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Measurement of kinematic quantities
through simple experiments
 1 Measurement of velocity
 2 Measurement of force
 3 Measurement of acceleration
 ４ Appendix=the motion of tops, especially
“gyroscope”
1 Measurement of velocity
(1) Using a stop watch and a measure
a body moving distance L [m], time taken t [s], the velocity v [m/s]
Ｌ
ｖ= ｔ
①
Exercise1 L=10m, required time t=0.50s, ask the velocity of the vehicle.
V=( 10 )/( 0.50 )=( 20 )m/s
t : large ⇒ v : average
t : small ⇒ v : instant
(2) Using a points recording timer
(ⅰ)measuring time very short as possible
(ⅱ) marking points on a body
periodically at very short time
Fig.1 marking points
Fig.2 points recording timer
Fig.3 principle of points recording timer
iron bar
iron core
carbon
paper
spring
paper
tape
Using 50Hz
AC supply
coil
AC
⇒(ⅰ) number of vibration of the iron bar per second is 50.
⇒(ⅱ) periodic pointing number per second is 50.
⇒(ⅲ) the time between adjacent points becomes (1 / 50) [s].
Now, the time duration between points can be set (1 / 50), or (1 / 10)
seconds .
Exercise 2 period time is (1/10)seconds.
paper tape
Nr.Point
0 1 2
3 4
5
Length cm 0 2.5 5.0 7.5 1 0.0 12.5
Asking for the velocity of the body.
0-1 v={(2.5−0.0)/100}/(1/10)= 0.25m/s
1-2 v={(5.0−2.5)/100}/(1/10)= 0.25m/s
2-3 v=
Caution for this equipment.
(ⅰ) To set the side of the paper tape chemicals painted upwards.
(ⅱ) Discharge electrode may be corrupted by putting a tape from the other side.
(ⅲ) Turn on the power switch after all have been set.
Now, the time duration between points can be set (1 / 50), or (1 / 10)
seconds .
Exercise 2 period time is (1/10)seconds.
paper tape
Nr.Point
0 1 2
3 4
5
Length cm 0 2.5 5.0 7.5 1 0.0 12.5
Asking for the velocity of the body.
0-1 v={(2.5−0.0)/100}/(1/10)= 0.25m/s
1-2 v={(5.0−2.5)/100}/(1/10)= 0.25m/s
2-3 v= {(7.5–5.0)/100}/(1/10)=0.25m/s
Caution for this equipment.
(ⅰ) To set the side of the paper tape chemicals painted upwards.
(ⅱ) Discharge electrode may be corrupted by putting a tape from the other side.
(ⅲ) Turn on the power switch after all have been set.
Experiment 1 To measure the velocity of linear motion of the hand .
Installation points recording timer, paper tape, ruler measure
Procedure
points timer (1/10)sec
a hand
after setting, switching ON, and pulling the tape
Nr.of points
０
length(cm)
difference(cm)
distance(m)
velocity(m/s)
１
０
２
３
４
Experiment 1 To measure the velocity of linear motion of the hand.
Installation points recording timer, paper tape, ruler measure
Procedure
points timer (1/10)sec
a hand
after setting, switching ON, and pulling the tape
Nr.of points
０
length(cm)
difference(cm)
distance(m)
velocity(m/s)
１
２
9.5
０
9.5
0.095
0.95
３
20.2
10.7
0.107
1.1
４
Experiment 2
Motion of hovering soccer ball.
rotating fan
Points on paper tape
air
lined up almost evenly.
⇓
constant velocity linear motion
or, uniform motion.
Points recording timer
ball
paper tape
layer
(3) Using “Be-Spe”
Fig.5 Be-Spe
Fig.4 principle
sw1
sw2
two light switches
(ⅰ) When the body has interrupted one⇒the timer switch is ON
(ⅱ) next it interrupts the other one ⇒ the timer switch is OFF
(ⅲ) Internal computer calculates and shows the value of the
velocity.
Experiment 3 To measure the velocity of small steel balls
moving inside the tube.
Installation
“Be-Spe”, transparency tube, small steel ball
Be-Spe
Height
velocity 1st
m/s
velocity 2nd
m/s
velocity 3rd
m/s
5 cm
height
10 cm
20 cm
Experiment 3 To measure the velocity of small steel balls
moving inside the tube.
Installation
“Be-Spe”, transparency tube, small steel ball
Be-Spe
Height
velocity 1st
m/s
velocity 2nd
m/s
velocity 3rd
m/s
height
5 cm
10 cm
0.9
20 cm
1.3
1.8
2 Measurement of force
(1)
dynamics trucks
Dynamics truck
wears a four wheels
whose axle is held in
ball bearings, so wheel
rotation is very smooth.
Rolling friction is
1 / 10 or less
Dynamic friction.
close to the
constant motion
Fig.6
(2)The third law of motion = law of action and reaction
Force is that a body operates to other body. Ａ
a body A operates a force to a body B,
⇔ the body B operates a force to the body A.
Fig.7 action, reaction
Ｂ
Experiment4 To depart two trucks. Making two trucks confront each
other.
Releasing the coil spring ⇒ Two trucks detach with the same speed.
Correctly speaking, detach
with same acceleration.
Naturally, it causes in the
case mass of trucks are equal.
(3) To measure forces by a spring balance
In many cases we use a scale or a spring balance.
Especially a spring balance is often used. For example they
are used when checking action reaction law.
Fig.9
M
g
Fig.8
0.98N
Drawing together
100gw
F=Mg
Each force equals
gravity force for the body of 100 g ⇒ 0.98 [N]
⇒ roughly equals 1 [N]
100g
3 Measurement of acceleration
(1) Acceleration
Acceleration is said the variation of the velocity vector divided by
the time at a extremely short time. ａ =
𝚫ｖ
𝚫ｔ
ｖ𝟐 −ｖ𝟏
=
𝚫ｔ
Experiment5 (demonstration) An experiment of acceleration display
Installation acceleration display, plane board
Both downward and upward
(ⅰ) instant value of acceleration
(ⅱ) direction of acceleration
downward
②
(2)
Constant acceleration motion
If the velocity of a body becomes to v1 from v0
in a very short time Δt, the acceleration is
𝑎=
𝚫𝐯
𝚫𝐭
=
𝐯𝟏 −𝐯𝟎
𝚫𝐭
③
If you can measure xi, body position, at extremely short cycle
time Δt each, you can calculate the velocity. For example, if
you can measure x0, x1, x2 ,
𝐯𝟏𝟐 =
𝐱𝟐 − 𝐱𝟏
𝚫𝐭
,
𝐯𝟎𝟏 =
𝐱𝟏 − 𝐱𝟎
𝚫𝐭
Next, assuming this velocity varies between the time Δt,
then the acceleration a in the time is got.
𝐯𝟏𝟐 − 𝐯𝟎𝟏
𝑎=
𝚫𝐭
④
Experiment6
To do the same construction as Exp.1, and
pull the paper tape at a accelerated velocity. Provided that
period time is (1/10) s.
nr.of points
length
ｍ
distance
velocity m/s
difference
acceleration m/s2
0
1
0
2
3
Experiment6
To do the same construction as Exp.1, and
pull the paper tape at a accelerated velocity. Provided that
period time is (1/10) s.
nr.of points
length
ｍ
0
1
0
distance
0.105
velocity m/s
1.05
2
0.105
0.260
0.155
1.55
difference
0.55
acceleration m/s2
5.5
3
(3) To check out the relation between force and acceleration=the
second law
The acceleration is proportional to the force and inversely
proportional to mass.
Experiment7-1 pulling twice of power? and also, make truck mass be twice?
installation dynamics truck(0.50kg), spring balance, plane board,
weight(0.25kg✕2), points timer (period time is 1/10 s)
spring balance
truck
points timer
Continue pulling the truck by ways of 3 type following.
(ⅰ) Pull the truck by the balance with the dial at 0.50[N].
(ⅱ) Pull the truck by the balance with the dial at 1.0[N].
(ⅲ) Put 2 weights upon the truck, and pull with the dial at 1.0[N].
Though you pull the trucks by the balance with the dial constantly,
of course, as the trucks will be accelerated, you should make the
balance move the same movement as the trucks.
From the paper tape to calculate velocity and acceleration.
(ⅰ)one truck, 0.50N
Number
0
Distance
m
Difference
0
velocity
difference
accelerati
on
average
1
2
3
(ⅱ)one truck, 1.0N
0
0
1
2
(ⅲ)two weights, 1.0N
3
0
0
1
2
3
Though you pull the trucks by the balance with the dial constantly,
of course, as the trucks will be accelerated, you should make the
balance move the same movement as the trucks.
From the paper tape to calculate velocity and acceleration.
(ⅰ)one truck, 0.50N
1
(ⅱ)one truck, 1.0N
(ⅲ)two weights, 1.0N
Number
0
Distance
m
Difference
0
0.048 0.059 0.066
0.045 0.063 0.083
0.035 0.044
0.053
velocity
0.48
0.45
0.35
0.53
2
3
0.048 0.107 0.173
0.58
difference
0.10
accelerati
on
average
1.0
0.9
0.66
0
1
2
3
0
0
0.045
0.108
0.191
0
0.63
0.83
0.18
0.20
1.8
2.0
0.8
1.9
1
2
0.035
3
0.079 0.132
0.44
0.09
0.09
0.9
0.9
0.9
Experiment7-2 By gravity imposed on weight, to pull trucks.
It is hard pulling with constant force.
making gravitational force of weight pull a truck.
But approximately and provisionally proportional.
Installation points timer(period time 1/10 s), 50g-weight, pulley,
plane board, strap(fishing line),
timer
pulley
weight
truck
Then make the weight pull the truck in the three types below.
(ⅰ) Pull with 50g-weight
(ⅱ) Pull with 50g-weight✕2
(ⅲ) Pull a truck and 2 weights on it with 50g-weight✕2
From the paper tape to calculate velocity and acceleration.
(ⅰ) 50g-weight
Number
0
Distance
0
Difference
Velocity
Difference
Acceleratio
n
Average
1
2
(ⅱ) 100g-weight
3
0
0
1
2
(ⅲ) 2 weights 100g
3
0
0
1
2
3
Then make the weight pull the truck in the three types below.
(ⅰ) Pull with 50g-weight
(ⅱ) Pull with 50g-weight✕2
(ⅲ) Pull a truck and 2 weights on it with 50g-weight✕2
From the paper tape to calculate velocity and acceleration.
(ⅰ) 50g-weight
(ⅱ) 100g-weight
(ⅲ) 2 weights 100g
Number
0
Distance
0
Difference
0.048
0.059
0.066
0.045
0.063
0.083
0.035 0.044
0.053
Velocity
0.48
0.58
0.66
0.45
0.63
0.83
0.35
0.53
1
3
0.048 0.107
Difference
0.10
Acceleratio
n
1.0
Average
2
0.9
0.8
0.173
0
1
2
0.045
0
0.108
0.18
0.20
1.8
2.0
1.9
3
0.191
0
0
1
0.035
2
0.079
0.44
0.09
0.09
0.9
0.9
0.9
3
0.132
(4) To determine the value of the gravitational acceleration
The value can be obtained by doing the following way.
Though we can ask the value by easier method in the Exp.9.
Experiment8(demonstration) To make a
weight attached a paper tape free fall,
and to measure the distances of points.
(ⅰ) Set a paper tape through a points
timer (period time 1/10 s).
(ⅱ) Attach the paper tape end to a
weight.
(ⅲ) Fall the weight free.
(ⅳ) Calculation.
timer
weight
(5)
To determine the gravitational acceleration by fall
distance and velocity
If you free fall at the field of gravitational acceleration g,
as a = g, v0 = 0
v2 = 2 g x exists,
Therefore g = v2 / 2 x ⑤
Namely, at a point x [m] fallen
if you measure the velocity v [m / s],
g can be obtained by a calculation.
Experiment 9 Using Be-Spe
small ball
To measure the velocity of
a steel ball at the point
where the ball have fallen
a certain distance.
Installation Be-Spe,
transparency tube, small ball
x
tube
distance x
reached velocity v
⇒ g = v2 / 2 x
x [m]
1st
2nd
Be-Spe
v
v [m/s]
g [m/s2]
Experiment 9 Using Be-Spe
small ball
To measure the velocity of
a steel ball at the point
where the ball have fallen
a certain distance.
Installation Be-Spe,
transparency tube, small ball
x
tube
distance x
reached velocity v
⇒ g = v2 / 2 x
x [m]
Be-Spe
v
v [m/s]
g [m/s2]
1st
0.50
3.1
9.6
2nd
0.60
3.4
9.6
(4) The top supported at center of gravity or a gyroscope
Even on the Earth, the tops Fig.26 gyroscope
supported at center of gravity
are intact because those tops
are not subjected
to the moment of force.
This is the principle of
“gyroscope”
or“gyrocompass”.
Its axis of rotation is
permanently constant, so it
points the relative changing
of direction of the bodies
nearby, for example
latitude and longitude,
and a position
of an airplane and a robot.
Experiment15 To operate “space top” to make
sure the pan-tilt motion and the gyro effect.
Spacetop or Chikyuu-koma is the equipment
that is so much simplified from a gyroscope.
(ⅰ) Rotating the space top, applying force
to the axis of rotation and checking pantilt motion.
(ⅱ) Rotating the top, holding the circle part
with two fingers like the Fig., and tilting
the gimbal, then you will receive the
force perpendicular to the action original.
(ⅲ)If you can support (ⅲ)
the circle part with
bearing, fulcrum, or
swivel, the top will
be “gyroscope”.
How do you realize it?
(ⅱ)