APPLIED MECHANICS
Mass Moment of Inertia of Flywheel
FACULTY OF ENGINEERING & INFORMATION TECHNOLOGY (Sep-Dec 2015)
Introduction: A heavy fywheel is set in rotation by a mass attached to a string wrapped around the axle o the fywheel. The orce exerted by the alling mass is related to the torque, Γ, and the rate o change o angular velocity o the wheel, that is, the angular acceleration, α. The moment o inertia, , is the constant o proportionality between these two variables and depends on the mass and e!ective radius o the rotating ob"ect. The above law can be veri#ed by using various masses and measuring the resulting acceleration or each as a unction o the net accelerating torque. THEORY $lywheel is a mechanical device with signi#cant moment o inertia used as a storage device or rotational energy. energy. $lywheels resist changes in their rotational speed, which helps steady the rotation o the shat when a fuctuating torque is exerted on it.
$or a thin solid dis% o fywheel, the mass moment o inertia is shown in &quation ', where m ( mass, r ( radius o the fywheel
)&quation
'*
$or rotational motion, +ewtons second law )see &quation -* can be adopted to describe the relation between the applied torque, T and angular acceleration, .
)&quation -*
T = I.
+ote that or constant angular acceleration, the angular displacement o a rotating ob"ect can be obtained rom &quation )&quation *
( t / 0 t-
Apparatus Needed:
• • •
$lywheel apparatus. A set o weights. A stopwatch and ruler.
rocedure:
'. 1ecord the measurements o the radius o torque pulley ) r p* and fywheel )r *, as well as the mass o the fywheel ) m*. -. 2ound a cord around the torque pulley and ta%e a load hanger o %nown weight and hang it at the ree end o the cord. . 3lace a load on the load hanger and hold the load in position 4. Ad"ust the fywheel so that the arrow mar%ed on it aligns with the arrow mar%ed on the rig. 5. 6et the stopwatch to 7ero. 8. 1elease the load while simultaneously pressing the stopwatch button.
9. Ater ' revolution, simultaneously.
stop
the
fywheel
and
the
stopwatch
:. 1ecord the time ta%en or the fywheel to rotate ' revolution. ;. 1epeat the experiment twice to get an average value o time ta%en or the fywheel to rotate ' revolution. '<.
1epeat 6teps = ; or another 4 di!erent sets o load.
''. 1epeat the experiment by attaching the small dis% and the ring to the fywheel. 1esults and >iscussion? Without the disk: R p = 7.5 mm Total load! " on tor#ue pulley $N%
Applied Tor#ue $Nm% &"' r p
'<
Time ta(en $)ec% t*
t+
t,
A-era. et
<.<95
5.'8
4.';
4.'5
4.5
-<
<.'5
-.-'
-.--
-.9
-.
<
<.--5
'.9<
'.5'
'.84
'.8
4<
<.
'.
'.4
'.85
'.5
5<
<.95
'.'
'.'9
'.-
'.-
With the disk R p = 20mm Total load! " on tor#ue pulley $N%
Applied Tor#ue $Nm% & " ' r p
'<
<.-
Time ta(en $)ec% t*
t+
t,
A-era. et
4.44
5.<:
5.-
4.;
-<
<.4
-.-5
-.9
-.-5
-.
<
<.8
'.;'
'.8;
'.99
'.:
4<
<.:
'.59
'.<
'.45
'.4
5<
'
'.-4
'.5
'.-;
'.
/alculation:
$rom the experiment, i an ob"ect rotates 8< degrees around a circle radius r )' revolution* that mean is
$rom the equation we can simpliy to @ ( -. Thereore ' revolution is -radians. 2e are going to #nd the angular acceleration o the pulley we will using this equation 1
2
θ= ω0 ∙t + ∙ α ∙ t 2
2e can see in this experiment we did only one revolution or all the 2
set o weights .so it is determined that the θ= ω0 t +
α . t 2
=2 π ,, i we want
to #nd the angular acceleration we should simpliy the equation to be 2
α .t 2
so
=2 π – ω t 0
α =2
2π
– ω0 t 2
t
2e can see that there wasnt an initial orce on the fywheel when the experiment was being done. 6o the initial angular velocity ) the fywheel when t ( < is also
ω0
( <
ω
0
¿
o
α =
Beans that?
4 π 2
t
The result or angular acceleration showing by the ollowing table o calculated result The result with the small dis(: Total load !" A-era.e $t% A-era.e$t+% on tor#ue pulley 012 '< -4.<' +1, -< 5.-; < .-4 *13 4< '.;8 *10 5< '.8; *1, The result without the small dis(: Total load !" on tor#ue pulley '< -< < 4< 5<
A-era.e $t%
014 +1, *15 *14 *1+
A-era.e$t+%
-<.-5 5.-; -.58 -.-5 '.44
An.ular acceleration
<.5-.: .:: 8.4' 9.44 An.ular acceleration
<.8-.: 4.;' 5.5; :.9
Cradient (
m=
1− 0.8 y 2 − y 1 = =0.1942 x 2 − x 1 7.44 −6.41
Dased on the graph shown above, mass moment o inertia o $lywheel is ound by calculating the slope o the graph which is
0.1942
)experimental value* The ormula or #nding the theoretical mass moment o inertia o $lywheel is I =
mr 2
2
2
(
12.1 kg × 0.125 m 2
3ercentage error
¿
¿
=0.0945 %gm-
( Experimentalvalue −t h eoretical value ) Experimental value
(0.1942 −0.0945 ) 0.1942
× 100
( 5.'F
Cradient (
m=
y 2 − y 1 0.25 −0.15 = =0.0637 3.8 − 2 x 2 − x 1
E '<
Dased on the graph shown above, mass moment o inertia o $lywheel is ound
by
calculating
the
slope
o
the
graph
which
is
0.0637
)experimental value* The ormula or #nding the theoretical mass moment o inertia o $lywheel is I =
mr 2
2
6.55
(
3ercentage error
kg × ( 0.03 5 ) 2 2
= 0.00411 %gm-5-
¿
( Experimentalvalue −t h eoretical value ) E '<
¿
(0.0637 −0.0041 ) × 100 =¿ ;.5F ( 0.0637 )
6iscussion: Gur goal in this experiment is to measure the moment o inertia o a fywheel in regard to the weights that are loaded to the wheel. As we can see rom our experiment results and conclusion the fywheel is moving ast when we move the dis%. However, the results are not stable because o many reasons as the time ta%ing in the right time o start and #nish o ' revolution. Adding to the next reason is the fywheel and the stopwatch were simultaneously stopped and getting the write records o the revolution. Those reasons ma%e us have big number o error percentage.
Results with the dis(:
$rom the diagram which we draw it is explain and clearing the relationship between the torque and angular o acceleration. And we #nd the gradient is ) <.';4-* or the #rst table which is with the dis% and we ound out that the percentage error is )5.'F*. Results without dis(:
n the second table which we ound the gradient is )<.<89* and ater we calculate the percentage o error is ( );.5F* .
The percentage o error was almost nothing because o the accuracy in calculating time )t*, it indicates that the data obtained )t* and the calculations were ;
/onclusion: &ven though our calculations were prIcised, we should %eep in mind that there are some important actors a!ected our results. $or example, in this experiment we neglected the riction orces between the rope and the wheel, the one between the wheel and the axle and the air resistance orces. n addition, the time readings werenJt perect because we got di!erent values in di!erent tries and we only too% the average. we want to get a perect reading then we should get a digital stop watch instead o using the normal stop watch. At the end o the day, we can say that we achieved our goal and that our calculations and results are o%.
