Work: constant force
Suppose a constant force F acts on a body while the object moves over a distance d. Both the
force F and the displacement d are vectors who are not necessarily pointing in the same direction
(see Figure 7.1). The work done by the forc
Angular Momentum of Rotating Rigid Bodies
Suppose we are dealing with a rigid body rotating around the z-axis. The linear momentum of
each mass element is parallel to the x-y plane, and perpendicular to the position vector. The
magnitude of the angular mo
Angular Momentum
The angular momentum L of particle P in Figure 12.7, with respect to the origin, is defined as
This definition implies that if the particle is moving directly away from the origin, or directly
towards it, the angular momentum associated w
Applications
Sample Problem 5-7
Figure 5.6 shows a block of mass m = 15 kg hanging from three cords. What are the tensions in
these cords ?
The mass m experiences a gravitational force equal to mg. Since the mass is at rest, cord C must
provide an opposin
Avogadro constant
The laws of classical thermodynamics do not show the direct dependence of the observed
macroscopic variables on microscopic aspects of the motion of atoms and molecules. It is
however clear that the pressure exerted by a gas is related t
Calculation of rotational inertia
To calculate the moment of inertia of a rigid body we have to integrate over the whole body
If the moment of inertia about an axis that passes through the center of mass is known, the
moment of inertia about any other axi
Center of mass
The motion of a rotating ax thrown between two jugglers looks rather complicated, and very
different from the standard projectile motion discussed in Chapter 4. Experiments have shown
that one point of the ax follows a trajectory described
Circular Motion
Suppose the motion of an object as function of time can be described by the following relations:
(8)
The equations in (8) describe a periodic motion: the position of the object at time t and at time t +
T are identical. The period of the p
Collisions in One-Dimension: Inelastic
If no external forces act on a system, its momentum is conserved. However, kinetic energy is not
always conserved. An example of an inelastic collision is a collision in which the particles stick
together (after the
Collisions in One-Dimension
Consider the collision shown in Figure 10.1. If there are no external forces acting on this system
(consisting of the two masses) the total momentum of the system is conserved. The first class of
collisions we will discuss are
Collisions in Two-Dimensions
Suppose a mass m1, with initial velocity v1i, undergoes a collision with a mass m2 (which is
initially at rest). The particles fly of at angles [theta]1 and [theta]2, as shown in Figure 10.4. Since
no external forces act on th
Conduction
Consider the slab of material shown in Figure 17.3. The left end of the beam is
maintained at a temperature TH; the right end of the beam is maintained at a temperature T C. As
a result of the temperature difference heat will flow through the s
Conservation of Angular Momentum
If no external forces act on a system of particles or if the external torque is equal to zero, the total
angular momentum of the system is conserved. The angular momentum remains constant, no
matter what changes take place
Conservation of energy
In the presence of non-conservative forces, mechanical energy is converted into internal energy
Uint (or thermal energy):
[Delta]Uint = - Wf
With this definition of the internal energy, the work-energy theorem can be rewritten as
wh
Constant angular acceleration
If the angular acceleration a is constant (time independent) the following equations can be used
to calculate [omega] and [theta] at any time t:
Note that these equations are very similar to the equations for linear motion.
P
Acceleration
The velocity of an object is defined in terms of the change of position of that object over time. A
quantity used to describe the change of the velocity of an object over time is the acceleration a.
The average acceleration over a time interv
18.6.1. Molar Heat Capacity at Constant Volume
Suppose we heat up n moles of gas while keeping its volume constant. The result of
adding heat to the system is an increase of its temperature
Here, CV is the molar heat capacity at constant volume , Q is the
Work
Suppose a system starts from an initial state described by a pressure p i, a volume Vi, and
a temperature Ti. The final state of the system is described by a pressure p f, a volume Vf, and a
temperature Tf. The transformation from the initial state t
Work: variable force
In the previous discussion we have assumed that the force acting on the object is constant (not
dependent on position and/or time). However, in many cases this is not a correct assumption. By
reducing the size of the displacement (for
2.1. Position
The position of an object along a straight line can be uniquely identified by its distance from a
(user chosen) origin. (see Figure 2.1). Note: the position is fully specified by 1 coordinate (that is
why this a 1 dimensional problem).
Figur
2.2. Velocity
An object that changes its position has a non-zero velocity. The average velocity of an object
during a specified time interval is defined as:
If the object moves to the right, the average velocity is positive. An object moving to the left h
2.4. Constant Acceleration
Objects falling under the influence of gravity are one example of objects moving with constant
acceleration. A constant acceleration means that the acceleration does not depend on time:
Integrating this equation, the velocity of
3.4. Multiplying Vectors - Vector Product
The vector product of two vectors and , written as x , is a third vector with the following
properties:
the magnitude of is given by:
where [phi] is the smallest angle between and . Note: the angle between and is
4.2. Velocity
The velocity of an object in two or three dimensions is defined analogously to its definition in
Chapter 2:
This equation shows that the velocity of an object in two or three dimensions is also a vector.
Again, the velocity vector can be dec
4.3. Acceleration
The acceleration of an object in three dimensions is defined analogously to its definition in
Chapter 2:
This equation shows that the acceleration of an object in two or three dimensions is also a vector,
which can be decomposed into thr
Projectile Motion
We will start considering the motion of a projectile in 2 dimensions. The coordinate system that
will be used to describe the motion of the projectile consist of an x-axis (horizontal direction)
and a y-axis (vertical direction). Assumin
4.1. Position
In Chapter 2 we discussed the motion of an object in one dimension. Its position was
unambiguously defined by its distance (positive or negative) from a user defined origin. The
motion of this object could be described in terms of scalars. T
8.1. Conservation laws
In this chapter we will discuss conservation of energy. The conservation laws in physics can be
expressed in very simple form:
" Consider a system of particles, completely isolated from outside influence. As the particles
move about
10.1. Introduction
In a collision, strong mutual forces act between a few particles for a short time. These internal
forces are significantly larger than any external forces during the time of the collision. The laws
of conservation of linear momentum and
15.1. Simple Harmonic Motion
Any motion that repeats itself at regular intervals is called harmonic motion. A particle
experiences a simple harmonics motion if its displacement from the origin as function of time
is given by
where xm, [omega] and [phi] ar