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LABORATORY WORK #2

STUDY OF MOMENTUM OF INERTIA

WITH MAXWELL’S PENDULUM

PhysicVirtualLab Software Package

LABORATORY WORK #2

STUDY OF MOMENTUM OF INERTIA WITH MAXWELL’S PENDULUM

1. AIM OF THE WORK

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

TASKS:

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

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

2. INTRODUCTION

2.1 Basic Rotational Quantities

The angular displacement is defined by:

For a circular path it follows that the angular velocity is

and the angular acceleration is

where the acceleration here is the tangential acceleration.

PhysicVirtualLabs Project Team, IITU

Copyright 2014

LABORATORY WORK #2

STUDY OF MOMENTUM OF INERTIA

WITH MAXWELL’S PENDULUM

PhysicVirtualLab Software Package

In addition to any tangential acceleration, there is always the centripetal acceleration:

The standard angle of a directed quantity is taken to be counterclockwise from the positive x axis.

Figure 1 -Basic Rotational Quantities

PhysicVirtualLabs Project Team, IITU

Copyright 2014

LABORATORY WORK #2

STUDY OF MOMENTUM OF INERTIA

WITH MAXWELL’S PENDULUM

PhysicVirtualLab Software Package

2.2 Angular Velocity

Angular velocity can be considered to be a vector quantity, with direction along the axis of rotation in the right-hand rule sense (Appendix 2). For an object rotating about an axis, every point on the object has the same angular velocity. The tangential velocity of any point is proportional to its distance from the axis of rotation. Angular velocity has the units rad/s.

or

Angular velocity is the rate of change of angular displacement and can be described by the relationship

and if v is constant, the angle can be calculated from

PhysicVirtualLabs Project Team, IITU

Copyright 2014

LABORATORY WORK #2

STUDY OF MOMENTUM OF INERTIA

WITH MAXWELL’S PENDULUM

PhysicVirtualLab Software Package

Figure 2 -Angular velocity can be considered to be a vector quantity,

with direction along the axis of rotation in the right-hand rule sense

2.3 Torque

A torque is an influence which tends to change the rotational motion of an object.

One way to quantify a torque is

Torque = Force applied x lever arm

The lever arm is defined as the perpendicular distance from the axis of rotation to the line of action of the force.

a)

b)

PhysicVirtualLabs Project Team, IITU

Copyright 2014

LABORATORY WORK #2

STUDY OF MOMENTUM OF INERTIA

WITH MAXWELL’S PENDULUM

PhysicVirtualLab Software Package

c)

Figure 3- Three examples of torque exerted on a wrench of length 20 cm

2.4 Newton's second Law: Rotation. Rotational and Linear Example

The relationship between the net external torque and the angular acceleration is of the same form as Newton's second law and is sometimes called Newton's second law for rotation. It is not as general a relationship as the linear one because the moment of inertia is not strictly a scalar quantity. The rotational equation is limited to rotation about a single principal axis, which in simple cases is an axis of symmetry.

A mass m is placed on a rod of length r and negligible mass, and constrained to rotate about a fixed axis. If the mass is released from a horizontal orientation, it can be described either in terms of force and acceleration with Newton's second law for linear motion, or as a pure rotation about the axis with Newton's second law for rotation. This provides a setting for comparing linear and rotational quantities for the same system. This process leads to the expression for the moment of inertia of a point mass.

PhysicVirtualLabs Project Team, IITU

Copyright 2014

LABORATORY WORK #2

STUDY OF MOMENTUM OF INERTIA

WITH MAXWELL’S PENDULUM

PhysicVirtualLab Software Package

In the setting, where gravity causes the mass to rotate downward, the descriptions must be equivalent. So we can express the angular quantities in terms of the linear quantities. This reinforces the basic definition of the moment of inertia of a point mass:

Substituting in the rotation equation gives:

,

(For a point mass)

2.5 Moment of Inertia Examples

Moment of inertia is defined with respect to a specific rotation axis. The moment of inertia of a point mass with respect to an axis is defined as the product of the mass times the distance from the axis squared. The moment of inertia of any extended object is built up from that basic definition. The general form of the moment of inertia involves an integral.

For a point mass the moment of inertia is just the mass times the radius from the axis squared. For a collection of point masses (figure 4a) the moment of inertia is just the sum for the masses.

For an object with an axis of symmetry (figure 4b), the moment of inertia is some fraction of that which it would have if all the mass were at the radius r.

Sum of the point mass moments of inertia (figure 4c)

 

 

 

PhysicVirtualLabs Project Team, IITU

Copyright 2014

LABORATORY WORK #2

STUDY OF MOMENTUM OF INERTIA

WITH MAXWELL’S PENDULUM

PhysicVirtualLab Software Package

Continuous mass distributions require an infinite sum of all the point mass moments which make up the whole (figure 4d). This is accomplished by integration over all the mass.

Figure 4 - Moment of Inertia Examples

As noted above for a point mass the moment of inertia is just the mass times the square of perpendicular distance to the rotation axis, I = mr2. That point mass relationship becomes the basis for all other moments of inertia since any object can be built up from a collection of point masses (figure 5).

PhysicVirtualLabs Project Team, IITU

Copyright 2014

LABORATORY WORK #2

STUDY OF MOMENTUM OF INERTIA

WITH MAXWELL’S PENDULUM

PhysicVirtualLab Software Package

Figure 5 - Common Moments of Inertia

3.EXPERIMENTS DESCRIPTION

Maxwell’s pendulum represents a disk, whose axis is suspended on two turning on it threads (figure 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 center of mass (center of gravity, center of inertia) and at the same time as the rotational motion around horizontal axes, passing through this center.

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

PhysicVirtualLabs Project Team, IITU

Copyright 2014

LABORATORY WORK #2

STUDY OF MOMENTUM OF INERTIA

WITH MAXWELL’S PENDULUM

PhysicVirtualLab Software Package

 

 

(1)

 

 

(2)

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):

(3)

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:

[

], (4)

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.

3.1 Description of experimental device.

The general view of Maxwell's pendulum is shown on figure 6. 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

PhysicVirtualLabs Project Team, IITU

Copyright 2014

LABORATORY WORK #2

STUDY OF MOMENTUM OF INERTIA

WITH MAXWELL’S PENDULUM

PhysicVirtualLab Software Package

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.

Figure 6 – Maxwell’s pendulum

3.1Procedure

3.1.1The path of the virtual lab, which has name “Maxwell.exe”.

3.1.2On the interface of the project you see the following four buttons:

“Theory”, “Tests for access”, “Test after lab” and “Lab”.

3.1.3Read the theory of the experiment, if you didn’t it yet.

3.1.4Tests of access and defends can be used by the recommendation of the teacher.

3.1.5By the clicking on the “Lab” you can set up the number of measurements.

After that you can start the experiment.

3.1.6The 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 Tabale1.

Table 1- Readings of the measurements

h,m

m,kg

t,

I,

<I>,

D0,m

d,m

Itheory, kg·m2

PhysicVirtualLabs Project Team, IITU

Copyright 2014

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