75 – 7.073 = 1.677J

Now, for the block to stop moving at this equilibrium position, the work done by the frictional force must be equal to KE of 1.677J

Thus,

F_f(x1) = 1.677J

F_f = μmgcos 41

Where, μ is the coefficient of friction;

So, F_f = μ(2.2 x 9.8 x 0.7547)

Ff = 16.271μ

Thus,

16.271μ x 0.5 = 1.677J

8.136μ = 1.677

μ = 1.677/8.136

μ = 0.206