Download Which tension is larger? 30° 45°

30°°
Which tension is larger?
45°°
Newton’s laws & applications, ch. 4-5
• Newton’s law & equilibrium concepts, & F = ma calculations
∑ F = ma
∑F
x
= max , ∑ Fy = may
• Equilibrium force sum = 0 ; Free body diagrams.
• Friction forces:
fk = µ k n kinetic is fixed;
f s ≤ µ s n static is an upper limit.
• Apparent weight;
accelerating vertically,
wapparent = mg + ma y
more generally,
wapparent = mg − ma
• Other forces: pulleys,
mv 2
circular motion (centripetal force: R )
A 20.0 cm-long horizontal spring is compressed by a
displacement of 5.0 cm from equilibrium, with a 20 kg
block attached to it, initially stationary. The surface is
frictionless. Just after release the block is found to
accelerate at a rate of 2.0 m/s2.
a) What is the spring constant?
b) What is acceleration as the block passes the
equilibrium point?
c) What is velocity as the block passes the point 2.0 cm
from equilibrium?
d) What power is provided by spring force at release? At
equilibrium point? At 2.0 cm from equilibrium?
Ch. 6-7: Work, Kinetic, Potential Energy
• Work converts between energy types.
W = F ⋅ s or Fs cos θ ; varying forces, W = ∫ Fx dx or
springs specific case,
∫ F ⋅ dl
F = kx ↔ W = 12 kx 2
2
• Work-Energy theorem: W = K 2 − K1 with K = 12 mv
dW
• Power: P =
is rate of doing work.
dt
mechanical version, P = F ⋅ v
• Potential energy: (conservative forces), ∆U = −Wc ;
gravity/springs: U = mgh , U = 12 kx 2
• Alternative Work-Energy theorem, ∆E = −Wother
• F vs. U: F = − dU
x
dx
; F = −  ∂U iˆ + ∂U ˆj + ∂U

∂y
∂y
 ∂x
∆K + ∆U = 0

ˆ
k  ≡ −∇U

Suppose this system is released with initial upward
velocity with magnitude vo. Find the velocity once
the left box reaches a position h below its starting
position. The ramp angle is θ, with no friction. Both
masses = M, and the pulley is massless.
vo
h
Suppose this system is released with initial
upward velocity with magnitude vo. Find the
acceleration, and the tension in the string.
Show your work. The ramp angle is θ, with no
friction. Both masses = M, and the pulley is
massless.
vo
h
Suppose this system is released with initial
upward velocity with magnitude vo. Find the
acceleration, and the tension in the string.
Show your work. The ramp angle is θ, with no
friction. Both masses = M, and the pulley is
massless.
vo
h
• Total work done to
mass on ramp?
• How high will
lower mass rise?
Find the center of mass.
30 m
50 kg
40 m
20 kg
30 kg
Ch. 8: Impulse, momentum, collisions
Momentum,
Impulse:
p = mv
J = F ∆t or
(inertia of moving object)
∫
2
1
Fdt
gives momentum change: J = p2 − p1
Momentum Conservation:
total
P = p A + pB + ...
conserved for closed systems.
Elastic vs. inelastic collisions: K conserved for elastic
m1r1 + m2 r2 + ...
Center of mass, rcm =
m1 + m2 + ...
CM velocity: CM moves as particle with momentum P = Mvcm
(also CM has important role for acceleration of solid rotors)
Find the center of mass velocity, if the lower two masses are
stationary.
30 m
50 kg
15 m/s
40 m
20 kg
30 kg
Exam 2 formulas
Review questions
Before the collision, 70 kg ball is stationary. Afterward, the 30
kg ball is stationary and 70 kg ball is moving to the right.
70 kg
30 kg
v
(a) Is this collision elastic?
(b) Find the final speed of the 70 kg ball.
Potential Energy:
a) Find an unstable equilibrium position.
b) For what region does this potential produce negative Fx?
c) Find turning points for a particle in this potential
with total E = 1.0 J.
U (J)
2
0
x(m)
Q5.2
A cable attached to a car holds
the car at rest on the frictionless
ramp (angle α).
The ramp exerts a normal force
on the car. How does the
magnitude n of the normal force
compare to the weight w of the
car? A. n = w
B. n > w
C. n < w
D. not enough information given to decide
A5.2
A cable attached to a car holds
the car at rest on the frictionless
ramp (angle α).
The ramp exerts a normal force
on the car. How does the
magnitude n of the normal force
compare to the weight w of the
car? A. n = w
B. n > w
C. n < w
D. not enough information given to decide
A5.2
A cable attached to a car is
accelerating up the frictionless
ramp (angle α).
Same result?
The ramp exerts a normal force
on the car. How does the
magnitude n of the normal force
compare to the weight w of the
car? A. n = w
B. n > w
C. n < w
D. not enough information given to decide
2 masses collide and stick together. Ramp height is 2.5 m.
Find velocity at the top?
20 kg
15 m/s
50 kg
Sudden elastic collision between ball and
block.
What is maximum angle of rise after
collision?
(a) same as original θ.
(b) larger.
(c) smaller.
A suitcase is sent sliding up a ramp with friction, at an
angle of θ from horizontal, with initial velocity vo.
(a) Draw a proper force diagram.
(b) If the suitcase goes a distance d along the ramp before
stopping, what is coefficient of friction?
(c) Will the acceleration going down the ramp be the same
as going up?
Suppose the pendulum bob has mass of 15 kg.
Find the tension in the string at position Q if at the
release point P the string has an angle of 30°
with the vertical, and the string has length 20 cm.
P
R
Q
Suppose the ball of mass M is
attached to a string and a
massless horizontal spring pivoted
at the center. The ball moves at a
constant speed in such a way that
the radius R includes an extension
of the spring (spring constant k = 1
N/m) so that the spring pulls with a
force Mg/2. The angle is such that
the string has length 2R.
• Find the string tension and v.
• What work is done by the string
and spring forces during the
circular motion?
A conservative force and a non-conservative
force act on an object. The work done by
the non-conservative force is negative and
yet the object’s speed increases.
(i) T or F: The potential energy increases and
the mechanical energy decreases.
(ii) T or F: The potential energy decreases and
the mechanical energy increases.
Two constant forces F1 = −3.0 N iˆ and F1 = 4.0 N ˆj
are the only ones acting on an object of
mass of 2.0 kg. What is the net work done
on the object as it moves from a point with
coordinates (7.0 m, 8.0 m) to a point with
coordinates (9.0 m, 8.0 m)?
a. 0 J
b. -6 J
c. +6 J
d. -10 J
e. +10 J
f. cannot be determined since we do not know
whether the forces are conservative.
A person lifts a heavy load from the floor to a
vertical height of 2.0 m in 3.0 seconds, then
places it on a shelf. If he/she had done this
more slowly in 6 seconds, the work on the
load would have been:
a twice as great
b four times as great
c the same
d half as great
Suppose the boxes are static, with masses MA < MB.
The ramp angle is θ, and there is no friction from A to
ramp, but sufficient static friction between A and B to
maintain a static situation.
What is the tension? The friction force?
2 masses m and 2m have initially equal
speeds in opposite directions. Spring is
massless.
(a) What are final velocities?
(b) At maximum spring compression what
are the velocities of the two masses?
(c) What is maximum spring compression?
m
2m
Two massless springs have
spring constant K, and
masses are both m/2.
a) What is compression of
each spring at equilibrium?
b) Suppose upper mass is
held at a vertical position
corresponding to the
unstretched lengths of the
two springs. What is z2?
A weighs 3w, B weighs w. A slides down
at constant speed. Friction coefficient is
same for A to ramp and for A to B. Find
the coefficient of friction.
Given 850N and 750N boxes,
(a) tension in two ropes if there is no
friction and a = 4.9 m/s2?
(b) Same question, if kinetic friction
coefficient µ = 0.10?