Mechanics Departmental Eam Last updated November 213 1. Two satellites are moving about each other in circular orbits under the influence of their mutual gravitational attractions. The satellites have comparable but not equal masses. The period of their rotation is τ and the radius of their orbit is ro. The motion is suddenl stopped. Then, the particles are released with zero velocit and allowed to fall into each other. a. Prior to the stopage of the circular motion what is the period of rotation τ in terms of their masses, radius of orbit and fundamental constants? b. After the stopped and released, how much time passes before the two satellites collide? Epress our results in terms of τ. Your results ma involve an integral, which ou do not need to evaluate. 2. A projectile eperiences a drag force proportional to its velocit (use the proportionalit constant b). a. Write down the equation of motion for a purel vertical trajector. b. Solve the equation of motion. c. What is the maimum height the projectile reaches? What is the result in the limit b? b? 3. Show that the surface of a liquid contained in a bucket which rotates about its smmetr ais at angular velocit ω is a paraboloid of revolution. 4. A projectile is fired from the origin of the coordinate sstem with initial velocit u in a direction making an angle θ with the horizontal. Compute the time needed for the projectile to cross a line passing through the origin and making an angle θ 1<θ with the horizontal. 5. The force F is given b 2 2 2 2 ( ) ( ) F = a 3z ˆi + 3a z ˆj 6 az k. ˆ a. Show that F is conservative. b. Find the potential energ function associated with this force. 6. Find Hamilton's equations of motion of a bead of mass m sliding on a frictionless wire under the influence of gravit. The wire has parabolic shape and rotates with constant angular velocit ω about the z ais. z ω wire gẑ ϕ m
7. Assume the pulles are weightless and the three objects are hung as shown. Is this in equilibrium? If not, find the direction and the magnitude of the acceleration of the 5 kg object. 8. A uniform door is to be hung using two hinges as shown. Where should the hinges be placed (Y1=? Y2=?) so that the horizontal force eerted on the top hinge is minimum? 9. A spherical water drop falls in a saturated water vapor atmosphere in the homogeneous gravitational field near the earth's surface. Due to condensation its mass increases at a rate that is proportional to its surface. a. Assume an initial radius ro = r(t=) and determine the radius of the drop at time t >. b. Obtain and solve the equation of motion. The initial conditions for position and velocit of the drop at time t= are z= and v=. c. Compare our result for ver large times ( t ) with a free falling drop of constant mass. Hint: The linear inhomogeneous differential equation ' + g() = h() with (o) = o is solved b G() o G(t) { o } () = e + h(t)e dt G(): = g(t) dt. o
1. A particle of mass m is constrained to move on the surface of a clinder defined b 2 + 2 = R 2. The particle is acted on b an attractive central force given b F = kr = k i ˆ+ j ˆ+ zk ˆ. Find and solve the equations of motion of the particle. 11. The solid clinder shown is connected b means of a spring, k = 3 N/m, to a rigid wall. The mass and radius of the clinder are 5 kg and 5 cm, respectivel. a. If the clinder rolls without slipping, describe the motion which results from an initial displacement of =3 cm =.3 m and an initial velocit of +.2 m/s as shown. b. What is the amplitude of the motion? c. What is the maimum velocit of the clinder? d. At what time after the initiation at t= of the motion will the magnitude of the velocit first be a maimum value? 12. A particle of mass m moves under the action of a central force whose potential is V(r) = Kr 3, (K>). a. For what energ and angular momentum will the orbit be a circle of radius a about the origin? b. What is the period of this circular motion? c. If the particle is slightl disturbed from this circular motion, what will be period of small radial oscillations about r=a? 13. There is friction between masses m 1 and m 2 and between m 1 and the tabletop. The coefficient of static friction for both is µ s, and of kinetic friction for both is µ k. a. Calculate the acceleration of the sstem. b. Under what conditions (i.e., relationships of the masses) does m 2 sta on top of m 1? c. Describe what happens phsicall (motion of each block) in the following limits: 1) M 2) µ s, µ k 3) m2
14. A homogeneous sphere (radius r, mass m, rotational inertia I = 2/5 mr 2 ) rests on top of a semi-sphere (radius R) that is fied in space. At time t= the sphere starts rolling under the influence of gravit. At what angle φ does it leave the semi-sphere? 15. A particle of mass m is constrained to move on the surface of a clinder defined b 2 + 2 = R 2. The particle is acted on b an attractive central force given b F = kr = k i ˆ+ j ˆ+ zk ˆ. Find and solve the equations of motion of the particle. 16. Consider a block of mass M 1 sliding without friction down an inclined plane of angle θ where M 1 is attached via a weightless, fleible cord of length L to a hanging mass m 2. a. Epress T and V in terms of 1, 1, 2, and 2. b. Epress the equation of constraint. c. Using the method of Lagrange multipliers find the equations of motion. d. Discuss briefl all possible phsical solutions. 17. Describe the motion a particle in the following potentials U(). For each potential, give conditions for bounded and unbounded motion (if applicable) and epress the location of the turning point(s) as a function of the total energ E. a. ( 2a a U() A e 2e ) = (Morse potential Fig. 1a); U b. U() = o (Fig. 1b); 2 cosh α 2 c. U() = U tan a (Fig. 1c).
18. A block of mass M rests on a frictionless horizontal table and is connected to two fied posts b springs having spring constants, k1 and k2 respectivel a. If the block is displaced slightl from its equilibrium position, what is the vibration? Suppose that the block is vibrating with amplitude A and that, a is passing through its equilibrium position, a mass m is dropped verticall sticks to it. Find: (i) the new frequenc of vibration, (ii) the new amplitude of vibration. 19. Consider the motion of a particle of mass m which is constrained to move on the surface of a cone having half angle α. The particle is subject to a gravitational force, mg, where g = gzˆ. z m α a. Using clindrical polar coordinates ( ρθ,, z) find the Lagrangian for this sstem using a coordinate sstem with the ape of the cone at the origin and gravit being along the z ais, i.e., g = gzˆ. b. Assuming that the particle moves in a circular orbit atρ=ρ, show that the frequencies for small oscillations about the stable orbit position are given b 3 ω= sin α, 2 mρ where is the angular momentum and ρ is the radius of the stable circular orbit.
2. Consider a particle of mass m that is constrained to sta on a smooth spherical surface under gravit, such as a small mass sliding around the inside a smooth spherical bowl. Use spherical coordinates to solve this problem, with the origin at the center of the sphere. Use ϕ and θ as our generalized coordinates. a. Derive the Lagrangian for the sstem. b. Find an equation of motions. c. Find an effective force and effective potential U eff (θ ). d. Find the equilibrium position θ. Is it a stable equilibrium point? e. Find the frequenc of small oscillations about the equilibrium point. 21. Find the work done to move a particle of mass m along a semicircle of radius a b a force F = -kd which alwas points at (a,), and D is the cord distance from the particles position and the point (a,). 22. A simple pendulum of length l with mass m is pivoted above a point that moves both horizontall and verticall in the plane of the pendulum with constant acceleration a= aˆ+ aˆ. Gravit acts on the mass in the ˆ direction. Ignore the mass of the string and pivot point. ϕ l a a. Derive the equations of motion for ϕ () t where ϕ () t the angle made with respect to the ais. Hint: the equations of motion should reduce to the class eample for a =. b. Describe the motion when a = g and a =. c. Describe the motion when a = g/2 and a =. d. Determine the equilibrium position ϕ = ϕ. e. Determine the frequenc of small oscillations ω about ϕ = ϕ. f. What is ω for a = and a =. Does this answer make sense? m
23. A particle of mass m attached to a rod of negligible mass is pivoted at point A and attached to two similar springs (of spring constant k). In equilibrium the value of θ is zero and the equilibrium length of each spring is o. Neglect the effects of gravit. a. For small oscillations, find the Lagrangian of this sstem in terms of θ and the contents given in the diagram above. b. Find the angular frequenc of oscillation. 24. The CO 2 molecule is described as the linear combination O=C=O. The mass of the carbon atom is 1.995 1-26 Kg and that of the ogen atom is 2.658 1-26 Kg. There are three normal vibration frequencies (eigenvalues) of, 3.75 1 11, and 7.2 1 11 Hertz. a. What is the approimate force constant between the atoms in Newtons/meter? b. Calculate the three sets of eigenvectors associated with the three eigenvalues. c. Describe what these eigenvectors tell ou about the vibration modes. 25. Two masses, m1 and m2, are connected b springs of spring constants k1 and k2 and natural lengths L1 and L2 as shown in the figure. Point P is fied and O1 and O2 mark the equilibrium positions of the springs. Suppose at t=, m1 is displaced a distance a1 from O1 and m2 is displaced a distance a2 from O2, and then both masses are released. Find the position 1(t) of m1 and position 2(t) of m2 for all t >. Assume no friction or eternal forces.
26. A planar annulus of uniform mass M, inner diameter a, and outer diameter b lies in the plane with its center at the origin. A small point mass m ( m M ), originall at the origin, is displaced a small distance h along the z-ais where h/ a 1 and h/ b 1, and then let go with zero initial velocit. Find the frequenc of small oscillations ω the point mass makes about z=. z b a m h M 27. Obtain the Lagrangian and equations of motion for the double pendulum shown in the figure, where the lengths of the pendula are 1 and 2 with corresponding masses m 1 and m 2. 28. A blob of putt having mass m 1 is incident upon a meter stick of mass m 2 which lies on a frictionless surface initiall at rest as shown below. The putt collides with the meter stick and sticks to it. Find the translational velocit and rotational velocit of the sstem after the collision takes place.
29. A mass is suspended from a spring and is free to move in all directions. Without the mass, the equilibrium length of the spring is. a. How man degrees of freedom are there? b. Write down the equations of motion and note an conserved quantities. c. Describe the normal modes phsicall and give the frequencies of small oscillations. 3. Eplain in our own words what do we mean b (NOTE: on the actual eam onl some of the questions below might appear): a. An inertial reference frame. b. A noninertial reference frame. c. Give two eamples of inertial and noninertial reference frames. d. Formulate Newton's laws. Are these alwas valid? e. The rocket is accelerated using constant force (alwas). Will its acceleration increase if during the process some fuel is getting burned (total mass decreases)? Wh? f. A rocket is accelerated keeping an acceleration constant. In which case the velocit of the rocket after a time t will be larger: case A) no mass is lost (ver ideal case of course), case B) half of the mass is lost (fuel burned). Wh? g. A center of mass. h. A central force. i. Kinetic energ - work theorem. j. Potential energ. k. Conservative forces or conditions for a force to be conservative. l. Simple harmonic motion (SHM); give two eamples of SHM. m. Underdamping, critical damping, and overdamping of a harmonic oscillator. n. Mechanical resonance and the qualit factor. o. Transient and long term solution for a driven harmonic oscillator. p. Fourier series and its applications for a driven harmonic oscillator. q. Gradient operator and its relation to forces and potentials. r. Corriolis and centrifugal forces. s. An inertia tensor. t. Lagrangian and Hamiltonian of a mechanical sstem. u. Hamilton's principle of least action.