A ball enters a frictionless track spiraling upward with a speed vo. The radius of the track is half a meter and it has a pitch (ѳ) of 15°.

(a) If the initial speed of the ball is 12 m/s, how high vertically will it rise?

(b) Compare your answer to part (a) with the height the ball would rise if it had been thrown vertically upward, into freefall, with the same initial speed. Explain your results.

(c) If you wish the ball to make exactly 5 revolutions before momentarily coming to rest, how fast must it initially be moving when entering the spiral track?

(d) What assumptions or simplifications did you make in part (c) about the total distance traveled that leaves your answer very close but not exact (friction is not the answer)? Use Related Quantities in conjunction with Order of Magnitude sense-making to justify your assumptions/simplifications.

A horizontal spring with a spring constant of 190 N/cm is compressed 6.3 cm. A wooden block with a mass of 1.5 kg is placed in front of and in contact with the spring. When the spring is released it pushes the block, which slides on a frictionless horizontal surface for some distance. The block then slides up a frictionless incline of 27 above the horizontal and comes to a momentary stop before sliding back down. The system is the spring, the block, the incline, and the Earth. Ignore air resistance.

A) Sketch the situation and label all variables for: i. the spring and block before the spring is released ii. the block sliding on the horizontal surface iii. the block sliding up the incline

B) List all known quantities.

C) Draw the free body diagram for the block sliding up the incline.

D) What is the potential energy of the spring before it is released?

E) What is the kinetic energy and the speed of the block as it slides on the horizontal surface after the spring has pushed it?

F) At what height does the block stop on the incline?

G) If the incline were rough, how would the stopping height of the block compare to the stopping height when the incline is frictionless? Explain using energy.

People who do very detailed work close up, such as jewelers, often can see objects clearly at a much closer distance than the normal 25.0cm. What is the power of the eyes of a woman who can see an object clearly at a distance of only 8.75cm? Assume a distance from the eye lens to the retina of 2.00cm

power: 61.43diopters -right

What is the size of the image of a 1.50mm object, such as lettering inside a ring, held at this distance? Enter your answer as a positive number if the image is upright and negative if it is inverted.

image size at : −0.23- wrong

What would the size of the image be if the object were held at the normal 25.0cm ?

image size at : −0.34 - wrong

A particle with speed V1= 75 m/s makes a glancing elastic collision with another particle that initially is at rest. Both particles have the same mass. After the collision, the struck particles moves off at an angle that is 45 degrees above the line along V1. The second particle moves off at 45 degree below this line. The speed of the struck particle after the colllision is approximately.

A: 38 m/s

B: 82 m/s

C: 64 m/s

D: 47 m/s

E: 53 m/s

Consider a thin spherical shell located between r = 0.49a₀ and 0.51a₀. For the n = 2, l = 1 state of hydrogen, find the probability for the electron to be found in a small volume element that subtends a polar angle of 0.11° and an azimuthal angle of 0.25° if the center of the volume element is located at: θ=5°, ϕ=35°.

1.Consider a negative point charge. Sketch electric field lines, including their direction.

2 Calculate electric field of 1 electron at a distance of 0.1 nanometer away from it. Express your answer in SI units.

3 Consider a point charge of 1 C and calculate its electric field at a distance of 1 m.

Consider a sheet of paper with a thickness of 0.1 mm and a dielectric constant of 3. Suppose we would like to make a 1 F capacitor using this sheet as a spacer between two metallic plates.

Suppose we took out the paper dielectric (while keeping somehow the distance between the plates the same).

1.26: What is the charge after the paper sheet is out completely?

1.27: What is voltage across the capacitor after the paper sheet is out completely?

1.28: How much work does it take to remove the sheet? Explain the sign of your answer.

1.29: How will the answer to 1.28 change is the capacitor remains connected to the 1V battery during the sheet removal.

A small solid sphere of mass M₀, of radius R₀, and of uniform density ρ₀ is placed in a large bowl containing water. It floats and the level of the water in the dish is L. Given the information below, determine the possible effects on the water level L, (R-Rises, F-Falls, U-Unchanged), when that sphere is replaced by a new solid sphere of uniform density.

The new sphere has mass M = M₀ and density ρ < ρ₀

The new sphere has radius R > R₀ and density ρ = ρ₀
The new sphere has radius R > R₀ and density ρ < ρ₀

The new sphere has density ρ = ρ₀ and mass M < M₀

The new sphere has radius R = R₀ and mass M < M₀

The new sphere has mass M = M₀ and radius R < R₀

Let's figure out the energy and momentum associated with that motion.

For the purposes of this problem, treat the Earth as a solid, uniform sphere with mass 5.97×10²⁴ kg and radius 6.37×10⁶ m , and assume that the Earth's orbit around the sun is circular with a radius of 1.5×10¹¹ m .

Part A

What is the angular kinetic energy of the Earth due to its orbit around the sun?

Part B

What is the magnitude of the Earth's angular momentum due to its orbit around the sun?

Part C

What is Earth's angular kinetic energy due to its rotation around its axis?

Part D

What is the magnitude of the Earth's angular momentum due to its rotation around its axis?

Part E

Remember that energy and momentum are always conserved (though energy can change forms). In other words, if you start with a certain amount of energy and momentum, you must end with the same amount of energy and momentum. By conservation of energy and momentum, the values you've calculated in this problem must have come from somewhere.

Which of the following best explains where the Earth's angular kinetic energy and momentum came from?

Remember that energy and momentum are always conserved (though energy can change forms). In other words, if you start with a certain amount of energy and momentum, you must end with the same amount of energy and momentum. By conservation of energy and momentum, the values you've calculated in this problem must have come from somewhere.

Which of the following best explains where the Earth's angular kinetic energy and momentum came from?

Two identical twins hold on to a rope, one at each end, on a smooth, frictionless ice surface. They skate in a circle about the center of the rope (the center of mass of the two-body system) and perpendicular to the ice. The mass of each twin is 78.0 kg. The rope of negligible mass is 4.0 m long and they move at a speed of 4.90 m/s.

(a) What is the magnitude, in kg · m²/s, of the angular momentum of the system comprised of the two twins?

(b) They now pull on the rope and move closer to each other so that the rope between them is now half as long. Determine the speed, in m/s, with which they move now.

(c) The two twins have to do work in order to move closer to each other. How much work, in joules, did they do? This is the same as the change in kinetic energy.

A 30 kg projectile is launched from the ground at an initial velocity of 300 m/s at an angle of 45 degrees above the horizontal. If air resistance is ignored, determine the following:

a. The projectile's speed at 2000 meters above the ground.

b. The total amount of energy the object has at 3000 m.

c. The maximum height of the projectile.

d. The maximum distance the projectile travels horizontally.

1. Provide an example of an inertial frame of reference and a non-inertial frame of reference. Explain the difference.

2. Using the Michelson-Morley experiment as an example, explain why classical mechanics was unable to explain natural phenomena.

3. Using at least one of Einstein's "thought-experiments", explain how special relativity addresses how it is possible for observers in two different inertial reference frames to “disagree” about time and distance intervals.

4. Describe how special relativity explains the conditions under which classical mechanics breaks down. (When would you, as an observer begin to notice the effects of time dilation and length contraction?)

5. In the early 20th century, the law of conservation of mass was replaced by the law of conservation of mass-energy. Why was this change needed, and how does E=mc² relate to the special theory of relativity?

a) The operator aˆ satisfies the equation [aˆ, aˆ†] = n. Prove that n is a real number. [Hint: Apply the Hermitian conjugation operation “†” to the above equation.]

b) Consider the Hamiltonian Hˆ = h ̄ωˆa†ˆa, where ω > 0 is a real parameter. Prove that Hˆ is Hermitian.

c) Let |ψ〉 be an eigenstate of the above Hamiltonian with the energy ε. Use the commutation relations from part a) to prove that aˆ†|ψ〉 is also an eigenstate of Hˆ. Find the energy of the state aˆ†|ψ〉. You do not need to prove that aˆ†|ψ〉 ≠ 0.

d) Let n > 0. Use the commutation relations to find the ground state energy of Hˆ .