A point particle slides frictionlessly down a sphere, starting from rest a the top(θ=0). Show that the particle leaves the sphere when cos θ= 2/3. Use lagrange multiplier instead of newtonian mechanics

A very long coaxial cable consists of a solid cylindrical inner conductor of radius 3.8mm , surrounded by an outer cylindrical conductor with inner radius 7mm , and outer radius 9.3mm . The region between the two conductors is filled with a waxlike insulating material to keep the conductors from touching each other. The inner conductor carries a current 4.65A , into the page, while the outer conductor carries a current 4.6A , out of the page.

Part A

Use Amp�re's law to derive an expression for the magnetic field as a function of r(the distance from the central axis) at points between the two conductors (R1<r<R2i). Use this expression to find the magnetic field at 4.8mm .

Use Amp�re's law to derive an expression for the magnetic field as a function of r(the distance from the central axis) at points inside the outer conductor (R2i<r<R2o). Use this expression to find the magnetic field at 7.1mm .

Use into page as negative and out of page as positive current.

Comment

Hint 1. Ampere's Law

Ampere's Law states:  ?B??s=?0?Ienclosed Therefore you need to figure out what the enclosed current is for this case.

Hint 2. Enclosed current

The enclosed current is the current in the inner conductor + that portion of the current enclosed in the outer conductor. Use Itotal/Atotal=Ienclosed/Aenclosed Don't forget Atotal=pi?(R22o?R22i) and there is a corresponding expression for Aenclosed.

A uniform thin rod of length 0.4 m and mass 0.5 kg can rotate in a horizontal plane about a vertical axis on the left end of the rod. The rod is at rest when a 10.0-g bullet traveling in the horizontal plane of the rod is fired into the right end of the rod at an angle 90o with the rod. The bullet lodges in the rod and the angular velocity of the rod is 10 rad/s immediately after the collision. The rotational inertia of a rod about axis through center perpendicular to length L is (1/12)*ML2.

Find the rotational inertia of the rod-bullet system when the bullet lodges in the rod. Keep 5 decimal places.

What is the angular momentum after the collision? Keep 4 decimal places.

Find the bullet’s initial velocity.

What mass of steam at 100∘C must be added to 1.90 kg of ice at 0∘C to yield liquid water at 15 ∘C? The heat of fusion for water is 333 kJ/kg , the specific heat is 4186 J/kg⋅C∘ , the heat of vaporization is 2260 kJ/kg .

When a potential difference is applied to the piece of metal wire, a current of 4.1 mA flows through it. if this wire is replaced with a silver wire having twice the diameter of the original wire, how much current will flow through the silver wire?

Assume that the lengths of the potential difference across both wires are the same. ( the resistivity of the original metal is 1.68x10^-2 ohm . m, and the resistivity of silver is 1.6x10^-8 ohm . m )

Suppose that a particle with charge q and mass m is initially at the origin with zero velocity in the presence of external electric and magnetic fields given by

E = (Ex, 0, Ez),

B = (0, 0, Bz).

(1) Find x(t) and draw the trajectory the particle follows in space.

A pair of radio transmitters separated by 118m along a north-south line each transmit an in-phase 5.55MHz wave. Using small angle approximations, the smallest angle (with respect to the easterly direction from the pair of transmitters) of destructive interference is closest to

A.16°

B.19°

C.6.4°

D.9.7°

E.13°

A basketball is thrown vertically up from a 1.6‑m height with an initial speed v₀ = 23 m/s. At what speed will the ball hit the ground?

The speed of the ball, v₁ =  

How long will it take for the ball to reach the ground?

The time of flight, t₁ =  

Now the basketball is thrown vertically down from the same 1.6‑m height and with the same initial speed v₀ = 23 m/s. First, check your physical intuition and complete the following two statements:

If the initial velocity is directed down, the final speed of the ball (Select an answer: decreases, increases, stays the same, do not know) .

If the initial velocity is directed down, the time of the free fall (Select an answer: decreases, stays the same, increases, do not know) .

Now, with the new conditions, what will be the speed of the ball just before it hits the ground?

The speed of the ball, v₂ =  

How long will it take for the ball to reach the ground?

The time of flight, t₂ =  

Four lawn sprinkler heads are fed by a 1.7 cm diameter pipe. The water comes out of the heads at an angle of 40 ∘ to the horizontal and covers a radius of 7.8 m .

A) What is the velocity of the water coming out of each sprinkler head? (Assume zero air resistance.)

Express your answer to two significant figures and include the appropriate units.

B) If the output diameter of each head is 2.8 mm , how many liters of water do the four heads deliver per second?

Express your answer using two significant figures.

C) How fast is the water flowing inside the 1.7 cm diameter pipe?

Express your answer to two significant figures and include the appropriate units.

You put a can of water into a fire. The can heats up, and the water remains liquid. Neglecting the tiny change in volume due to thermal expansion, the change in the internal energy of the water is __________the heat absorbed by the water.

Two harmonic waves of pulsation m and amplitude a overlap at a point P. It is observed that also the resulting wave has amplitude equal to a.
> Calculate the phase shift Δφ between the two waves;
> Calculate the phase shift of the resulting wave with respect to the original wave it has
minor initial phase. If the phase shift of the two harmonics is 180 °, how much the resulting wave amplitude would become?
data:
1. amplitude of the waves: α
2. phase displacement Δφ = 180 °
Unknows:to calculate
1. phase displacement Δφ = ?
2. Initial phase of the resulting wave: Φο =?
3. Amplitude of the resulting wave A =?

A police siren of frequency is attached to a vibrating platform. The platform and siren oscillate up and down in simple harmonic motion with amplitude and frequency . (a) Find the maximum and minimum sound frequencies that you would hear at a position directly above the siren. (b) At what point in the motion of the platform is the maximum frequency heard? The minimum frequency? Explain.

Can you explain part B in details?thank you

Question 2: In a steam power plant operating according to the ideal intermediate superheated Rankine cycle, water vapor enters the high pressure turbine at 8 MPa pressure and 500 ^oC temperature and expands to 3 MPa pressure. Then, the steam is re-heated to a temperature of 500 ^oC in the boiler and expands to 20 kPa condenser pressure in the low pressure turbine.

I. Show the cycle in the T-s diagram and explain for what purpose intermediate superheating is done.

II. Calculate the total turbine work.

III. Calculate the thermal efficiency of the cycle.