Masses coupled through springs is a model for the behavior of atoms in a crystal. Consider a one-dimensional crystal, what if we have a crystal made of two kinds of atoms with different masses? Consider a one-dimensional lattice made up of alternating large and small masses, M and m, respectively. Assume periodic boundary conditions. (a) Calculate the normal modes of this lattice (you should find two normal modes). (b) Describe the relative motion of the two types of masses for each mode.

Neutrons are born out of fission at a high energy and must be slowed down to the low energy region, called the “thermal energy” region, to cause another fission. A material called a “moderator” is used to slow the neutrons down, or “moderate” them, by elastic scattering collisions.The average neutron energy coming out of fission of thermal neutrons is 1.98 MeV in kinetic energy. What is the number of collisions required to bring its average kinetic energy to 0.1 eV if it elastically scatters off of the following moderators:

a. Hydrogen-1

b. Hydrogen-2

c. Carbon-12

d. Uranium-238

A bullet is fired through a board 10.0cm thick in such a way that the bullet’s line of motion is perpendicular to the face of the board. If the initial speed of the bullet is 4.00 x 10^2 m/s and it emerges form the side of the board with a speed of 3.00 x 10^2 m/s, find (a) the acceleration of the bullet as it passes through the board and (b) the total time the bullet is in contact with the board.

Question 1:

(a) A gymnast jumps from a platform at vertical position s_y = 1.0m above a trampoline at s_y = 0 m. The gymnast’s vertical velocity at

the moment when she leaves the platform (time t = 0) is  v_y = 3.0 m/s. The positive vertical direction is upwards. Assume that the effects of air resistance and friction can be neglected touches the trampoline at time t= t₁

(i) Sketch a graph of the gymnast’s vertical position versus time between t=0 and t1. Label s_y =0m, t=0,and t₁ on your graph.

(ii) Sketch a graph of the gymnast’s vertical velocity versus time from the moment she leaves the platform until the moment she touches the trampoline. Label v_y = 0 ms^-1,

t = 0, and t₁ on your graph.

(iii) Calculate the vertical velocity of the gymnast at the instant she touches the trampoline.

(iv) The trampoline’s canvas is connected to its frame by a number of springs. One of these springs has a spring constant of 2000 N m^-1 Calculate the strain potential energy of the spring when it is extended by 0.15 m from its equilibrium position. Assume that this extension obeys Hooke’s Law.

(v) The gymnast now does a somersault and rotates with an average angular velocity of 6 rad s^-1.Angular momentum is a conserved quantity and the gymnast had zero angular momentum when she jumped from the platform. Explain how this is possible.

(b) A coach standing in the gym hits a drum which emits a sound with a frequency of 110 Hz. The speed of sound in air is 340 m s .

(i) An athlete is running straight towards the coach with velocity 20.0 km h^-1. What frequency will the athlete hear? (3 marks )

(ii) x defines the distance from one edge of the drum’s membrane to the opposite edge, passing through the centre of the membrane. A standing wave on the drum membrane can be described byy(x, t) = 2A cos(ωt) sin(kx)

where y is the vertical displacement of the drum’s membrane at position x and time t. Briefly explain what A,ω, and k signify in this equation.

A neutron in a nuclear reactor makes an elastic, head-on collision with the nucleus of a carbon atom initially at rest. (a) What fraction of the neutron's kinetic energy is transferred to the carbon nucleus? (The mass of the carbon nucleus is about 12.0 times the mass of the neutron.) (b) The initial kinetic energy of the neutron is 1.30 10-13 J. Find its final kinetic energy and the kinetic energy of the carbon nucleus after the collision. neutron J carbon nucleus J

A person walks into a room that has, on opposite walls, two plane mirrors producing multiple images. Find the distances from the person to the first three images seen in the left-hand mirror, when the person is 9.00 ft from the mirror on the left wall and 10.0 ft from the mirror on the right wall.

(a) What is the magnitude of the tangential acceleration of a bug on the rim of a 12.0-in.-diameter disk if the disk accelerates uniformly from rest to an angular speed of 77.0 rev/min in 3.70 s?
m/s²

(b) When the disk is at its final speed, what is the magnitude of the tangential velocity of the bug?
m/s

(c) One second after the bug starts from rest, what is the magnitude of its tangential acceleration?
m/s²

(d) One second after the bug starts from rest, what is the magnitude of its centripetal acceleration?
m/s²

(e) One second after the bug starts from rest, what is its total acceleration? (Take the positive direction to be in the direction of motion.)

Consider if you were to vary the number of washers suspended on the end of the string, how would this affect the centripetal force? How would increasing the number of washers at the end of the string affect the periodic time of moving mass? Generally, explain the effect of moving mass and radius on centripetal force.

Please give explantations

A heavy sled is being pulled by two people as shown in the figure. The coefficient of static friction between the sled and the ground is μs = 0.571, and the kinetic friction coefficient is μk = 0.395. The combined mass of the sled and its load is m = 306 kg. The ropes are separated by an angle φ = 23°, and they make an angle θ = 29.7° with the horizontal.

Assuming both ropes pull equally hard, what is the minimum rope tension required to get the sled moving?

If this rope tension is maintained after the sled starts moving, what is the sled\'s acceleration?

An airplane is heading due south at a speed of 680 km/h .

A. If a wind begins blowing from the southwest at a speed of 90 km/h (average), calculate the velocity (magnitude) of the plane relative to the ground.

B. Calculate the velocity (direction) of the plane relative to the ground.

C. Calculate how far from its intended position will it be after 18 min if the pilot takes no corrective action.

A rod AB with A at the origin rotates such that the angle between the rod and the x?axis is theta = pi (8t

A charge q1 = 5.00 nC is placed at the origin of an xy-coordinate system, and a charge q2 = -4.50 nC is placed on the positive x axis at x= 4.00 cm . If a third charge q3 = 8.00 nC is now placed at the point x= 4.00 cm , y= 3.00 cmcm find the x and y components of the total force exerted on this charge by the other two charges. Find Fx, Fy, and find the magnitude and direction of F.