Apparent Weight

1
Apparent Weight

• Riding in a elevator why does our weight appear
  to change when we start up (increase) and slow
  down (decrease)?
• Our sensation of weight change is due to a force
  exerted on our feet by the elevator floor
  (normal force N). If force greater we feel
  heavier and vice versa.

2
Apparent Weight

• Eg. Upward accelerating elevator

As accelerating, there must be a net upward
force.
(2nd law) Fnet N W m a
W
But our true weight W m g
Apparent weight N W ma
N
N m (g a) (i.e. heavier)
If lift accelerating downwards (or decreasing
upwards)
N m (g a) (ie. lighter)
3
Free-Falling

• When you jump off a wall, or throw a ball or drop
  a rock in a pool, the object is free-falling ie.
  falling under the influence of gravity.
• Question What happens to our apparent weight in
  free-fall?
• Nasty Exp Cut elevator wires so its downward
  acceleration a g (i.e. free-fall)!
• Apparent weight N m (g a)
• But a g, so N 0 i.e. no normal force.
• Weightless is zero apparent weight.
• Everything is falling at same rate, so no normal
  force is needed to support your weight.

4
Free-Falling

• Ex Aircraft flying in a parabolic path can
  create weightless conditions for up to 30 s!
• Spacecraft / astronauts in orbit are weightless
  as they (and the spacecraft) are continuously
  free-falling towards the Earth!!

5
Circular Motion (chapter 5)

• So far we have focused on linear motion or
  motion under gravity (free-fall).
• Question What happens when a ball is twirled
  around on a string at constant speed?
• Ans Its velocity continuously changes in
  direction.
• This implies
• The velocity change is caused by an acceleration.
• By Newtons 2nd law an acceleration requires a
  force!

6
Circular Motion (chapter 5)

• By Newtons 2nd law an acceleration requires a
  force!
• Big questions
• What is the nature of this force / acceleration?
• What is the relationship between the acceleration
  and the velocity of the ball and the radius of
  curvature?

7

• In the absence of gravity, tension provides the
  only force action on the ball.
• This tension causes ball to change direction of
  velocity.

Instantaneous velocity vector changing in
direction but its magnitude stays constant.
8

• Question What happens if you let go of the
  string?

Answer Ball travels in direction of last
instantaneous vector. (Newtons 1st law)
9

• Lets imagine you are on a kids roundabout

Question Why do we feel an outward force if
its not really there?
Your body naturally wants to move this way
(Newtons 1st law)
You must pull inwards

• However, to keep you in circular motion you must
  apply a force inwards to change your direction.

10

• Your pulling inwards creates the sensation that
  the roundabout is pushing you outwards!

11
Centripetal Acceleration

• The force (tension) causes an acceleration that
  is directed inwards towards center of curvature.
• ie. The string is continuously pulling on the
  ball towards the center of curvature causing its
  velocity to constantly change.
• This is called centripetal acceleration (ac).

12
Centripetal Acceleration

• Centripetal acceleration is the rate of change
  in velocity of an object due to a change in its
  velocity direction only.
• It is always perpendicular to the velocity
  vector and directed towards the center of
  curvature.
• There is NO such thing as centrifugal (ie
  outward) force.

13
Nature of ac
v2
v1 v2 same speed
v1
Accn.
r
Acceleration is in direction of ?v.
14

• Dependencies of ac
• 1. As speed of object increases the magnitude of
  velocity vectors increases which makes ?v
  larger.

?v
?v
v1
Therefore the acceleration
v1
increases.
v2
v2
15

• Dependencies of ac

2. But the greater the speed the more rapidly
the direction of velocity vector changes
Large angle change
Small angle change
Slow
Fast
increases
16
Dependencies of ac

• 3. As radius decreases the rate of change of
  velocity increases as vector direction changes
  more rapidly.

d
d
Large radius
Small radius
Same distance (d) moved but larger angle change.
r
Result increases as radius
decreases.
17
Summary

• Points 1 and 2 indicate that the rate of change
  of velocity will increase with speed.
• Both points are independent of each other and
  hence ac will depend on (speed)2.

18
Summary

• Point 3 shows that ac is inversely proportional
  to radius of curvature (i.e. ).

2
v
m/s2 (towards center of curvature)
Thus

a
c
r
i.e. Centripetal acceleration increases with
square of the velocity and decreases with
increasing radius.
19
Example Ball on a string rotating with a
velocity of 2 m/s, mass 0.1 kg, radius0.5 m.

• What forces can produce this acceleration?
• Tension
• Friction
• Gravitation attraction (planetary motion).
• Nuclear forces
• Electromagnetic forces
• ?

20

• Lets consider the ball on a string again
• If no gravity

Center of motion
T
T
m
ac
m v2 r
T m ac

• Ball rotates in a horizontal plane.

21

• Lets consider the ball on a string again

With gravity
String and ball no longer in the same horizontal
plane.

• The horizontal component of tension (Th)
  provides the necessary centripetal force. (Th
  mac)

• The vertical component (Tv) balances the
  downward weight force (Tv mg).

22
Stable Rotating Condition
Th T cos ?
Tv T sin ? mg
Th T cos ?
m v2 r
As ball speeds up the horizontal, tension will
increase (as v2) and the angle ? will reduce.
23
Stable Rotating Condition
Thus, as speed changes Tv remains unaltered
(balances weight) but Th increases rapidly.
24
Unstable Condition

• Tv no longer balances weight.

• The ball cant stay in this condition.

Ex. again Ball velocity 2 m/s, mass 0.1 kg,
radius0.5 m.
Centripetal force Fc mac 0.1 x 8 0.8 N
Thus, horizontal tension (Th) 0.8 N.
Now double the velocity
Centripetal
Fc mac 0.1 x 32 3.2 N
Thus, the horizontal tension increased 4 times!
25
Summary

• A centripetal force Fc is required to keep a body
  in circular motion
• This force produces centripetal acceleration that
  continuously changes the bodys velocity vector.
• Thus for a given mass the needed force
• increases with velocity 2
• increases as radius reduces.

26

• Example The centripetal force needed for a car
  to round a bend is provided by friction.
• If total (static) frictional force is greater
  than required centripetal force, car will
  successfully round the bend.
• The higher the velocity and the sharper the
  bend, the more friction needed!

Ff
Ff

• As Fs µs N - the friction depends on surface
  type (µs).
• Eg. If you hit ice, µ becomes small and you fail
  to go around the bend.
• Note If you start to skid (locked brakes) µs
  changes to its kinetic value (which is lower) and
  the skid gets worse!
• Moral Dont speed around tight bends!
  (especially in winter)

27
Motion on a Banked Curve
N
Nv mg

• The normal force N depends on weight of the car W
  and angle of the bank ?.
• There is a horizontal component (Nh) acting
  towards center of curvature.
• This extra centripetal force can significantly
  reduce amount of friction needed

Nh
?
Wmg
v2 rg

• If tan ? then the horizontal Nh
  provides all the centripetal force needed!

• In this case no friction is necessary and you
  can safely round even an icy bend at speed

• Ice skaters cant tilt ice so they lean over to
  get a helping component of reaction force to
  round sharp bends.

28
Vertical Circular Motion
Ball on String
Ferris Wheel
Feel pulled in and upward
T
N
T gt W
W
N gt W
Wmg
Bottom of circle

• Centripetal acceleration is directed upwards.

• Total (net) force is thus directed upwards

Fnet N - W mac Napparent weight (like in
elevator)
Thus N W mac i.e. heavier/larger tension
29
N
N lt W
Component of W provides tension
W
Feel thrown out and down
T
W
T
N
T gt W
N gt W
Wmg
W
Top of circle

• Weight only force for centripetal acceleration
  down.

N W m ac i.e. lighter / less tension
If W m ac ? feel weightless (tension T0)
(larger r, higher v)
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31

• Example The centripetal force needed for a car
  to round a bend is provided by friction.
• If total (static) frictional force is greater
  than required centripetal force, car will
  successfully round the bend.
• The higher the velocity and the sharper the
  bend, the more friction needed!

Ff
Ff

• As Fs µs N - the friction depends on surface
  type (µs).
• Eg. If you hit ice, µ becomes small and you fail
  to go around the bend.
• Note If you start to skid (locked brakes) µs
  changes to its kinetic value (which is lower) and
  the skid gets worse!
• Moral Dont speed around tight bends!
  (especially in winter)