More Applications of Newtons Laws

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Lecture 12

• More Applications of Newtons Laws

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Example Bucket

• A stone of mass m sits at the bottom of a
  bucket. A string is attached to the bucket and
  the bucket is made to move in circles. What is
  the minimum speed that the bucket needs to have
  at the highest point of the trajectory in order
  to keep the stone inside the bucket?

R
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• If v increases, N needs to be larger (if v
  becomes too large, since N is also the force on
  the bucket by the stone, the bottom of the bucket
  might end up broken)

• If v decreases, N needs to be smaller. But at
  some point, N will become zero! This is the
  condition for the minimum speed

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Example Curve
A car of mass m with constant speed v drives
through a curve of radius R. What is the minimum
value of the coefficient of static friction
between the tires and the road for the car not to
slip?
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EXAMPLE Incline and pulley, with friction
Same system, but µS 0.2 and µk 0.1 What
happens when the system is released?
2m
m
35?
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Does the system move at all?
Maximum static friction force
It does not move!
T
2m
T
µS 0.2 µk 0.1
m
35?
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ACT Magnitude of friction
What is the magnitude of the static friction
force in the system we just studied?

• 0.15 mg
• 0.33 mg
• 0.40 mg

35?
For the system to be at rest, fS needs to cancel
out the net force without friction
Option C is beyond the maximum possible
value. Option B would produce a net force to the
left along the incline!
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Example Box on truck

• A box with mass m 50 kg sits on a truck. The
  coefficients of friction between the box and the
  truck are µK 0.2 and µS 0.4.

What is the maximum acceleration that the truck
can have without the box slipping?

• 2.0 m/s2
• 3.1 m/s2
• 3.9 m/s2
• 4.9 m/s2
• 9.8 m/s2

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a
fS MAX mBa MAX
fS mBa N-W 0
Answer C
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Drag forces
For solid-fluid relative motion, friction force
(called drag force or resistance) depends on
the relative speed
k and D depend on the geometry and the materials.
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Terminal speed
Acceleration of a suitcase that falls from a
plane
Eventually, fD mg , so a 0!
When this happens, the system has reached its
terminal speed
This is how parachutes work!
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EXAMPLE Pulling yourself up
A kid with mass m 30 kg has designed a rough
elevator to get to his tree-house. Its made of a
seat of mass M 5 kg, a rope and a pulley. To
use the elevator, you sit on the seat and pull on
the rope as shown below. How strong is the kid
pulling if the elevator is moving at constant
speed?
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ACT Pulling somebody up
If, instead, a friend pulled with tension T 172
N on the loose end of the rope, the elevator
would
A. Move exactly as before. B. Make it faster to
the tree-house C. Not go up.
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The net force on the elevatorkid system is now
Fnet T - W But T W/2, so with this tension
the net force on the system ?and its acceleration
? point down!
Force on the friend!
The friend needs to pull at least twice as hard!
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EXAMPLE Box on another box

• A box of mass m1 1.5 kg is being pulled by a
  horizontal string with tension T 45 N. It
  slides with friction (µK 0.50, µS 0.70) on
  top of a second box of mass m2 3.0 kg, which in
  turn sits on a frictionless floor. Find the
  acceleration of box 2.

µK 0.5
a1
T
m1
a2 ?
m2
frictionless
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m1
m2
For box 2 fK µKN m2a2
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The magnitude of the tension did not play any
role! The tension just needs to be large enough
so the boxes cannot move together.
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EXAMPLE Box on another box (2)

• Same problem (m1 1.5 kg,T 45 N, µK 0.50,
  m2 3.0 kg), but now the string makes an angle ?
  15 with the horizontal. Find the acceleration
  of box 2.

µK 0.5
a1
T
?
m1
a2 ?
m2
frictionless
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N
T
?
fK
m1
W1
m2
fK
For box 2 fK µKN m2a2
N lt m1 g
From box 1, N m1g Tsin? 0
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ACT Car on a bump

• The pavement on Grand Ave. is higher along the
  center of the street than along the sides. So
  when you drive along 13th street and across Grand
  Ave., your car goes over a small hill. We can
  estimate the bump to have a radius of curvature
  of 30 m. What is the maximum speed your car
  should have if your wheels are to stay in contact
  with the ground all the time?

A. 12 m/s B. 17 m/s C. 20 m/s
R
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Large v -gt Small N Small v -gt Large N
W N may
Smallest N N 0
17.1 m/s 60 mi/h
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EXAMPLE Accelerometer

• A car has a constant acceleration of 2 m/s2. A
  small ball of mass m 0.5 kg attached to a
  string hangs from the ceiling.
• Find the angle ? between the string and the
  vertical direction.

a
?
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• Free-body diagram

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In components

• x Tx ma Tsin? ma
• y Ty ? W 0 Tcos? ? mg 0

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• Tsin? ma
• Tcos? ? mg 0

2 equations, 2 unknowns
Note ? does not depend on the mass of the ball!
For a 2 m/s2 , ? 12 Check For a 0
(constant speed), ? 0
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EXAMPLE Double incline

• A box of mass m1 1 kg sitting on a double
  incline is attached to another box of mass m2 2
  kg sitting on the other side of the incline by an
  ideal string that goes through an ideal pulley.
  The angles between the inclines and the
  horizontal are ?1 30 and ?2 45. If the
  blocks are moving to the left and ?k 0.2, what
  is the acceleration of the system?

m1
m2
?2
?1
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1 T - W1x m1 a 2 W2x T m2 a
m1 1 kg m2 2 kg ?1 30 ?2 45 ?k 0.2
m1
m2
?1
?2
No friction case
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1 T - W1x fK1 m1 a 2 W2x T fK2 m2 a
where fK1 µk N1 µk m1 g cos?1 where fK2
µkN2 µkm2 g cos?2
0.15 g 1.5 m/s2
m1 1 kg m2 2 kg ?1 30 ?2 45 ?k 0.2
m1
m2
?1
?2
With friction
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Example Climber
A 49 kg rock climber is climbing a chimney
between two rock slabs. The coefficient of static
friction between her shoes and the rock is 1.2
between her back and the rock it is 0.80. She has
reduced her push against the rock until her back
and her shoes are on the verge of slipping. What
is her push against the rock?

• 480 N
• 240 N
• 400 N
• 600 N
• 720 N

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Answer B