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The University of Alabama at Birmingham (UAB)
Department of Physics
PH 461/561 – Classical Mechanics I – Fall 2005
Assignment # 10 Due: Tuesday, November 29
- A particle of mass m moves in one dimension under a restoring force − kx , a
linear resistance − bx &^ , and a time-dependent force F 0 cos ω t. The quantities
k , b , F 0 , and ω are all positive constants, with
k = b^2 4 m. (a) Find x ( ) t and show that it may be understood as the sum of a transient term that vanishes for t → ∞ , and a steady state term that dominates the motion when t → ∞. (b) Graph the behavior of the amplitude of the steady state term as a function
of the frequency ω , for various values of the damping factor γ ≡ b 2 m.
(c) Discuss the effect of the damping factor γ on the sharpness of the
resonance. (d) Graph the behavior of the phase difference between the steady state term
and the force F 0 cos ω t as a function of the frequency ω , for various
values of the damping factor γ.
(e) Discuss the physical meaning of the phase difference dependence on ω ,
with particular attention to the cases when ω → 0 (low frequency regime)
and ω → ∞ (high frequency regime).
- Given a force F = Fx i + Fy j + Fz k , with components
F x = a y^2 z 3 and F y = 2 a xyz 3
(a) Determine a component F z such that the force is conservative. (b) In this case, calculate the potential V(x,y,z) such that V(0,y,z) = 0.
- A particle of mass m moves in three dimensions under a time-dependent force whose components are F x = a , F y = bt, F z = ct^2
where a, b, c are positive constants and t is the time. Provide the following: (a) A discussion of the degrees of freedom of the system. (b) An identification of any constraints to the motion of the particle. (c) The differential equations of motion in a suitable coordinate system. Is this force conservative? Why? (e) Assuming that at t = 0 the particle had the following initial conditions: r 0 =0 and v 0 = v 0x i , determine the motion of the particle (i.e., Find r (t) )
- A particle of mass m moves in three dimensions as it is subjected to the force
F mg k
r r = − , where g is the acceleration due to gravity. Neglecting air resistance provide the following:
(a) A discussion of the degrees of freedom of the system. (b) An identification of any constraints to the motion of the particle. (c) The differential equations of motion in a suitable coordinate system. (d) Find the motion (i.e., solve the equations of motion) assuming initial conditions r 0 = 0
r , v 0 ≠ 0
r . (e) Find an analytical expression for the trajectory of the particle. (f) Find, in terms of the given initial conditions, the maximum height the particle reaches. (g) Find the range of the particle (i.e., the maximum linear distance the particle reaches on the x-y plane).
- Re-work problem 2 above assuming that in addition to the force of gravity F
r , the particle is also subject to a linear air resistance Fair = − b dr dt ( )
r (^) r .
- Consider a particle of mass m in three dimensions, subject to a restoring force that
may be expressed in Cartesian coordinates as F Fxi Fyj Fzk
r r r^ r = + + , where F (^) x = − kxx ; Fy =− kyy ; Fz =− kzz .(The positive constants k (^) x , ky , kz may or may not be equal).
(a) Find an expression for the potential energy of the particle. (b) Find an expression for the total mechanical energy of the particle. (c) Is this force conservative? Why?
- A particle of mass m is constrained to move in two dimensions under the force
F = − kr
r (^) r where k is a constant and r
r is the position vector of the particle with respect to the origin. Provide the following:
(a) A discussion of the degrees of freedom of the system. (b) An identification of any constraints to the motion of the particle. (c) The differential equations of motion in a suitable coordinate system. (d) Find the motion (i.e., solve the equations of motion) assuming initial position r 0 (^) = y j 0 r r and initial velocity v 0 (^) = v i 0 r r , where i , j
r r are the unit vectors in the x , y directions. (e) Find an analytical expression for the trajectory of the particle.
a. Find the horizontal and vertical components of the electron acceleration in regions I, II, III, and IV.
b. Find the horizontal and vertical components of the electron velocity in regions I, II, III, and IV.
c. Calculate the vertical deflection Y on the tube screen with respect to the initial direction of propagation of the electrons.
l L
d
I II III IV
Y
P V 0
V
a
F
screen
Trajectory of electrons (electron beam)
A A
D
D
l L
d
II IIII IIIIII IVIV
Y
P V 0
V
a
F
screen
Trajectory of electrons (electron beam)
A A
D
D
