Theory#2

Notice that the horizontal force being applied to the crate acts through the rope. The

exerted by the floor on the crate.

We can now apply Newton’s second law in component form to the crate. The only force acting in the x direction is . Applying to the horizontal motion gives

No acceleration occurs in the y direction because the crate moves only horizontally. Therefore, we use the particle in equilibrium model in the y direction. Applying the y component of Equation 7 yields

That is, the normal force has the same magnitude as the gravitational force but acts in the opposite direction.

modeled as a particle under constant acceleration in the x direction, and the equations of kinematics from Chapter 2 can be used to obtain the crate’s position x and velocity vx as functions of time.

example, suppose a book is lying on a table and

you push down on the book with a force as in Figure 5.9. Because the book is at rest and

Problem-Solving Strategy

APPLYING NEWTON’S LAWS

The following procedure is recommended when dealing with problems involving

Newton’s laws:

1.Conceptualize. Draw a simple, neat diagram of the system. The diagram helps establish the mental representation. Establish convenient coordinate axes for each object in the system.

2.Categorize. If an acceleration component for an object is zero, the object is

3.Analyze. Isolate the object whose motion is being analyzed. Draw a free-body diagram for this object. For systems containing more than one object, draw separate freebody diagrams for each object. Do not include in the free-body diagram forces exerted by the object on its surroundings.

Find the components of the forces along the coordinate axes. Apply the appropriate model from the Categorize step for each direction. Check your dimensions to make sure that all terms have units of force.

Solve the component equations for the unknowns. Remember that you generally must have as many independent equations as you have unknowns to obtain a complete solution.

4.Finalize. Make sure your results are consistent with the free-body diagram. Also check the predictions of your solutions for extreme values of the variables. By doing so, you can often detect errors in your results.

ONE BLOCK PUSHES ANOTHER

WEIGHING A FISH IN AN ELEVATOR

ATWOOD MACHINE

Part # 2

ROTATION OF A RIGID OBJECT

Basic Rotational Quantities. The angular displancment 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.

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.

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.

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

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.

Moment of Inertia. Moment of inertia is the name given to rotational inertia, the rotational analog of mass for linear motion. It appears in the relationships for the dynamics of rotational motion. The moment of inertia must be specified with respect to a chosen axis of rotation. 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.

Common Moments of Inertia

Newton's 2nd 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.

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.

Moment of Inertia, General Form. Since the moment of inertia of an ordinary object involves a continuous distribution of mass at a continually varying distance from any rotation axis, the calculation of moments of inertia generally involves calculus, the discipline of mathematics which can handle such continuous variables. Since the moment of inertia of a point mass is defined by

then the moment of inertia contribution by an infinitesmal mass element dm has the same form. This kind of mass element is called a differential element of mass and its moment of inertia is given by

Note that the differential element of moment of inertia dI must always be defined with respect to a specific rotation axis. The sum over all these mass elements is called an integral over the mass.

Usually, the mass element dm will be expressed in terms of the geometry of the object, so that the integration can be carried out over the object as a whole (for example, over a long uniform rod).

Having called this a general form, it is probably appropriate to point out that it is a general form only for axes which may be called "principal axes", a term which includes all axes of symmetry of objects. The concept of moment of inertia for general objects about arbitrary axes is a much more complicated subject. The moment of inertia in such cases takes the form of a mathematical tensor quantity which requires nine components to completely define it.