PHYS 380: STUDY GUIDE FOR PART I.

ONE DIMENSIONAL PARTICLE MOTION

Outline:

 TOPIC Homework Problems 1 Review of elementary problems 1,2,3 2 Limiting Cases 4 a Taylor series 3 Review of general equation of motion in 1-D: F(x, dx/dt, t) = m d2x/dt2 4 Review of numerical approximation methods 5 F(t) = m d²x/dt²           or: F(t) = m dv/dt 5 6 F(dx/dt) = m d²x/dt²    or: F(v) = m dv/dt 6 7 F(x) = m d²x/dt² 7,8,9 a conservative forces and potential energy b solving motion problems c energy diagrams d oscillations near stable equilibrium 8 Free fall problems a with air resistance near earth's surface 10 b without air resistance from great height 9 2nd order linear differential equations 11,12 a homogenous: simple harmonic oscillator with friction b inhomogeneous: forced harmonic oscillator 10 Principle of superposition 13

Homework: (collected)

Problem #1: Do the computer homework program: Motion Graphs on Vol. 1 (Your best score counts. In this case you do not have to worry about limiting cases or approximate numerical solutions.)

Problem #2: Do the computer homework program: Gravitational Deflection (Trajectories) on Vol. 1. (Your best score counts. In this case you do not have to worry about limiting cases or approximate numerical solutions.)

Problem #3: Do the computer homework program: Newton's 2nd Law on Vol 1. (Your best score counts. In this case you do not have to worry about limiting cases or approximate numerical solutions.)

Problem #4: Derive the first 3 non-zero terms of the Taylor series expansion for:

a) cos(x) ;

b) 1/Ö [1+x] ;

c) eax .

For each case, be sure to demonstrate with at least two values of x how close the first three terms of the series come to giving the "correct" (calculator) answer.  You do not have to worry about limiting cases for this problem.

Problem #5: Given that the force on a particle of mass 5 kg initially at rest at the location xo = 0 m is  F(t) = 10 Nt * sin2( [2 rad/sec]*t).  Suggestion:  in this problem use the symbol m for the value of 5 kg, Fo for the value 10 Nt, and use the symbol w for the value of 2 rad/sec.  Only use the numerical values when plotting the graphs for x(t) and v(t).

a) find v(t);

b) find x(t);

c)  Since the Taylor series expansion for sin(q) to first order gives sin(q) » q, find x(t) and v(t) for a particle of mass, m, initially at rest at the location xo = 0 m given F(t) = 10 Nt  *  ([2 rad/sec]*t)2 .

d)  Plot v(t) from part a with the v(t) from part c over a range of times from t = 0 seconds to t = 1 second; then plot x(t) from part b with the x(t) from part c over a range of times from t = 0 seconds to t = 1 second.

e)  Since the average of sin2(q) = ½, find x(t) and v(t) when a constant force of F(t) = 5 Nt is applied, that is, F(t) = ½ Fo .

f)  Plot v(t) from part a with the v(t) from part e over a range of times from t = 0 seconds to t = 100 seconds; then plot x(t) from part b with the x(t) from part e over a range of times from t = 0 seconds to t = 100 seconds.

HINT: 0òt sin²(w t) dt = 1/2 (t - [sin(w t) cos(w t)/w ]). Extra 10% if you can derive the result for the above integral.

Problem #6: Assume a plane of mass m initially at rest is accelerated by a jet engine that develops a constant thrust of Fo and the jet encounters an air resistance of FAR(v) = –bv2.

a)  Find v(t) for the plane.

b)  Is there a terminal velocity for the plane under these circumstances?  If so, what is it (in terms of the parameters given)?

HINT: 0òa (du / [1-u²]) = 1/2 ln[1+a] - 1/2 ln[1-a]. Extra 10% if you can derive the solution for the above integral.

Problem #7:  Find x(t) and v(t) for a particle of mass, m, projected up from the earth’s surface with the escape speed for the earth.  Neglect air resistance.

Problem #8: Consider a particle of mass m that has the potential energy function:  PE(x) = ax2 – bx3 , where a and b are positive constants.

a)  Graph PE(x) versus x.  (Your choice for values for a and b as long as both a and b are positive.  Warning: be careful with your choice of values for a and b and with your range of x when plotting this graph – if your range of x is too big you may miss the “well”.  Your values of a and b may affect the width and depth of the “well” also.)

b)  Is there a potential energy “well” near the origin?  If so, find the critical velocity for the particle when it is at the position x=0 for the particle to be able to escape from this well.

c)  Find the force, F(x) that is associated with this potential energy.

Problem #9: Consider a potential energy of the form: PE(x) = -a/x6 + b/x12  where a and b are positive constants.  (This is one model for the potential energy for the force between two atoms in a diatomic molecule.)  Assume one molecule is very heavy and doesn’t move much, and the other molecule is very light in comparison with mass, m, and does move in one dimension (x) based on the force due to the above potential energy function.

a)  Graph PE(x) versus x.  (Your choice for values for a and b as long as both a and b are positive.  Warning: be careful with your choice of values for a and b and with your range of x when plotting this graph – if your range of x is too big you may miss the “well”.  Your values of a and b may affect the width and depth of the “well” also.)

b)  Find the equilibrium distance between the two molecules (the equilibrium position of the lighter atom).

c)  Find the frequency of small oscillations of the light atom about it’s equilibrium position in the molecule.

HINT: ò (dy / Ö [1-y²]) = arcsin(y) ; or HINT: let sin(A) = x Ö [k/2 E]

Problem #10.  Use the method of successive approximation to find the first order expression for v(t) for the case of dropping an object from near the earth with air resistance FAR = -bv2.

Problem #11:   Find x(t) and v(t) for a particle of mass, m, subject to a spring force of –kx and a damping force of –bv if it is displaced an initial distance of xo from the equilibrium position.  Do this for the OVERDAMPED case.

Problem #12:   Find x(t) and v(t) for a particle of mass, m, subject to a spring force of –kx (but no damping force) and an applied force, F(t) = Fo sin(wt).

Problem #13: Fourier Series: Find the expressions for the coefficients for the Fourier Series for a square wave.

Study Questions: (for the test, not for collected homework)

1. Be able to recognize cases where the Force is a constant, and set up the appropriate equations of motion (i.e., v = vo + at and x = xo + vot + 1/2at²)

2. Be able to perform a Taylor series expansion of any function of one variable.

3. Be able to

1. set up the differential equation;
2. reduce this equation to an integration (or integrations) to find the equations of motion [ i.e., x(t) and v(t) ]; for the following cases:
1. F(t)
2. F(v)
3. F(x)
4. F(x,v,t)

4. Be able to graph a potential energy function, V(x); and when given the total energy, E, be able to describe qualitatively what the motion will look like for this situation.

5. Be able to look at possible solutions for a particular problem and decide which are reasonable and which are definitely not correct:

1. by using dimensional analysis;
2. by looking at limiting cases:
1. x(t=0) = ? and v(t=0) = ?
2. x(t=¥ ) = ? and v(t=¥ ) = ?
3. x(t) = ? and v(t) = ? as any of the parameters goes to zero or goes to infinity.