GT4601 GT4601A Moment of Inertia Tester
Moment of inertia is a measure of the inertia of a rigid body and is a physical quantity that characterizes the rigid body. The magnitude of the moment of inertia is related to the mass of the object, but also to the position and mass distribution of the shaft (ie, shape, size, and density). Simple, uniform mass distribution, can directly calculate its moment of inertia around a specific axis. But in engineering practice, we often encounter a large number of complex shapes, and the mass distribution is not uniform, the theoretical calculation will be extremely complicated, usually using experimental methods To determine.
The measurement of the moment of inertia generally causes the rigid body to move in a certain form. By characterizing the relationship between the physical quantity and the moment of inertia of the motion feature, the conversion measurement is performed. There are various methods for measuring the moment of inertia of the rigid body. The three-line pendulum method is An experimental method with good physical ideas, which has the advantages of simple equipment, intuitive, and convenient testing.
ã€Purpose】
1. Learn to measure the moment of inertia of an object with a three-line pendulum.
2. Learn to measure the period of periodic motion by cumulative amplification.
3. Verify the parallel axis theorem of moment of inertia.
[Experimental Apparatus]
1, GT4601 moment of inertia tester 1
2, experimental rack 1 set
3, the ring 1
4, cylinder 2
[Experimental principle]
Figure 1 is a schematic view of the three-line pendulum experimental device. The upper and lower discs are horizontal and suspended on the beam. Three symmetrically distributed isometric suspension lines connect the two disks. The upper disk is fixed and the lower disk can be wound around the center. The axis OO` is used for torsional pendulum motion. When the angle of rotation of the lower disk is small and the air resistance is omitted, the motion of the torsion pendulum can be approximated as a simple harmonic motion. According to the law of conservation of energy and the law of rotation of the rigid body, the object can be derived around the central axis OO' Moment of inertia (see the appendix of this experiment for the derivation process).
(4-1)
The meaning of each physical quantity in the formula is as follows: m0 is the mass of the lower disc; r and R are the distances of the upper and lower suspension points from the center of the respective disc; H0 is the vertical distance between the upper and lower discs during balance; T0 is the simple disc movement for the lower disc Cycle, g is the acceleration of gravity (in Hangzhou area) ).
Place the object to be tested of mass m on the lower plate, and make the axis of the rigid body to be measured coincide with the OO' axis. Measure the vertical distance between the pendulum motion period T1 and the upper and lower discs. The total moment of inertia of the rigid body and the lower disc against the central axis OO' axis is:
(4-2)
If the suspension is not caused by the weight change, there is H≈H0. Then, the moment of inertia of the object to be tested around the central axis is:
(4-3)
Therefore, by measuring the length, mass and time, the moment of inertia of the rigid body around a certain axis can be obtained.
The parallel axis theorem can also be verified by the three-line pendulum method. If the mass of mass m is rotated through its centroid axis, the moment of inertia is Ic. When the axis of rotation moves parallel to the distance x (as shown in Figure 2), the object is aligned with the new axis OO. The moment of inertia of ` is I00=Ic+mx2. This conclusion is called the parallel axis theorem of moment of inertia.
During the experiment, two cylinders of the same mass and mass distribution are placed symmetrically on the lower disc (there are two small holes in the lower disc). In the same way, two small cylinders are measured. And the rotation period Tx of the lower coil center axis OO`, then the moment of inertia of each cylinder to the central axis OO` can be found:
(4-4)
If the distance x between the center of the small cylinder and the center of the lower disk and the radius Rx of the small cylinder are measured, the parallel axis theorem can be obtained.
(4-5)
Comparing the size of Ix and I`x, we can verify the parallel axis theorem.
ã€laboratory apparatus】
Three-line pendulum, level gauge, stopwatch, meter ruler, vernier caliper, physical balance and object to be tested.
[instrument operation]
1. Turn on the power, the program preset period is T=30 (digital display), that is: the number of times the ball passes back and forth through the photogate is T=2n+1 times.
2, according to specific requirements, if you want to set 50 times, first press "set" to unlock, then press up (or down) to change the period T, and then press "set" to lock, at this time, you can press the execute button to start timing. The signal light flashes continuously, that is, the timing state. When the number of cycles of the object passing through the photogate reaches the set value, the digital display will display the specific time, the unit is “seconds.†When the “50†cycle is required, no need to reset, just press "Back" can return to the number of cycles "50" that was just executed last time, and then press the "Execute" button, you can time the second time.
(When the power is turned off and on, the program is preset 30 times from the beginning, and the above steps must be repeated)
[Experimental content]
1. Using a three-line pendulum to measure the moment of inertia of the ring pair passing through its center of mass and perpendicular to the torus axis.
2. Verify the parallel axis theorem with a three-line pendulum. The main points of the experimental steps are as follows:
1) Adjust the lower plate level: place the level between any two suspension lines on the lower plate, adjust the three knobs on the small disc, and change the length of the three suspension lines until the lower level.
2) Measure the movement period T0 of the empty coil center axis OO`: gently rotate the upper disc to drive the lower disc to rotate, so as to avoid the sway of the three-wire pendulum during the torsion pendulum movement. Note that the rotation angle of the torsion pendulum is controlled within 5°. The cumulative amplification method measures the period of the torsional pendulum motion (using a stopwatch to measure 30 to 50 times of time, and then finding its motion period, why not measure one cycle directly?). When measuring time, start counting when the lower plate passes the equilibrium position. And silently read 5, 4, 3, 2, 1, 0, when the number reaches "0", the stop table is started, so that there is a preparation process of counting, and not a few cycles.
3) Measure the period T1 of the common rotation of the ring to be tested and the lower plate: place the ring to be tested on the lower plate, pay attention to make the centers of the two coincide, and measure the period T1 of their movement together in the same way.
4) Measure the period T x of the rotation of the two small cylinders (symmetrically placed) and the lower disc.
5) Measure the distances a and b between the three hanging points of the upper and lower discs, and then calculate the distance r and R from the hanging point to the center (the radius of the circumscribed circle of the equilateral triangle)
6) Measurement of other physical quantities: measure the vertical distance H0 between the two discs by using a meter ruler and the spacing of the small holes of the two small cylinders 2x; measure the inner and outer diameters 2R1, 2R2 and small of the ring to be tested with a vernier caliper The diameter of the cylinder is 2Rx.
7) Record the quality of each rigid body.
[Data and results]
1, experimental data record
Lower quality
Ring quality to be tested
Cylinder mass
Note: The radius of the upper suspension hole of this experiment is r=44.0mm, and the radius of the hem of the hem is R=90.0mm.
Table 4-1: Accumulated Method Period Data Record Reference Table
The time unit required to swing 50 times (seconds)
Lower plate
Lower ring plus ring
Lower plate plus two cylinders
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
average
average
average
cycle
T0= S
T1= S
Tx= S
Table 4-2: Reference table for data measurement of multiple lengths
project
frequency
Upper plate spacing
Upper plate spacing
Ring to be tested
Small cylinder diameter
Place a small cylinder with two small holes
Outer diameter
Inner diameter
1
2
3
4
5
average
2. Calculate the measurement result of the ring to be tested, and compare it with the calculated value of the theoretical value, and find the relative error and discuss it. The formula for calculating the moment of inertia of the ideal circle around the central axis is
3. Find the moment of inertia of the cylinder around its own axis and calculate the theoretical value {
}Compare, verify the parallel axis theorem.
[thinking questions]
1. Why do you have to keep the lower plate level when using the three-wire pendulum to measure the moment of inertia of the rigid body?
2. During the measurement process, the following discs are shaken, does it have an effect on the measurement of the period? If there is any impact, how should it be avoided?
3. After the three wires are placed on the object to be tested, is the swing period necessarily greater than the rotation period of the empty disk? why?
4. When measuring the moment of inertia of a ring, if the rotation axis of the ring does not coincide with the axis of the lower disk, what effect does it have on the experimental results?
5. How to use a three-line pendulum to measure the moment of inertia of an object of any shape around an axis?
6. The three-wire pendulum is damped by air during the swing, and the amplitude is getting smaller and smaller. Does its cycle change? Does it have a big impact on the measurement results? why?
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