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Measured weight of a ramp with sliding crate.
A crate, of mass m, slides with negligible friction down along a ramp of mass M. The
upper surface of this ramp makes an angle 9 with its horizontal lower surface. The ramp
remains at rest, lying on a scale anchored to the ground.
What is the magnitude of the vertically downward force exerted on the scale by the ramp
(i.e., what is the weight measurement indicated by the scale) while the crate is sliding down
along the ramp? Express your answer in terms of the known quantities m, M, 6, and the
magnitude g of the gravitational acceleration due to the earth.
Particle sliding back and forth.
A particle, of mass m, starts from rest at the point A and slides down along the straight
path AB making an angle 6 with the horizontal. The coefficient of kinetic friction between the
particle and this straight portion of the path is u,. The particle then continues sliding along
the rest of the path, where the coefficient of friction is zero, passing through its lowest point
C and coming momentarily to rest at the point D. The particle then slides back through the
point C until it comes again momentarily to rest at the point E.
Answer the following questions and indicate your'reasoning. The quantities assumed to
be known are m, 9, u, the magnitude g of the gravitational acceleration, and the heights h^,
hg, and hp of the points A, B, and D above the lowest point C of the path.
(a) What is the work done on the particle by the gravitational force due to the earth while
the particle moves from the point A to the point D?
(b) What is the work done on the particle by the friction force, due to contact with the
path, while the particle moves from the point A to the point D?
(c) What is the speed of the particle at the lowest point C?
(d) What is the height (above the point C) of the point E where the particle is again
momentarily at rest?
Clear Description
Available information
Diagram(s) with verbal summaries.
Useful symbols.
Situation described at successive special and typical times.
Goal
Goal specification.
Analytic Description
For every interesting system at every interesting time, draw a diagram
summarizing available information about the following:
Mass
Motion
Position
Velocity
Acceleration
Forces
Every long-range force (by specified object)
Every contact force (by specified touching object)
AN (bysurface)
Example:
Block B at time t
m
B
a
mg (by earth)
F (by string)
Applying a principle
Which <principle?> applied to which <system?>
For Newton's principle ma = Ftot
at which <time?>
for component along which <direction?>
For energy principle AE = W0ther
between which <times?>
Mechanics principles
Newton's principle (Utility: Relating position or velocity to time}
{total force}]
Ftot := £ F.j
where
ma = Ftot
[~
Energy principle futility: Relating speed and position, without reference to time}
AE = Wother
where
WotherS
{work done by all interactions not included in the energy E}
Motion descriptors
Basic descriptors:
Position vector
Velocity
Acceleration
v := dr/dt
a := dv/dt
Circular motion
Acceleration components (for circle or arc of radius r)
Along velocity: av = dv/dt.
Toward center: ac = v2/r.
Interaction descriptors
Force
Force on particle by other particles: F ma (specifies how accel. is influenced by other particles).
F: - FJ
Reciprocity (Newton's 3rd law):
{equal magnitudes, opposite directions}
Work-energy descriptors
Kinetic energy: K : = \ m v2
W:=/FdrF
Work:
Potential energy (if work is path-independent):
{Point A, standard position S}
UA := WAs
Energy:
Force laws
Long-range forces
Gravitational force (near surface of the earth)
(g « constant).
F = mg
Force on particle of mass m:
g « 9.8 m/s2 downward near surface of earth.
(h « height above standard position).
U = mgh
Potential energy:
Contact forces
Force by string
Direction:
Magnitude:
F along and toward string (opposes elongation).
F * 0 if string is taut; F = 0 otherwise.
Force by solid surface
N = perpendicular "normal force"
F=N+f
f = parallel "friction force"
f
N perpendicular away from surface (opposes penetration).
N * 0 if touching surface; N = 0 otherwise.
N
Direction:
Magnitude:
f
Direction:
Magnitude:
If no relative motion:
If relative motion:
f opposes relative motion of contact point.
f < u,s N
f = \IL. N
{jis = static coefficient of friction}.
{u.k = kinetic coefficient of friction}.