Labs / Lab2
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LABORATORY WORK #2
STUDY OF MOMENTUM OF INERTIA
WITH MAXWELL’S PENDULUM
PhysicVirtualLab Software Package
sec kg·m2 kg·m2
3.1.7Calculate the momentum of inertia I for every value of time t using the formula (4).
3.1.8Calculate the mean value <I> and estimate its absolute ΔI and relative
uncertainty using the Student’s method with probability λ = 0,95.
3.1.9Calculate the theoretical value of the moment of inertia of the pendulum Itheory using the following formula:
[ |
( |
) ( |
)] , |
(5) |
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.
3.2.10 Compare Itheory with the value of <I>.
4QUESTIONS:
4.1Give the definition for Maxwell’s pendulum? Why do we call it pendulum?
4.2Formulate momentum of inertia for Maxwell’s pendulum.
4.3What will be the moment of inertia respect to the fixed axis?
4.4What will be the mechanical energy of the body in translational and rotational motion? In what case mechanical energy conserves?
4.5Why formula of momentum of inertia for Maxwell’s pendulum J includes diameter of the pendulum’s rod, but not the diameter of the disk? Does J depend on diameter of the disk?
4.6How pendulum’s period of oscillation will change, if mass of pendulum increases and its geometrical size remains the same?
4.7From what does fall time of the Maxwell’s pendulum with fixed height h depend?
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Copyright 2014
LABORATORY WORK #2
STUDY OF MOMENTUM OF INERTIA
WITH MAXWELL’S PENDULUM
PhysicVirtualLab Software Package
4.8How does center of mass of the pendulum move? From what does its acceleration depend?
4.9What is the mechanical energy of the pendulum at the upper position? At the lower position? What is the relation between these values?
4.10Point out reasons of the uncertainty in experiment and deviation of experimental and theoretical values.
4.11What is the reason for damping of Maxwell’s pendulum oscillations?
4.12Give the definition for mechanical energy conservation law.
4.13What mechanical transformations take place during the process of
Maxwell’s pendulum oscillations?
4.14What is the analogy between main characteristics of translational and rotational motion?
4.15Evaluate error in calculation of Maxwell’s pendulum momentum of inertia.
5REFERENCES:
5.1Raymond A. Serway,John W. Jewett Physics for Scientists and Engineers, 2006.
5.2Irodov I.E. Fundamental laws of mechanics, 2001.
5.3http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html.
PhysicVirtualLabs Project Team, IITU
Copyright 2014
LABORATORY WORK #2
STUDY OF MOMENTUM OF INERTIA
WITH MAXWELL’S PENDULUM
PhysicVirtualLab Software Package
Appendix
Rotational-Linear Parallels
PhysicVirtualLabs Project Team, IITU
Copyright 2014
LABORATORY WORK #2
STUDY OF MOMENTUM OF INERTIA
WITH MAXWELL’S PENDULUM
PhysicVirtualLab Software Package
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
PhysicVirtualLabs Project Team, IITU
Copyright 2014
LABORATORY WORK #2
STUDY OF MOMENTUM OF INERTIA
WITH MAXWELL’S PENDULUM
PhysicVirtualLab Software Package
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.
Examples of integration to get moment of inertia
PhysicVirtualLabs Project Team, IITU
Copyright 2014
LABORATORY WORK #2
STUDY OF MOMENTUM OF INERTIA
WITH MAXWELL’S PENDULUM
PhysicVirtualLab Software Package
Rod Moment Calculation
The moment of inertia calculation for a uniform rod involves expressing any
mass element in terms of a distance element dr along the rod. To perform the integral,
it is necessary to express eveything in the integral in terms of one variable, in this
case the length variable r. Since the total length L has mass M, then M/L is the
proportion of mass to length and the mass element can be expressed as shown.
Integrating from -L/2 to +L/2 from the center includes the entire rod. The integral is
of polynomial type:
Moment of Inertia: Cylinder
The expression for the moment of inertia of a solid cylinder can be built up from the moment of inertia of thin cylindrical shells.
Using the general definition for moment of inertia:
The mass element can be expressed in terms of an infinitesmal radial thickness dr by
PhysicVirtualLabs Project Team, IITU
Copyright 2014
LABORATORY WORK #2
STUDY OF MOMENTUM OF INERTIA
WITH MAXWELL’S PENDULUM
PhysicVirtualLab Software Package
Substituting gives a polynomial form integral:
PhysicVirtualLabs Project Team, IITU
Copyright 2014
LABORATORY WORK #2
STUDY OF MOMENTUM OF INERTIA
WITH MAXWELL’S PENDULUM
PhysicVirtualLab Software Package
PhysicVirtualLabs Project Team, IITU
Copyright 2014