(a) consider a 1-dimensional monatomic lattice of atomic spacing, a. by consideration of a simple harmonic (hooke's law) approach between nearest neighbours only, show that the relationship between the angular frequency, o, of a lattice disturbance and its wave number, k, (the dispersion relation) can be written in the form b2. o-n sin(ka/2) (the identity cos(ka)- 1-2 sin°(ka/2) may be useful.) [10 marks] (b) show that m can be written as om= 2(w/m)^, with m the atomic mass and u the force constant between neighbours. describe, with the aid of a sketch of o versusk, the physical significance of o indicate the first brillouin zone on your sketch [4 marks] (e) for solid argon the maximum phonon energy is 6 mev. given that argon has a mass number of 40 and that the mass of the proton (and neutron) is 1.67 x 10 kg, calculate the force constant in units of kgs2. (e=1.60 x 10*19; a 1.05 x 104 js) 3 marks (d) with the dispersion relation given in part (a), show that the group and phase velocities are equal in the long wavelength limit.

Many machines- including inclined planes such as ramps- increase the strength of the force put into the machine but decrease the distance over which the force is applied. true or false

For problems 1-3, consider a simple dc brush motor with a permanent magnet stator. with an an consider a simple dc brush motor with a permanent magnet ae supply voltage of 100 v, the no-load speed of the motor is 1000 rpm. this motor can provide a torque of 1.9 n-m at 800 rpm drawing a current of 2 a current are both zero at no load.) 1. with an armature supply voltage of 100 v, the motor is operated at 900 rpm. what is the motor torque? what is the mechanical power delivered by the motor? what is the current draw? what is the electrical power input? what is the energy efficiency of the motor? with an armature supply voltage of 100 v, the motor is operated at 1100 rpm (where it acts as a generator). what torque input is required? what mechanical power is required what current is generated? what is the electrical power generated? what is the energy efficiency of the motor acting as a generator? 2. with an armature supply voltage raised to 120 v, the motor is operated at 1000 rpm. what is the no-load speed? what is the motor torque? what is the mechanical power delivered by the motor? what is the current draw? what is the electrical power input what is the energy efficiency of the motor? 3.

How much force was applied to an object if was moved 2 meters and the work done on the object was 8 joules? a. 0.25 n b. 4 n c. 6 n d. 16 n