Problem Solving Steps

1
Problem Solving Steps
1. Geometry drawing trajectory, vectors
, coordinate axes
free-body diagram, 2. Data
a table of known and unknown quantities,
including implied data. 3. Equations ( with
reasoning comments ! ), their solution in
algebraic form, and the final answers in
algebraic form !!! 4. Numerical calculations and
answers. 5. Check dimensional, functional,
scale, sign, analysis of the answers and
solution.
2
Formula Sheet PHYS
218 Mathematics p
3.14 1 rad 57.30o 360o/2p volume of sphere
of radius R V (4p/3)R3 Quadratic equation
ax2 bx c 0 ?
Vectors and trigonometry


Calculus
3
Chapters 1 - 3 Constants g 9.80 m/s2,
Mearth 61024 kg, c 300 000 km/s, 1 mi
1.6 km 1-Dimensional Kinematics
3- or 2-Dimensional Kinematics
Equations of 1-D and 3-D kinematics for constant
acceleration
Circular motion
4
Chapters 4 7
Dynamics
Energy, work, and power
Chapters 8 11
Momentum
Rotational kinematics ? d?/dt, a d?/dt, s
r ?, vtan r?, atan ra, arad ac
r?2 Constant acceleration ? ?0at ?
?0(?0?)t/2, ? ?0?0tat2/2, ?2 ?022a?? I
Simiri2, IIcmMrcm2, KRI?2/2,
EMvcm2/2Icm?2/2U, WR ?td?, PRdWR/dtt? Rota
tional dynamics t Fl Fr sinf,
Rigid body rotating around a symmetry axis Iaz
tz ,
Equilibrium
Pressure p F-/A
5
Exam Example 16 The Ballistic Pendulum (example
8.8, problem 8.43)
y
A block, with mass M 1 kg, is suspended by
a massless wire of length L1m and, after
completely inelastic collision with a bullet
with mass m 5 g, swings up to a maximum height
y 10 cm.
L
Vtop0
Find (a) velocity v of the block with the
bullet immediately after impact (b) tension
force T immediately after impact (c) initial
velocity vx of the bullet.
Solution
(a) Conservation of mechanical energy KUconst
(b) Newtons second law
yields
(c) Momentum conservation for the collision
6
Exam Example 17 Collision of Two Pendulums
7
Exam Example 18 Head-on elastic collision
(problems 8.48, 8.50)
Data m1, m2, v01x, v02x Find (a) v1x, v2x
after collision (b) ?p1x, ?p2x , ?K1, ?K2 (c)
xcm at t 1 min after collision if at a moment
of collision xcm(t0)0
V01x
y
V02x
m1
X
m2
Solution In a frame of reference moving with
V02x, we have V01x V01x- V02x, V02x 0, and
conservations of momentum and energy
yield m1V1xm2V2xm1V01x ? V2x(m1/m2)(V01x-V
1x) m1V21xm2V22xm1V201x?
(m1/m2)(V201x-V21x)V22x (m1/m2)2(V01x-
V1x)2 ? V01xV1x(m1/m2)(V01xV1x)?
V1xV01x(m1-m2)/(m1m2) and V2xV01x2m1/(m1m2
) (a) returning back to the original
laboratory frame, we immediately find V1x
V02x(V01x V02x) (m1-m2)/(m1m2) and V2x
V02x (V01x V02x)2m1/(m1m2)
X
0
(a) Another solution In 1-D elastic
collision a relative velocity switches
direction V2x-V1xV01x-V02x. Together with
momentum conservation it yields the same answer.
(b) ?p1xm1(V1x-V01x), ?p2xm2(V2x-V02x) ?
?p1x-?p2x (momentum conservation)
?K1K1-K01(V21x-V201x)m1/2, ?K2K2-K02(V22x-V202
x)m2/2??K1-?K2 (Econst)

• xcm (m1x1m2x2)/(m1m2) and Vcm const
  (m1V01xm2V02x)/(m1m2)
• ? xcm(t) xcm(t0) Vcm t t
  (m1V01xm2V02x)/(m1m2)

8
Exam Example 19 Throwing a Discus (example 9.4)
9
Exam Example 20 Blocks descending over a massive
pulley (problem 9.83)
?
Data m1, m2, µk, I, R, ?y, v0y0
m1
R
Find (a) vy (b) t, ay (c) ?,a (d) T1, T2
Solution (a) Work-energy theorem Wnc ?K ?U,
?U - m2g?y, Wnc - µk m1g ?y , since FN1
m1g, ?KK(m1m2)vy2/2 I?2/2
(m1m2I/R2)vy2/2 since vy R?
x
m2
0
ay
?y
y
(b) Kinematics with constant acceleration t
2?y/vy , ay vy2/(2?y) (c) ? vy/R , a
ay/R vy2/(2?yR) (d) Newtons second law for
each block T1x fkx m1ay ? T1x m1
(ay µk g) , T2y m2g m2ay ? T2y -
m2 (g ay)
10
Combined Translation and Rotation Dynamics
Note The last equation is valid only if the axis
through the center of mass is an axis of
symmetry and does not change direction.
Exam Example 21 Yo-Yo has IcmMR2/2 and rolls
down with ayRaz (examples 10.4, 10.6) Find (a)
ay, (b) vcm, (c) T Mg-TMay
tzTRIcmaz ay2g/3 , TMg/3
y
ay
11
Exam Example 22 Race of Rolling Bodies (examples
10.5, 10.7 problem 10.22)
y
Data IcmcMR2, h, ß
Find v, a, t, and min µs preventing from slipping
FN
x h / sinß
ß
x
Solution 1 Conservation of Energy
Solution 2 Dynamics (Newtons 2nd law)
and rolling kinematics aRaz
fs
v22ax
12
Equilibrium, Elasticity, and Hookes Law
Conditions for equilibrium
Exam Example 23 Ladder against wall (example
11.3, problem 11.10)
d/2
h
x
Static equilibrium
y
State with is equilibrium but is not static.
Data m, M, d, h, y, µs Find (a) F2, (b) F1, fs,
d
?
Strategy of problem solution
(c) yman when ladder starts to slip

• (0)
• Choice of the axis of rotation
• arbitrary - the simpler the better.
• (ii) Free-body diagram
• identify all external forces and
• their points of action.
• (iii) Calculate level arm and
• torque for each force.
• (iv) Solve for unknowns.

Solution equilibrium equations yield
(a) F2 Mg mg (b) F1 fs Choice of
B-axis (no torque from F2 and fs) F1h mgx
Mgd/2 ? F1 g(mxMd/2)/h fs (c) Ladder starts
to slip when µsF2 fs, x yd/h ? µsg (Mm)
g (mymand/hMd/2)/h ?