__Experiment 1__

Rotational Equilibrium

(Mass Moment of Inertia)

Objective:

To experimentally measure the mass moment of inertial of a solid disc

Equipment:

A solid disc and axle, vertical holding rod, a few meters of inextensible string, a stop watch or two electronic time gates, a weight hanger, assorted weights, and a calculator

Theory:

The condition for the rotational equilibrium of a rotating disc** **about its
centroidal axis** **is

ΣT =
Iα**.**

I is the mass moment of
inertia of the solid disc about its centroidal axis (the axis that is normal to it
and passes through its center** **of mass)**.** Of course, the center
of mass of a uniform solid disc is the same point as its geometric center**.**

In this experiment a solid disc that can spin about
its centroidal axle will be given angular acceleration by applying a constant
tangential force to its outer edge for a short period (t_{1})**.**
The disc accelerates during t_{1}**.**
After removing the force at the end of period t_{1},
the disk will decelerate due to friction (mainly at its axle) and comes to stop
within a period (t_{2})**.** By measuring
t_{1} and t_{2}, the
angular acceleration of the disc in each phase can be calculated**.** A
schematic diagram of one possible apparatus setup is shown in
Fig. 1.

The hanging mass M_{2}
exerts a constant tangential force F = M_{2}g
onto to outer edge of the small disc** **or cylinder** **
(Fig. 2)**.** When M_{2} moves from
A to B during period
t_{1}, it accelerates the disc**.** The linear
acceleration a of M_{2}
during the falling time t_{1}can easily be
calculated using the distance y = AB.

If
α_{1}
is the angular acceleration of the attached discs,
it is related to a by a =
R_{1}α_{1}
from which α_{1}
can be calculated**.**

α_{2},
the angular deceleration of the disc during the stopping period, can
also be calculated from time t_{2
}or the angular displacement θ_{2} measured by
counting the number of turns during deceleration**.**

Let ΣT
be the net torque acting on the discs causing it to rotate about its centroidal axis**.**
With friction present at the axle, ΣT
= T_{applied} - T_{friction}**.**

The applied torque is T_{applied}
= F∙R_{1} where F = M_{2}g
is the tangential force and
R_{1} is the radius of the disc** **about which
the string is wrapped**.**

Fig. 1

Fig. 2

__
Procedure:__

1) Gather the appropriate
equipment as per the equipment list**.**

The trick is to measure the
falling time (t_{1})
of weight M_{2}g from
A to B in order to calculate its
falling acceleration, a**.** Do all necessary
preparations to set the apparatus as shown and

2) Let M_{2}
(The weight hanger including the added weights) fall and pull the string that
rotates the disc**.** M_{2} will have
linear acceleration, a_{1},
while the disc gains angular acceleration,
α_{1}**.** The two
are related by a_{1}
= Rα_{1}.

3) Calculate
a_{1}
from the travel time (t_{1})
and the falling distance y = AB. Time t_{1}
can
either be measured by a stop watch or an electronic time gate**.**

The distance
y = AB can be measured by a ruler or a tape measure**.**

4) Calculate
α_{1}**.**

5) Calculate the applied torque
T_{applied}
= F∙R_{1} in which F = M_{2}g.

6) Calculate final velocity,
v_{f}
,
of the falling mass and the ω_{f}
of the solid disc**.** These are the final speeds at the end of the
acceleration period**.** As soon as the weight hits the ground or
passes point B, it must no
longer pull the string and the disc must no longer accelerate**.**
The arrangement must be made for this to happen**.** Make sure that
Point B is the end of the acceleration period after which the
rotating disc starts to decelerate and comes to stop**.**

7) While a group member is
measuring the falling time, another member must count the number of turns the
disc makes before it comes to stop**.** This means the number of turns
during the slowing down period**.**

8) Convert the number of turns
made to radians and calculate the angular deceleration,
α_{2}
, of the disc during the slowing down period**.**

9) Treat I as an unknown, and use
α_{2
}to calculate the frictional torque, T_{friction}**.**
The frictional torque, T_{friction}, will
therefore be in terms of the unknown, I**.**

T_{friction}
= -** **Iα_{2}
(Why?)

10) Apply the formula
ΣT =
Iα_{1}
(during the acceleration period) to solve for the unknown or the measured value of I**.**

11)
Calculate the accepted value of I directly from
I
= 0.5MR^{2 } + 0.5M_{1}R_{1}^{2} by using the
masses and radii of the discs**. **If masses are not given, calculate
them for each disc using M = *ρ*V where *
ρ* is the mass density of the material of the discs**.**

12)
calculate the %error on I**.**

13)
Repeat the experiment for two more different values of M_{2}, the
falling weight**.**

14)
Average the 3 values you found for I in step 10 and
calculate a final %error on I**.**

15) Calculate the maximum angular momentum of the disc during the experiment based on the value of I in Step 14.

**
Data:**

**
Given: **

g = 9**.**81 m/s^{2}**.**

** Measured****:**
M, M_{1},
M_{2},
t_{1}, t_{2}
or θ_{2},
R, and
R_{1}

Trial | M (kg) | M_{1 }
(kg) |
M_{2 }
(kg) |
t_{1}
(s) |
t_{2}
(s) |
θ_{2}
(turns) |
θ_{2}
(radians) |
R (m) |
R_{1}
(m) |

1 | |||||||||

2 | |||||||||

3 | |||||||||

__Comparison
of the results:__

Provide the percent error
formula used and the calculated percent errors in each case as well as the
percent error on the average value of I**.**

__Conclusion:__

State your conclusions of
the experiment**.**

__
Discussion:__

Provide a discussion if
necessary**.**