Let’s look at some gravitational potentials and the density profiles that can generate them.

i) For ρ(r) = ρ◦r ◦^2/r^2 , what is the corresponding gravitational potential?

ii) For Φ(r) = −GM/[b + √ (b ^2 + r ^2)], what is the corresponding density profile?

iii) For Φ(R, z) = Φ◦ln[ (R ◦^2+R ^2+(z ^2/q^2))/R◦^2 ], what is the corresponding density profile ρ(R, z)? In this expression, the constant q controls the axis ratio of the potential, where q = 1 would be spherically symmetric and q < 1 means that the potential is “flattened” a bit along the z-axis. Does something strange happen along the z axis if q < 0.7 ?

In this exercise, the first of a series, we will make connections between the physics you have been learning (in this case, kinematics) and how it is used by people in their work and research. Today we consider an example from kinesiology research, based on a 2000 paper from the Journal of Measurement in Physical Education and Exercise Science (nota bene: you do not have to read this paper, or any of the hyperlinks to follow the exercise – it is only provided here for your interest).

Jumping is an activity common to many sports, such as basketball, volleyball, and others. An important part of training to improve jumping ability is tracking your progress: how else will you decide if your chosen routine is working? Researchers in the field , as well as professional athletes and their coaches , use a variety of techniques to accurately measure the vertical jump height. One approach is straightforward: just "see" how high you jump! To do it accurately, however, you need high speed cameras, a way of calculating the position of the body's centre of mass, balancing torques... very complicated. Or you can use the old "reach" method, having subjects jump and touch the highest point they can reach with a chalk-covered palm – but now you have to correct for differential arm length, swing timing... rather crude.

However, with your knowledge of kinematics, you can get away with as little as a stopwatch, saving valuable time and money. Below we will get you to follow in the footsteps of pioneering kinesiologists and exercise scientists and re-invent two of these physics-based methods. (By the way, this kind of use of mechanics in biology is called biomechanics.)

a) (Please refrain from looking at the rest of the question until you have given this part some thought: not to worry, no wrong answers here!) Given your knowledge of kinematics, what variable would you want to measure to be able to calculate someone's vertical jump height? Please explain in a few words how you would go about doing this.

b) Suppose you could measure the time of flight ?flight. How would you calculate the vertical jump height ℎ? (Unsurprisingly, this is called the time-of-flight method . Calculate the predicted height for the following times-of-flight: 0.827 s (LeBron James), 0.846 s (Michael Jordan), 0.864 s (Wilt Chamberlain), 0.53 s (the author of this exercise).

c) What about if you measured the vertical take-off velocity ?=(0,??) instead? (This is known as the impulse-momentum method , because of the way the take-off velocity is calculated using a force platform). Calculate the predicted height for the following take-off velocities: 3.91 m/s (LeBron James), 4.00 m/s (Michael Jordan), 4.09 m/s (Wilt Chamberlain), 2.60 m/s (the author of this exercise).

d) Suppose you have both numbers: would you expect them to be consistent, i.e., to give the same predicted vertical jump height? Give your reasons. How might you check for consistency?

e) Use the two datasets from the previous parts of this problem to check consistency. Were you right? If you found an inconsistency, list some of the reasons behind this inconsistency. If not, explain why you should expect them to be consistent.

One of the advantages of skates is that they allow you to push against ice, which
typically has low friction. Let’s consider an hockey player on frictionless ice who is
going to shoot a slap shot. The player weighs Mp = 100kg. He is going to fire the
puck, which weighs Mpuck = 0.5kg with a velocity of v = 50(m/s). How fast does the
hockey player recoil in the direction opposite to the puck’s trajectory. If the force he
exerts on the puck occurs over a time, ∆t = 0.1s what is the average force needed to
create the impulse that accelerates the puck to that velocity?

Please explain carefully so that I can understand it!

Two charged metal balls of a different size, radius R1 is bigger than R2, and then connected by a wire. Which ball gets the larger charge? What about charge density? Prove the answer.

Uniqueness theorem

Prove that the solution to the Laplace’s equation in a spatial region is unique if the potential is specified on the surface of the region.

A charge of -2.705 μC is located at (2.830 m , 4.456 m ), and a charge of 1.460 μC is located at (-2.757 m , 0).

Find the electric potential at the origin. the answer to this was 154 V.

Here is the part I dont understand. Need (x,y) coordinates

there is one point on the line connecting these two charges where the potential is zero. Find this point.(x,y)

Express your answers using three decimal places separated by a comma.

Please show work and answer in (x,y)

True/False:

1. Gravity is an example of inverse square behavior.
2. A grindstone is rotating at constant rotational speed. There is a net torque on it.
3. A lever multiplies a force but does not redirect it.
4. On a rotating merry go round one child sits halfway out between the axis and the edge. Another child sits on the edge. Both kids have the same rotational speed.
5. Angular momentum is a vector.

If X, Y and Z are three arbitrary vectors, prove these identities:

a. (X×Y).Z = X.(Y×Z)

b. X×(Y×Z) = (X.Z)Y – (X.Y)Z

c. X.(Y×Z) = -Y.(X×Z)

How can a force (e.g., a muscle force or a gravitational force) produce a translatory and a linear force simultaneously?

1 .Identify type of image represented in the following graphic:

Group of answer choices

Real

Virtual

2. Identify type of image represented in the following graphic:

Group of answer choices

Virtual

Real

3. Identify the direction that the refracted beam will bend:

Group of answer choices

Towards the normal

Away from the normal

4. The capability of a prism that to separate white light into its constituent colors is due to the slight variation of index of refraction as a function of wavelength. This property is referred to as:

Group of answer choices

Dispersion

Funky Colors

Diffusion

Total internal Reflection

5. True or false: A ray of light within a medium of higher index of refraction comes to an interface with a medium of lower index of refraction at an angle greater than the critical angle. There is no light transmitted into the medium of lower index of refraction

Group of answer choices

True

False

6.You are standing exactly halfway between to perfectly parallel mirrors. You see your reflection in the mirror that you are facing, but you notice a second (and many more) reflection that appears even further away.   How much further away is the second reflection image from the first reflection image in the mirror you are facing.

Group of answer choices

10 feet from first reflection

• 1 Light year from first reflection

15 feet from first reflection

5 feet from first reflection

7.(True or false) Parallel rays that pass through a thin lens will be be directed through the lens focal point.

Group of answer choices

True

False

8.(True or false) The law of reflection is mathematically expressed with Snell's law: n 1 sin ⁡ θ 1 = n 2 sin ⁡ θ 2

Group of answer choices

True

False

Suppose a photon is absorbed by the electron in a hydrogen atom in an n= 2 state. What wavelength should the photon have to enable the electron to transition to the n= 4

state? Once the photon is absorbed, what are the various wavelengths of photons that could be emitted by the atom? (Use Bohr approximation).

A 54.0-kg woman stands at the western rim of a horizontal turntable having a moment of inertia of 490 kg · m² and a radius of 2.00 m. The turntable is initially at rest and is free to rotate about a frictionless, vertical axle through its center. The woman then starts walking around the rim clockwise (as viewed from above the system) at a constant speed of 1.50 m/s relative to the Earth. Consider the woman–turntable system as motion begins.

(a) Is the mechanical energy of the system constant?

Yes/No

(b) Is the momentum of the system constant?

Yes/No

(c) Is the angular momentum of the system constant?

Yes/No

(d) In what direction and with what angular speed does the turntable rotate?

Magnitude: ___ rad/s

Direction: Clockwise/Counterclockwise

(e) How much chemical energy does the woman's body convert into mechanical energy of the woman–turntable system as the woman sets herself and the turntable into motion?
___ J

A spherical shell is cut into two halves that remain forming the
sphere, but are electrically isolated from each other at a negligible distance. The nortern hemisphere is at potential +V_0 and the southern hemisphere is at potential -V_0. Obtain the potential V (r, θ, ϕ) inside and outside of the sphere.

Figure out

how to test what happens when the light rays come from underwater. Compare your ideas about why

things look different under water to how light rays appear to “bend.”.