MEASURE ROTATIONAL INERTIA: ENERGY METHOD                           Exp. #20

(Alward/Harlow web File: "inertia.doc" 3-23-04)           1.5 hours

Name: _____________________ Partners: _______________________Section No. _______

Equipment Needed: Rotational Inertia Cradle    Ring    Disk     Large Table Clamp

2-meter pole    2 Utility Clamps    2 Pasco pulleys     Paper Clip    5-1 g masses   5-2 g masses

1-10 g masses    1-5 g mass    1-200 g mass    50-g Beck weight hanger     Vernier Caliper

2 meter sticks      Stopwatch

I.  Theory and Purpose

In this exercise the student will use conservation of energy principles to measure the rotational inertia of an object.

A cradle supporting and object will be caused to rotate by means of a falling mass.  This arrangement is shown in the figure.  The system will be initially at rest.  A small compensating mass hanging at the end of the string will cancel the effects of friction (this compensating mass will be discussed later).  With the compensating mass in place and friction canceled, the final total mechanical energy of the system will equal the initial total mechanical energy.  Thus,

E = E0                                                       (1)

˝ mv2 + ˝ Iw2 = mgh             (2)

Determining v:

The falling mass starts from rest and has a final speed v.  The final speed is twice the average speed, which is equal to the distance traveled, h, divided by the travel time, t:

v = 2 (h/t)                                (3)

Determining w:

The tangential speed of a point on the side of the string drum is the same as the speed of any point along the length of the string, which is the same as the speed, v, of the falling mass.  The radius of the drum is r, so the angular velocity, w, is given by Eqn. 4:

w = v/r                        (4)

Determining I:

The rotational inertia, I, can be found as soon as one measures the fall distance, h, and the time of fall, t.  Just use Equation 3 to find v, then use Equation 4 to find w, then substitute the values of m, v, w, g, and h into Equation 1 to solve for I.

First compensate for friction:

With the cradle empty, begin hanging small masses on a paper clip attached to the end of the string.  Give the cradle a gentle twist to start it moving; if it continues to move without stopping and with constant speed, then the frictional forces will have been canceled.  Keep adding mass until the cradle and mass move with approximately constant speed (zero acceleration).

Measure h:

On top of the compensating mass already at the end of the string, place an additional 20 grams at the end of the string.  Pre-select some fall distance for the hanging mass: from a point near the top pulley, to a point near the ground.  Record this distance as h in Table I.  Note: this distance is not the one meter distance from pulley to table shown in the figure on Page 1.

Measure r:

Use the vernier calipers to measure the diameter d, of the string drum.  Divide by two to obtain the radius of the drum.  Record both values in Table I.

Measure Fall Time:

Release the cradle and start the stopwatch simultaneously. Record the time of fall through the distance h.  Brake the cradle before the mass strikes the ground; this also prevents the string from unraveling and getting caught underneath the drum and becoming knotted.

Repeat the timing for a total of five times; record the times of fall in Table I.

 h =                               m d =                               m r = d/2 =                            m

 Trial Number Time, t      (seconds) 1 2 3 4 5

 Average t =                    s v= 2 (h/t) =                       m/s w = v/r =                        rad/s

 m = 20 grams = 0.020 kg   Substitute the values above for m, v, w, and h into Equation 2:       ˝ mv2 + ˝ Iw2 = mgh         to solve for the rotational inertia of the cradle.  Record this value below as IC .                                                                                                                                                                                             IC =                   kg m2

III.  The Ring

The rotational inertia of a ring is given in Equation 5:

IR = ˝ MR (R12  + R22 )                        (5)

MR is the mass of the ring, and R1 and R2 are the ring’s inner and outer radii, respectively.  The mass of the ring is 4.3 kg.  This value is recorded in Table II below.  Measure the inner and outer diameters, D1 and D2, then divide by two to obtain the radii.  Record values in Table II.

Calculate the ring’s rotational inertia using Equation 5.  This value will be referred to as the expected value.

TABLE II:   RING DATA

 Mass of the Ring:      MR =             4.3 kg

 Inner Diameter:         D1   =                   cm R1 = D1/2 =                cm =                      m Outer Diameter:        D2 =                    cm R2 = D2/2 =                cm =                      m

 Expected Rotational Inertia of the Ring (Eqn. 5):             I’R =                                        kg m2

Place the ring on the cradle.  Because of the greater weight placed on the drum’s bearings, the frictional force will be greater than it was without the ring on the cradle.  Thus, you will have to place additional compensating weights on the hanger.

Compensating for friction.  Add additional mass to the compensating mass already present at the end of the string and gently twist the ring to get it moving.  Continue adding mass until the hanging weight appears to fall with approximately constant speed.  Keep this mass at the end of the string, but ignore it in future calculations.

Measure the rotational inertia of the ring.   Place 200 grams on a 50-gram hanger (total: 250 grams) at the end of the string (not counting the compensating mass that is already there). Release the ring, start the stopwatch, and record in Table III the times of fall through the distance h.

Table III: The Ring

 h =                               m d =                               m (this is string drum diameter) r = d/2 =                            m

 Trial Number Time, t      (seconds) 1 2 3 4 5

 Average t =                    s v= 2 (h/t) =                       m/s w = v/r =                        rad/s

 m = 250 grams = 0.250 kg   Substitute the values above for m, v, w, and h into Equation 2:       ˝ mv2 + ˝ Iw2 = mgh         to solve for the rotational inertia, I,  of the Cradle + Ring.  Record this value below as IC+R .                                                                                                                                                                                             IC+R =                  kg m2

 IR = IC+R - IC =                                                    kg m2 I’R =                                                                    kg m2                        (From Table II. ) Percent Difference:     |IR - I’R| 100 % / I’R   =                                      %

IV.  The Disk

The rotational inertia of a disk is given in Equation 6:

ID = ˝ MDR2                         (6)

MD is the mass of the ring, and R is the disk’s radius.  The mass of the disk is 4.5 kg.  This value is recorded in Table IV below.  Measure diameter, D, then divide by two to obtain the radius. Record values in Table IV.

Calculate the disk’s rotational inertia using Equation 6.  This value will be referred to as the expected value.

TABLE IV:   DISK DATA

 Mass of the Disk:      MD =             4.5 kg

 Diameter:         D   =                   cm R = D/2 =                cm =                      m

 Expected Rotational Inertia of the Disk (Eqn. 6):             I’D =                                        kg m2

Place the disk on the cradle.

Compensating for friction.  Use the same compensating mass that was used for the ring.

Measure the rotational inertia of the disk.   Place 200 grams on a 50-gram hanger (total: 250 grams) at the end of the string (not counting the compensating mass that is already there). Release the disk, start the stopwatch,, and record in Table V the times of fall through the distance h.

Table V: The Disk

 h =                               m d =                               m (this is the string drum diameter) r = d/2 =                       m

 Trial Number Time, t      (seconds) 1 2 3 4 5

 Average t =                    s v= 2 (h/t) =                       m/s w = v/r =                        rad/s

 m = 250 grams = 0.250 kg   Substitute the values above for m, v, w, and h into Equation 2:       ˝ mv2 + ˝ Iw2 = mgh         to solve for the rotational inertia, I,  of the Cradle + Disk.  Record this value below as IC+D .                                                                                                                                                                                             IC+D =                  kg m2

 ID = IC+D - IC =                                                    kg m2 I’D =                                                                    kg m2                        (From Table IV. ) Percent Difference:     |ID - I’D| 100 % / I’D   =                                      %