Лабораторная работа


Maxwell’s pendulum represents a disk, whose axis is suspended on two turning on it threads (fig. 1). It is possible to study experimentally dynamics laws of translational and rotational motions of rigid body using this pendulum, as well as the main law of physics − the law of mechanical energy conservation.



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24 чел.


Objective :

study the laws of mechanics for the rigid body by the example of its planar motion

Main tasks:

1. Experimental determination of momentum of inertia for the Maxwell’s pendulum by its fall time;

2. Calculation of momentum of inertia of Maxwell’s pendulum using the theoretical formula.

  1.  Theory of experiment

Maxwell’s pendulum represents a disk, whose axis is suspended on two turning on it threads (fig. 1). It is possible to study experimentally dynamics laws of translational and rotational motions of rigid body using this pendulum, as well as the main law of physics − the law of mechanical energy conservation. Having rotated the pendulum, we raise the disk to height h and let go down without push, then the disk is starting to go down and at the same time to rotate around its horizontal axis. At the same time trajectory of all points of the disk lie in parallel plane (surface). Such motion of rigid body is called planar. It can be considered as the translational motion of the body, which is occurring with the velocity of centre of mass (centre of gravity, centre of inertia) and at the same time as the rotational motion around horizontal axes, passing through this center.

The equation of motion for the centre of gravity and rotation of pendulum relatively to mentioned axes has the following form:

where m is mass of the pendulum, I is momentum of inertia, а is acceleration of gravity center, ε – angular acceleration of the pendulum, Т is tension of the thread, r is radius of tube.

Taking into account, that the accelerations in this case are connected with each other by the relation, we obtain from formulae (1) and (2):

From the last relation comes that the center of mass of the pendulum moves with constant acceleration, which depends on the body’s momentum of inertia. This circumstance is the basis of this theory.

From the relation (3) with taking into account the formula of the path for uniformly accelerated motion h = at2/2, we obtain the calculation formula:

where D = D0 + d.

Thus, to determine the momentum of inertia of the Maxwell’s pendulum, it is necessary to measure time t of its fall from given height h, to define its mass m, and diameter of the tube D0 and thickness d of thread.

2. Description of experimental device.

The general view of Maxwell's pendulum is shown on fig. 1. This device consists of pendulum, electromagnet, two photoelectric sensors, electronic timer which is connected with the sensors. On the disk of the pendulum puts over one of the removable rings, that allows to change its mass and momentum of inertia of the pendulum. Electromagnet holds the pendulum at the upper position if the current is running through its winding. Length of the pendulums suspension (height h) is being changed by the millimeter scale, which is marked at the vertical column.

Fig. 1 – Maxwell’s pendulum

3. Procedure

1. The path of the virtual lab, which has name “Maxwell.exe”, is D:\ \Физика1 \ММФ_5. Start the program by clicking two times on the icon of the program.

2. On the interface of the project you see the following four buttons: “Theory”, “Tests for access”, “Test after lab” and “Lab”.

3. Read the theory of the experiment, if you didn’t it yet.

4. Tests of access and defends can be used by the recommendation of the teacher.

5. By the clicking on the “Lab” you can set up the number of measurements. After that you can start the experiment. In fig.2 you can see the window of the experiment.

Fig.2- The main window of the experiment

6. The number of measurement is changing automatically. In every experiment you should change the height of the pendulum. Diameter of the tube is D0 = 1 cm, thickness of the thread is 0,1 mm. The mass of the pendulum will be given for every student individual. Run the experiment and write down time of fall in milliseconds, but don’t change the mass of the pendulum. Write down the results of the measurements in Tab.1.

Table 1- Readings of the measurements

4. Analyses of results

1. Calculate the momentum of inertia I for every value of time t using the formula (4).

2. Calculate the mean value <I> and estimate its absolute ΔI and relative uncertainty using the Student’s method with probability   λ = 0,95.

3. Calculate the theoretical value of the moment of inertia of the pendulum Itheory using the following formula:

where the diameter of the disk is  Dd= 10 cm, the diameter of the ring is Dr=7 cm, mass of the disk is md= 35 gram , mass of the tube is m0= 5 gram , mass of the ring depends on total mass of the pendulum, which you have obtain from teacher, i.e. if for example mass of pendulum is m= 6 gram, then mass of ring is mr= 25 gram.

4. Compare Itheory with the value of <I>.

5. Questions

1. From what does fall time of the Maxwell’s pendulum with fixed height h depend?

2. How does center of mass of the pendulum move? From what does its acceleration depend?

3. What is the mechanical energy of the pendulum at the upper position? At the lower position? What is the relation between these values?

4. Point out reasons of the uncertainty in experiment and deviation of experimental and theoretical values.

Using the obtained results formulate the report, give answers to the questions mentioned above and make conclusions with pointing out the errors of “virtual experiment”.

Control questions

1. Give the definition for the main axis of inertia? Central axis? Show examples.

2. Give the definition for Maxwell’s pendulum? Why do we call it pendulum?

3. What’s moment of inertia?

4. Formulate momentum of inertia for Maxwell’s pendulum.  

5. Give the definition for mechanical energy conservation law.

6. What is the analogy between main characteristics of translational and rotational motion?

7. What caused the pendulum damping Maxwell?

Appendix 1

Rotational-Linear Parallels

Appendix 2

Rotation Vectors

       Angular motion has direction associated with it and is inherently a vector process. But a point on a rotating wheel is continuously changing direction and it is inconvenient to track that direction. The only fixed, unique direction for a rotating wheel is the axis of rotation, so it is logical to choose this axis direction as the direction of the angular velocity. Left with two choices about direction, it is customary to use the right hand rule to specify the direction of angular quantities.

Directions of Angular Quantities

        As an example of the directions of angular quantities, consider a vector angular velocity as shown. If a force acts tangential to the wheel to speed it up, it follows that the change in angular velocity and therefore the angular acceleration are in the direction of the axis. Newton's 2nd law for rotation implies that the torque is also in the axis direction. The angular momentum will also be in this direction, so in this example, all of these angular quantities act along the axis of rotation as shown.

Angular Momentum Change

A force tangential to the wheel produces a torque along the axis as shown (right hand rule). The change in angular momentum is therefore along the axis and the wheel increases in angular velocity. However, if the torque direction is perpendicular to the axis of the wheel the effect is very different. The change in angular velocity is perpendicular to the angular velocity vector, changing its direction but not its magnitude. The resultant motion of the wheel around a vertical axis is called precession.


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