Tutorial work - Tutorials (9) to practice problems with phys321a PDF

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Tutorials (9) to practice problems with PHYS321A...

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Tutorial #3 Problems (Chapter 4 (Fowles), and Mid-term preparation) Problem 1: A gun is located at the bottom of a hill of constant slope φ. Show that the range of the gun measured up the slope of the hill is

Where α is the angle of elevation of the gun, and that the maximum value of the slope range is

Problem 2: A bead slides on a smooth rigid wire bent into the form of a circular loop of radius b. If the plane of the loop is vertical, and if the bead starts from rest at a point that is level with the centre of the loop, f ind the speed of the bead at the bottom and the reaction of the wire on the bead at that point. Problem 3: Let a particle of unit mass be subject to a force x-x3. Where x is its displacement from the coordinate origin. a) Find the equilibrium point, and determine whether they are stable or unstable. b) Calculate the total energy of the particle, and show explicitly that it is a conserved quantity. Problem 4: Consider a particle of mass m and charge q>0, subject to an electric field along the x direction, E=e . The particle moves under the effect of air resistance that is quadratic in the velocity v. a) Write down the equation of motion for the particle. b) Find the magnitude and direction of the terminal velocity of the particle. c) Assume the initial conditions x0=y0=z0=0 and v0=y0=z0=0. Find the velocity and position as a function of time. Useful integrals: ∫ and ∫

Tutorial #5: Chapter 5 (Fowles & Cassiday) Problem 1: Consider a simple pendulum (mass m, length l) mounted inside a railroad car that is accelerating to the right with constant acceleration . Find the angle at which the pendulum will remain at rest erlative to the accelerating car and find the frequency of small oscillations about this equilibrium angle.

Problem 2: A particle of mass, m, is confined to move, without friction, in a vertical plane, with axes x horizontal and y vertically up. The plane is forced to rotate with constant angular velocity about the y axis. Find equations of motion for x and y, solve them and describe the possible motions.

Problem 3: A bucket of water is spinning about its vertical axis with angular velocity Ω. Show that once the water has settled in equilibrium (relative to the bucket), its surface will be a parabola. (Use cylindrical polar coordinates and remember that the surface is an equipotential under the combined effects of the gravitational and centrifugal forces.)

P321a Tutorial #6 Problem 1: Show that the gravitational force on a test particle inside a thin uniform spherical shell is zero a) by finding the force directly, b) by showing that the gravitational potential is constant.

Problem 2: A particle moving in a central field describes the spiral orbit . Show that the force law is inverse cube and that varies logarithmically with t.

Problem 3: Calculate Earth’s velocity of approach toward the sun, when Earth in its orbit is at an extremum of the latus rectum through the sun. Take the eccentricity of the earth’s orbit to be 1/60 and its semimajor axis to be 93000000 miles.

P321a Tutorial #7 Problem 1: Use the radial equation to determine the orbit of a free particle.

Problem 2: A particle of mass m moves with angular momentum l about a fixed force centre with F(r)=k/r3 where k can be either positive or negative. Sketch the effective potential energy for various values of k and describe the various possible orbits based on Ueff.

Problem 3: A spacecraft in a circular orbit wishes to transfer to another circular orbit of quarter the radius by means of a tangential thrust to move into an elliptical orbit and a second tangential thrust at the opposite end of the ellipse to move into the desired circular orbit. Find the thrust factors required.

P321a Tutorial #8 Problem 1: A system consists of three particles, each of unit mass, with positions and velocities as follows:

=j =i+j+k a) Find the position and velocity of the centre of mass. Find also the linear momentum of the system. b) Find the kinetic energy of the system c) Find the value of d) Find the angular momentum about the origin Problem 2: A block of wood rests on a smooth horizontal table. A gun is fired horizontally at the block and the bullet passes through the block, emerging with half its initial speed just before it entered the block. Show that the fraction of the initial kinetic energy of the bullet that is lost as frictional heat is ¾-1/4γ, where γ is the ratio of the mass of the bullet to the mass of the block (γ...