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]

　　　　　　　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

Fig.2 points recording timer

(2) Using a points recording timer

(ⅱ) marking points on a body periodically at very short time

(ⅰ)measuring time very short as possible

Fig.1 marking points

Fig.3 principle of points recording timer

 iron bar

carbon 　　　　　　　 iron core paper 　　　　　 spring 　　　 coil papertape 　　　　　　 AC

Using 50Hz ⇒(ⅰ) number of vibration of the iron bar per second is 50.AC supply ⇒(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/s1-2 v={(5.0−2.5)/100}/(1/10)= 0.25m/s2-3 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/s1-2 v={(5.0−2.5)/100}/(1/10)= 0.25m/s2-3 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.

{(7.5–5.0)/100}/(1/10)=0.25m/s

Nr.of points ０ １ ２ ３ ４length(cm) ０        difference(cm)

           

distance(m)            velocity(m/s)            

Experiment 1 To measure the velocity of linear motion of the hand .Installation 　 points recording timer, paper tape, ruler measureProcedure 　 points timer (1/10)sec

a hand 　 after setting, switching ON, and pulling the

tape

Nr.of points ０ １ ２ ３ ４length(cm) ０ 9.5 20.2    difference(cm)

  9.5

10.7      

distance(m)   0.095

0.107

     

velocity(m/s)   0.95

1.1      

Experiment 1 To measure the velocity of linear motion of the hand.Installation 　 points recording timer, paper tape, ruler measureProcedure 　 points timer (1/10)sec

a hand 　 after setting, switching ON, and pulling the

tape

Points recording timer

Experiment 2 　 Motion of hovering soccer ball.

rotating fan

Points on paper tape air layer lined up almost evenly. ⇓ constant velocity linear motion

or, uniform motion.

　　　　　

ball paper tape

(3) Using “Be-Spe”

Fig.4 principle       sw1 sw2 two light

switches

Fig.5 Be-Spe

(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

Height 5 cm 10 cm 　 20 cm

velocity 1st m/s

     

velocity 2nd m/s

     

velocity 3rd

　 m/s     

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

Height 5 cm 10 cm 　 20 cm

velocity 1st m/s

0.9

1.3

1.8

velocity 2nd m/s

     

velocity 3rd

　 m/s     

2 Measurement of force

Dynamics truck wears a four wheels 　whose axle is held in ball bearings, so

wheelrotation is very

smooth.

Rolling friction is 1 / 10 or less

Dynamic friction.   close to the constant motion

(1) 　 dynamics trucks Fig.6

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.

(2)The third law of motion = law of action and reaction Fig.7 action, reaction

Force is that a body operates to other body. a body A operates a force to a body B, ⇔ the body B operates a force to the body

A.

　　　　　Ａ　　　　Ｂ　　　　　　　　　　　　　　　　　

(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.8 

   

Drawing together   Each force equals

Fig.9 M g 0.98N 100gw

 

  F=Mg  

100g   gravity force for the body of 100 g ⇒ 0.98 [N]

⇒ roughly equals 1 [N]

3 Measurement of acceleration(1) Acceleration

Acceleration is said the variation of the velocity vector divided by the time at a extremely short time. 　 ②

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) 　 Constant acceleration motionIf the velocity of a body becomes to v1

from v0 in a very short time Δt, the acceleration is 　　　　　　　

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

𝚫𝐭Next, assuming this velocity varies between the time Δt, then the acceleration a in the time is got. 　　　　　

𝑎=𝐯𝟏𝟐−𝐯𝟎𝟏

𝚫𝐭④

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 0 　　　 1 　　　 2 　　　 3

length 　ｍ 　　　　　 0      

distance          

velocity m/s          

difference        

acceleration m/s2

       

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 0 　　　 1 　　　 2 　　　 3

length 　ｍ 　　　　　0

0.105 0.260  

distance   0.105 0.155

   

velocity m/s   1.05 1.55    

difference   0.55    

acceleration m/s2

  5.5    

(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 (ⅱ)one truck, 1.0N (ⅲ)two weights, 1.0N

Number 　0　

1 　2 　　 　　

　 3 　0 　　

　 1 　 2 　 3 　 0 1 　 2 　 3

Distance m

　0

      　 0       　 0      

Difference

                             

velocity                              

difference

                       

acceleration

                       

average      

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 (ⅱ)one truck, 1.0N (ⅲ)two weights, 1.0N

Number 　0　

1 　2 　　 　　

　 3 　0 　　

　 1 　 2 　 3 　 0 1 　 2 　 3

Distance m

　 0 0.048

0.107

0.173

　 0 0.045 0.108 0.191

　 0 0.035 0.079

0.132

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 0.44 0.53  

difference

  0.10     0.18 0.20     0.09 0.09  

acceleration

  1.0 0.8     1.8 2.0     0.9 0.9  

average 0.9 1.9 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 truck

    weight

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

　 1 　 2 　 3 　0

　 1 　 2 　 3 　0

　 1 　 2 　 3

Distance 　 0 　 0 　 0

Difference    

Velocity    

Difference  

Acceleration

 

Average

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

　 1 　 2 　 3 　0

　 1 　 2 　 3 　0

　 1 　 2 　 3

Distance 　 0 0.048 0.107 0.173 　 0 0.045 0.108 0.191 　 0 0.035 0.079 0.132

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 0.44 0.53  

Difference   0.10     0.18 0.20     0.09 0.09  

Acceleration

  1.0 0.8     1.8 2.0     0.9 0.9  

Average 0.9 1.9 0.9

(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.

timer(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.

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 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 　⑤Namely, 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

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

small ball

x tube　　

Be-Spe 　　　　 　　　　 v

  　　　 x [m] 　　 v [m/s] 　　 g [m/s2]1st      

2nd      

distance x reached velocity

v ⇒ 　 g = v2 / 2 x

Experiment 9 　 Using Be-Spe

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

small ball

x tube　　

Be-Spe 　　　　 　　　　 v

  　　　 x [m] 　　 v [m/s] 　　 g [m/s2]1st 0.50 3.1 9.6

2nd 0.60 3.4 9.6

distance x reached velocity

v ⇒ 　 g = v2 / 2 x

(4) The top supported at center of gravity or a gyroscope

Fig.26 gyroscope Even on the Earth, the tops supported at center of gravity are intact because those tops are not subjectedto 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

pan- tilt 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

the circle part with

bearing, fulcrum, or

swivel, the top will

be “gyroscope”. How do you realize

it?

(ⅱ)(ⅲ)