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Applications of Newton’s Laws of Motion
Equilibrium: no net force

F  0 
F
F
x
0
y
0
1.
Draw a simple diagram
2.
Draw a free body diagram for one body
3.
Identify interactions (force magnitudes and directions on that body)
4.
Choose coordinates and determine the components of each force
5.
Set algebraic sum of all x-components to zero to obtain an algebraic relation.
In a separate equation set all the y-components equal to zero.
6.
Repeat 2-5 for any other bodies in the problem.
7.
Use all the simultaneous equations from the step(s) 5 to solve for the
unknowns.
Phys211C5 p1
A 500 N gymnast begins climbing a 100 N rope.
what is the tension at the bottom of the rope?
what is the tension in the top of the rope?
Phys211C5 p2
A car engine of weight w is suspended from a chain linked to
two other chains as shown. One chain is fastened to the ceiling
and makes a 60° angle with respect to the ceiling. The other is
horizontal and attached to the wall. What is the tension in the
chains in terms of the weight of the engine?
60o
Phys211C5 p3
A car of weight w rests on a ramp and is held in place by a
cable, once the car’s brakes and transmission are released. If the
ramp is at an angle of a with respect to the horizontal, determine
the tension in the cable in terms of the car’s weight.
a
Phys211C5 p4
Tension over a frictionless pulley: A weight w1 slides on a
frictionless 15o slope. A cable attached to the weight is also
attached over a pulley to a counterweight w2 chosen to just
counter balance w1, so that the entire system moves at constant
velocity. Determine w2 in terms of w1.
w1
15o
w2
Phys211C5 p5
Dynamics:


 F  ma 
F
F
x
 ma x
y
 ma y
1.
Draw a simple diagram
2.
Draw a free body diagram for one body
3.
Identify interactions (force magnitudes and directions on that body)
4.
Choose coordinates and determine the components of each force
5.
Set algebraic sum of all x-components to max to obtain an algebraic relation.
In a separate equation set all the y-components equal to may.
6.
Repeat 2-5 for any other bodies in the problem.
7.
Use all the simultaneous equations from the step(s) 5 to solve for the
unknowns.
Phys211C5 p6
An iceboat (mass including rider is 200 kg) is initially at rest on a horizontal frictionless
surface. What force must be applied to the boat to accelerate it to a speed of 6.00 m/s in
4.00 s?
An iceboat (mass including rider is 200 kg) is initially at rest on a horizontal surface. What
force must be applied to the boat to accelerate it to a speed of 6.00 m/s in 4.00 s if the
opposing frictional force has a constant magnitude of 100 N?
Phys211C5 p7
An elevator and its load have a total mass of 800 kg. The elevator is originally moving
downward at 10.0 m/s. The elevator is brought to rest over a distance of 25.0 m. What is
the tension in the supporting cable?
A 50.0 kg woman stands on a scale while riding in the elevator in the previous example.
What is the reading on the scale while the elevator is coming to a stop? What is the reading
once the elevator is at rest?
Phys211C5 p8
A sled rides down a frictionless surface which makes an anglea with respect to the
horizontal. What is the acceleration of the sled?
A 4.0 kg cart riding on a horizontal frictionless track is attached to a .500 kg rope. If a
9.00 N force is applied to the other end of the rope, determine the acceleration of the rope
and cart and the tension in both ends of the rope.
Phys211C5 p9
A mass m1 slides on a frictionless horizontal surface. A
mass less cable attached to m1 is also attached over a
pulley to a counterweight m2. Determine the acceleration
of the masses and the tension in the cable.
m1
m2
Phys211C5 p10
A simple accelerometer is made by suspending a weight
by a string from the center of a level protractor. Given the
mass m of the suspended object and the angle b that the
string makes with respect to the vertical, determine the
acceleration a.
b
Phys211C5 p11
Force of friction
Friction
opposes motion
due to surfaces sticking together
Kinetic Friction: surfaces are moving relative to each other
a.k.a. Sliding Friction
Static Friction: surfaces are not moving relative to each other.
Static Friction prevents stationary objects from moving until
sufficient force has been applied.
Rolling Friction, etc
Applied Force
Phys211C5 p12
Coefficient of Friction
Frictional forces depend upon
how hard the surfaces are being pressed together
-> force perpendicular to the surface = normal force (FN)
the types of surfaces that are in contact
-> coefficient of friction
F f   s FN
F f   k FN
Material
wood on wood
wood on stone
steel on steel (smooth)
rubber tire on dry concrete
rubber tire on wet concrete
steel on Teflon
see also table 5-1
Static Friction
Kinetic Friction
static: s
0.5
0.5
0.74
1.0
0.3
0.04
kinetic: k
0.3
0.4
0.57
0.8
0.25
0.04
Phys211C5 p13
A 500 N crate is dragged by a rope across a horizontal surface. To start it moving takes 230
N of horizontal force. Once moving, it takes 200 N to keep it moving at constant speed.
What are the coefficients of friction?
What is the friction force if the crate is at rest and a horizontal force of 50 N is applied to it?
What is the force necessary keep the crate moving at constant velocity if the rope is pulled at
an angle of 30o above the horizontal?
Phys211C5 p14
A chute is being built along which crates are to be slid down at constant speed. The
coefficient of kinetic friction is k.
What angle should the chute make with respect to the horizontal?
What is the acceleration of a moving crate if the angle is actually greater than this
critical angle?
q
Phys211C5 p15
Fluid resistance: force on a body moving through a fluid (gas/liquid)
always opposes motion
low speeds: force is proportional to speed
fv
f = k v : k depends upon fluid, geometric size/shape of body
high speeds: force is proportional to speed2
f  v2
f = D v2 : D depends upon fluid, geometric size/shape of body
terminal velocity:
Fy  f t  mg  ma y  0
kvt  mg  0
vt 
mg
k
Dvt  mg  0
2
mg
vt 
D
Phys211C5 p16
Dynamics of Circular Motion
v2
recall arad 
R
now apply Frad = m arad
A .300 kg mass revolves uniformly on a frictionless horizontal surface. The mass is
attached to a .140 m cord tied to a pin at the center of the circle. What is the tension in
the string if the mass makes 2.00 revolutions per second?
Phys211C5 p17
A conical pendulum consists of a mass m suspended from a fixed length L of string. The
mass swings in a horizontal circle so that the string always makes an angle b with respect to
the vertical. How are the speed and the period related to the parameters specified above?
b
Phys211C5 p18
A circular curve of radius R is to have a flat roadbed. The coefficient of static friction
between the tires and the road is s. What is the maximum speed vmax, the car can go
around the corner without slipping?
phys
Phys211C5 p19
A circular curve of radius R is to have a banked roadbed so that for a speed v, the car
will not need any force of friction to make the curve. Relate the angle of bank to the
speed of the car and radius of the curve.
Phys211C5 p20
Motion in a Vertical Circle
mass on a string
minimum speed
At the top:
2
v
T1  mg  m 1
r
2
v
T1  m 1  mg
r
r
speed decreasing
h = 2r
speed increasing
At the bottom:
2
v
T2  mg  m 2
r
2
v
T2  m 2  mg
r
maximum speed
Critical speed: speed at top at which string goes slack (T=0)
2
T1  m
vc
 mg  0 vc  rg
r
+ applications to amusement park physics
Phys211C5 p21
Fundamental Forces
Gravitation (interaction between masses)
Electromagnetism (interaction between charges)
unified force: electrostatics + magnetism + …
for interacting protons, 35 orders of magnitude stronger than
gravity
Strong Interaction (aka the Strong Nuclear force)
short range
Weak Interaction (aka the Weak Nuclear force)
further successful unification: Electroweak theory
Strong + Electroweak = Grand Unified Theory (GUT)
all four unified = Theory of Everything (TOE)
Phys211C5 p22