Contemporary physics

of a Rigid Object about a Fixed Axis.

Newton’s third law, we must have F12 = - F21.

can be modeled as a particle of mass m moving

with a velocity v is defined to be the product of the mass and velocity:

(4.1)

(4.2)

Linear momentum is a vector quantity because it equals the product of a scalar quantity m and a vector quantity v. Its direction is along v, it has dimensions ML/T, and its SI unit is kg · m/s.

momentum provides a quantitative distinction between heavy and light particles

moving at the same velocity. For example, the momentum of a bowling ball moving at 10 m/s is much greater than that of a tennis ball moving at the same speed. Newton called the product mv quantity of motion; this is perhaps a more graphic description than our present-day word momentum, which comes from the Latin word for movement.

If a particle is moving in an arbitrary direction, p must have three components

linear momentum of a particle to the resultant force acting

on the particle. We start with Newton’s second law and substitute the definition of acceleration:

(4.3)

As m=const:

The time rate of change of the linear momentum of a particle is equal to the net force acting on the particle.

p1i and p2i are the initial values and p1f and

p2f the final values of the momenta for the two particles for the time interval during which the particles interact.

(4.5)

(4.4)

momentum, can be extended to any number of particles in

an isolated system. It is considered one of the most important laws of mechanics. We can state it as follows:

Whenever two or more particles in an isolated system interact, the total momentum of the system remains constant.

This law tells us that the total momentum of an isolated system at all times equals its initial momentum.
Notice that we have made no statement concerning the nature of the forces acting on the particles of the system. The only requirement is that the forces must be internal to the system.

(4.6)

acts on the particle.
According to Newton’s second law
(4.7)
(4.8)

force varies with time. The quantity on the right side

of this equation is called the impulse of the force F acting on a particle over the time interval ∆t=tf - ti. Impulse is a vector defined by

Equation 4.8 is an important statement known as the impulse–momentum theorem:

The impulse of the force F acting on a particle equals the change in the momentum of the particle.

the direction of the change in momentum. Impulse has the

dimensions of momentum—that is, ML/T. Note that impulse is not a property of a particle; rather, it is a measure of the degree to which an external force changes the momentum of the particle. Therefore, when we say that an impulse is given to a particle, we mean that momentum is transferred from an external agent to that particle.

Figure 4.2 (a) A force acting on a particle may vary in time. The impulse imparted to the particle by the force is the area under the force-versus-time curve. (b) In the time interval (t, the time-averaged force (horizontal dashed line) gives the same impulse to a particle as does the time-varying force described in part (a).

(a)

(b)

time, it is convenient to define a time-averaged force
where ∆t=tf

- ti.

(4.10)

(4.12)

The calculation becomes especially simple if the force acting on the particle is constant. In this case, and Equation 4.11 becomes

In many physical situations, we shall use what is called the impulse approximation, in which we assume that one of the forces exerted on a particle acts for a short time but is much greater than any other force present.

(4.11)

which two particles come close to each other and interact

by means of forces. The time interval during which the velocities of the particles change from initial to final values is assumed to be short. The interaction forces are assumed to be much greater than any external forces present, so we can use the impulse approximation.

Figure 4.3 (a) The collision between two objects as the result of direct contact. (b) The “collision” between two charged particles.

collision equals the total momentum of the system just after

the collision.
The total kinetic energy of the system of particles may or may not be conserved, depending on the type of collision. In fact, whether or not kinetic energy is conserved is used to classify collisions as either elastic or inelastic.
An elastic collision between two objects is one in which the total kinetic energy (as well as total momentum) of the system is the same before and after the collision. Collisions between certain objects in the macroscopic world, such as billiard balls, are only approximately elastic because some deformation and loss of kinetic energy take place. For example, you can hear a billiard ball collision, so you know that some of the energy is being transferred away from the system by sound. An elastic collision must be perfectly silent! Truly elastic collisions occur between atomic and subatomic particles.

energy of the system is not the same before and

after the collision (even though the momentum of the system is conserved).
Inelastic collisions are of two types.
When the colliding objects stick together after the collision, as happens when a meteorite collides with the Earth, the collision is called perfectly inelastic.
When the colliding objects do not stick together, but some kinetic energy is lost, as in the case of a rubber ball colliding with a hard surface, the collision is called inelastic (with no modifying adverb).
In most collisions, the kinetic energy of the system is not conserved because some of the energy is converted to internal energy and some of it is transferred away by means of sound. Elastic and perfectly inelastic collisions are limiting cases; most collisions fall somewhere between them.
The important distinction between these two types of collisions is that momentum of the system is conserved in all collisions, but kinetic energy of the system is conserved only in elastic collisions.

between two particles: (a) before collision and (b) after collision.
(a)
(b)
(4.13)
(4.14)

two particles: (a) before collision and (b) after collision.
(4.15)
(4.16)

in Equation 4.15 to obtain
(4.17)
(4.18)
(4.19)
To obtain our final result, we

divide Equation 4.17 by Equation 4.18 and obtain

are known.
(4.20)
(4.21)
Let us consider some special cases. If m1 =

m2, then Equations 4.20 and 4.21 show us that v1f = v2i and v2f = v1i .

That is, the particles exchange velocities if they have equal masses. This is approximately what one observes in head-on billiard ball collisions - the cue ball stops, and the struck ball moves away from the collision with the same velocity that the cue ball had.

0, and Equations 4.20 and 4.21 become
(4.22)
(4.23)
If m1 is much

greater than m2 and v2i = 0, we see from Equations 4.22 and 4.23 that v1f ≈ v1i and v2f ≈ 2v1i .

If m2 is much greater than m1 and particle 2 is initially at rest, then v1f ≈ -v1i and v2f ≈ 0.

when the system is isolated. For any collision of two

particles, this result implies that the momentum in each of the directions x, y, and z is conserved.

For such two-dimensional collisions, we obtain two component equations for conservation of momentum:

4.16 (conservation of kinetic energy) with v2i = 0 to

give

If the collision is inelastic, kinetic energy is not conserved and Equation 4.26 does not apply.

a light, rigid rod. (a) The system rotates clockwise when

a force is applied between the less massive particle and the center of mass. (b) The system rotates counterclockwise when a force is applied between the more massive particle and the center of mass. (c) The system moves in the direction of the force without rotating when a force is applied at the center of mass.

unequal mass on the x axis is located at xCM,

a point between the particles, closer to the one having the larger mass.

(4.27)

(4.28)

(4.29)

a distribution of small elements of mass ∆mi. The center

of mass is located at the vector position rCM, which has coordinates xCM, yCM, and zCM.

The center of mass of any symmetric object lies on an axis of symmetry and on any plane of symmetry.

a continuous mass distribution

is, no articles enter or leave the system, we obtain

the following expression for the velocity of the center of mass of the system:

where vi is the velocity of the ith particle. Rearranging Equation 4.34 gives

(4.34)

(4.35)

Therefore, we conclude that the total linear momentum of the system equals the total mass multiplied by the velocity of the center of mass. In other words, the total linear momentum of the system is equal to that of a single particle of mass M moving with a velocity vCM.

we obtain the acceleration of the center of mass of

the system:

Rearranging this expression and using Newton’s second law, we obtain

where Fi is the net force on particle i.

(4.36)

(4.37)

particles equals the total mass of the system multiplied by

the acceleration of the center of mass. If we compare this with Newton’s second law for a single particle, we see that the particle model that we have used for several chapters can be described in terms of the center of mass:

The center of mass of a system of particles of combined mass M moves like an equivalent particle of mass M would move under the influence of the net external force on the system.

(4.38)

θ between this reference line on the object and the

fixed reference line in space, which is often chosen as the x axis.

Figure 5.1 A compact disc rotating about a fixed axis through O perpendicular to the plane of the figure.

Rotation of a Rigid Object About a Fixed Axis

(5.1)

A particle on a rotating rigid object moves from A

to B along the arc of a circle. In the time interval ∆t = tf - ti , the radius vector moves through an angular displacement ∆θ = θf - θi.

The average angular speed

The instantaneous angular speed

(5.2)

(5.3)

is rotating about a fixed axis, every particle on the

object rotates through the same angle in a given time interval and has the same angular speed and the same angular acceleration.

(5.4)

(5.5)

rule for determining the direction of the angular velocity vector.

speed of the rigid object at time t = 0.
Equation

5.6 allows us to find the angular speed ωf of the object at any later time t.

t = 0.
Equation 5.7 allows us to find the angular

position θf of the object at any later time t.

obtain
This equation allows us to find the angular speed ωf

of the rigid object for any value of its angular position θf .

If we eliminate α between Equations 5.6 and 5.7, we obtain

(5.8)

(5.9)

about the fixed axis through O, the point P has

a tangential velocity v that is always tangent to the circular path of radius r.

(5.10)

That is, the tangential speed of a point on a rotating rigid object equals the perpendicular distance of that point from the axis of rotation multiplied by the angular speed.

object to the tangential acceleration of the point P by

taking the time derivative of v:

That is, the tangential component of the linear acceleration of a point on a rotating rigid object equals the point’s distance from the axis of rotation multiplied by the angular acceleration.

(5.11)

axis through O, the point P experiences a tangential component

of linear acceleration at and a radial component of linear acceleration ar. The total linear acceleration of this point is a = at + ar.

A point moving in a circular path undergoes a radial acceleration ar of magnitude v2/r directed toward the center of rotation .
Because v = rω for a point P on a rotating object, we can express the centripetal acceleration at that point in terms of angular speed as

(5.12)

(5.13)

z axis with angular speed ω.
(5.14)

as the moment of inertia I:
From the definition of moment

of inertia, we see that it has dimensions of ML2 (kg ·m2 in SI units). With this notation, Equation 5.14 becomes

(5.15)

Where KR is Rotational kinetic energy.

(5.16)

about O as F increases and as the moment arm

d increases. The component Fsinφ tends to rotate the wrench about O.

When a force is exerted on a rigid object pivoted about an axis, the object tends to rotate about that axis. The tendency of a force to rotate an object about some axis is measured by a vector quantity called torque τ (Greek tau).

(5.18)

counterclockwise about O, and F2 tends to rotate it clockwise.
(5.19)

in a circle under the influence of a tangential force

Ft. A force Fr in the radial direction also must be present to maintain the circular motion.

angular acceleration, and the proportionality constant is the moment of

inertia.

(5.20)

a different linear acceleration at , they all have the

same angular acceleration α.

So, again we see that the net torque about the rotation axis is proportional to the angular acceleration of the object, with the proportionality factor being I, a quantity that depends upon the axis of rotation and upon the size and shape of the object.

(5.21)

object rotates about an axis through O under the action

of an external force F applied at P.

The work done by F on the object as it rotates through an infinitesimal distance

where Fsinφ is the tangential component of F, or, in other words, the component of the force along the displacement.

(5.22)

as the object rotates about the fixed axis through the

angle dθ in a time interval dt is

(5.23)

that
the net work done by external forces in rotating a

symmetric rigid object about a fixed axis equals the change in the object’s rotational energy.

motion, as the cylinder rotates through an angle θ, its

center moves a linear distance s = Rθ.

(5.25)

(5.26)

a direction perpendicular to an axis through the instantaneous point

of contact P. In other words, all points rotate about P. The center of mass of the object moves with a velocity vCM, and the point P’ moves with a velocity 2 vCM .

modeled as a combination of pure translation and pure rotation.

elastic head-on collision between two particles: (a) before collision and

(b) after collision.

across a horizontal surface with an initial speed v. It

slides until it stops due to the friction force between the block and the surface. The same block is now projected across the horizontal surface with an initial speed 2v. When the block has come to rest, how does the distance from the projection point compare to that in the first case? (a) It is the same. (b) It is twice as large. (c) It is four times as large. (d) The relationship cannot be determined.

Quick Quiz 2 A car and a large truck traveling at the same speed make a head-on collision and stick together. Which vehicle experiences the larger change in the magnitude of momentum? (a) the car (b) the truck (c) The change in the magnitude of momentum is the same for both. (d) impossible to determine.