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Kinetics of Particles: Newton's Second Law of Motion, Study notes of Mechanics

University of the Philippines Open UniversityMechanics

An overview of newton's second law of motion and its application in particle kinetics. It explains the relationships between mass, force, and acceleration, and how to apply newton's second law to solve particle kinetics problems using different coordinate systems. The document also covers the analysis of central force motion problems using principles of angular momentum and newton's law of gravitation. The content includes objectives, explanations of key concepts, sample problems, and discussions on systems of units and equations of motion. This comprehensive coverage of particle kinetics and related topics could be useful for university students studying engineering, physics, or related disciplines.

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2022/2023

Uploaded on 04/16/2023

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ENGR. JOSHUA C. JUNIO
1
Module No. 3
Chapter3
KINETICS OF PARTICLES: NEWTON’S SECOND LAW OF MOTION
The forces experienced by the passengers on a roller coaster will depend
on whether the roller-coaster car is traveling up a hill or down a hill, in a
straight line, or along a horizontal or vertical curved path. The relation
existing among force, mass, and acceleration will be studied in this chapter.
Objectives
Explain the relationships between mass, force, and acceleration.
Apply Newton's second law of motion to solve particle kinetics problems
using different coordinate systems.
Analyze central force motion problems using principles of angular
momentum and Newton's law of gravitation.
3.1 Newton’s Second Law and Linear Momentum
In statics, we used Newton’s first and third laws of motion extensively to study
bodies at rest and the forces acting upon them. We also use these two laws in dynamics;
in fact, they are sufficient for analyzing the motion of bodies that have no
acceleration. However, when a body is accelerated–– that is, when the magnitude or the
direction of its velocity changes––it is necessary to use Newton’s second law of motion
to relate the motion of the body to the forces acting on it.
First law: If no net force acts on a particle, then it is possible to select a
set of reference frames, called inertial reference frames, observed from which the
particle moves without any change in velocity.
Second law: Observed from an inertial
reference frame, the net force on a particle is
proportional to the time rate of change of its linear
momentum: F = d[mv] / dt. Momentum is the product of
mass and velocity. This law is often stated as F =
ma (the net force on an object is equal to the mass
of the object multiplied by its acceleration)
Third law: Whenever a particle A exerts a force
on another particle B, B simultaneously exerts a
force on A with the same magnitude in the opposite
direction. The strong form of the law further
postulates that these two forces act along the same
line.
3.1.1 Newton’s Second Law of Motion
We can state Newton’s second law as follows:
If the resultant force acting on a particle
is not zero, the particle has an acceleration
proportional to the magnitude of the resultant
and in the direction of this resultant force.
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Module No. 3

Chapter 3

KINETICS OF PARTICLES: NEWTON’S SECOND LAW OF MOTION

The forces experienced by the passengers on a roller coaster will depend

on whether the roller-coaster car is traveling up a hill or down a hill, in a

straight line, or along a horizontal or vertical curved path. The relation

existing among force, mass, and acceleration will be studied in this chapter.

Objectives

Explain the relationships between mass, force, and acceleration.

Apply Newton's second law of motion to solve particle kinetics problems

using different coordinate systems.

Analyze central force motion problems using principles of angular

momentum and Newton's law of gravitation.

3.1 Newton’s Second Law and Linear Momentum In statics, we used Newton’s first and third laws of motion extensively to study bodies at rest and the forces acting upon them. We also use these two laws in dynamics; in fact, they are sufficient for analyzing the motion of bodies that have no acceleration. However, when a body is accelerated–– that is, when the magnitude or the direction of its velocity changes––it is necessary to use Newton’s second law of motion to relate the motion of the body to the forces acting on it. First law : If no net force acts on a particle, then it is possible to select a set of reference frames, called inertial reference frames, observed from which the particle moves without any change in velocity. Second law : Observed from an inertial reference frame, the net force on a particle is proportional to the time rate of change of its linear momentum: F = d [ mv ] / dt. Momentum is the product of mass and velocity. This law is often stated as F = ma (the net force on an object is equal to the mass of the object multiplied by its acceleration) Third law : Whenever a particle A exerts a force on another particle B , B simultaneously exerts a force on A with the same magnitude in the opposite direction. The strong form of the law further postulates that these two forces act along the same line. 3.1.1 Newton’s Second Law of Motion We can state Newton’s second law as follows: If the resultant force acting on a particle is not zero, the particle has an acceleration proportional to the magnitude of the resultant and in the direction of this resultant force.

The constant value obtained for the ratio of the magnitudes of the forces and accelerations is a characteristic of the particle under consideration; it is called the mass of the particle and is denoted by m. When a particle of mass m is acted upon by a force F , the force F and the acceleration a of the particle must therefore satisfy the relation When a particle is subjected simultaneously to several forces, the equation should be replaced by where ∑ F represents the sum or resultant of all the forces acting on the particle. 3.1.2 Linear Momentum of a Particle and its Rate of Change Suppose we replace the acceleration a in the equation by the derivative d v / dt. We have Since the mass m of the particle is constant, we can write this as The product m v is called the linear momentum , or simply the momentum , of the particle. It has the same direction as the velocity of the particle, and its magnitude is equal to the product of the mass m and the speed v of the particle (Fig. 12.3). The equation says, The resultant of the forces acting on the particle is equal to the rate of change of the linear momentum of the particle. The second law of motion was originally stated by Newton in this form. Denoting the linear momentum of the particle by L , we have

3.1.4 Equations of Motion Steps: Body: Define your system by isolating the body (or bodies) of interest. If a problem has multiple bodies, you may have to draw multiple freebody diagrams and kinetic diagrams. Axes: Draw an appropriate coordinate system (e.g., Cartesian, normal and tangential, or radial and transverse). Support Forces: Replace supports or constraints with appropriate forces (e.g., two perpendicular forces for a pin, normal forces, friction forces). Applied Forces and Body Forces: Draw any applied forces and body forces (also sometimes called field forces) on your diagram (e.g., weight, magnetic forces, a known pulling force). Dimensions : Add any angles or distances that are important for solving the problem. In statics problems, we deal with bodies in equilibrium, and the inertial term in Newton’s second law is zero. For dynamics problems, this is not the case. We utilize the kinetic diagram to visualize this term. Body: This is the same body as in the free-body diagram; place this beside the free body diagram. Inertial terms: Draw the m a term to be consistent with the coordinate system. Generally, draw this term in different components (e.g., max and ma y or man and mat ). If they are unknown quantities, it is best to draw them in the positive directions as defined by your coordinates.

Sample Problems:

  1. A 200 - lb block rests on a horizontal plane. Find the magnitude of the force P required to give the block an acceleration of 10 ft/s^2 to the right. The coefficient of kinetic friction between the block and the plane is Ό k = 0.25.

SOLUTION WILL BE PRESENTED DURING CLASS DISCUSSION

  1. A 0.5-kg fragile glass vase is dropped onto a thick pad that has a force deflection relationship as shown. Knowing that the vase has a speed of 3 m/s when it first contacts the pad, determine the maximum downward displacement of the vase.

SOLUTION WILL BE PRESENTED DURING CLASS DISCUSSION

  1. The two blocks shown start from rest. The horizontal plane and the pulley are frictionless, and the pulley is assumed to be of negligible mass. Determine the acceleration of each block and the tension in each cord.

SOLUTION WILL BE PRESENTED DURING CLASS DISCUSSION

  1. The 12 - lb block B starts from rest and slides on the 30 - lb wedge A , which is supported by a horizontal surface. Neglecting friction, determine ( a ) the acceleration of the wedge, ( b ) the acceleration of the block relative to the wedge.

SOLUTION WILL BE PRESENTED DURING CLASS DISCUSSION

3.2 Newton’s law of Gravitation 3.2.1 Angular Momentum of a Particle and Its Rate of Change Consider a particle P with a mass m moving with respect to a Newtonian frame of reference Oxyz. The linear momentum of the particle at a given instant is defined as the vector m v that is obtained by multiplying the velocity v of the particle by its mass m. The moment about O of the vector m v is called the moment of momentum, or the angular momentum , of the particle about O at that instant and is denoted by H O. Note that H O is a vector perpendicular to the plane containing r and m v and has a magnitude

Newton’s law of universal gravitation states that two particles of masses M and m at a distance r from each other have a mutual attraction of equal and opposite forces F and 2 F directed along the line joining the particles (Fig. 12.17). The common magnitude F of the two forces is where G is a universal constant, called the constant of gravitation. Experiments show that the value of G is (66.73 ±0.03) x10-^12 m^3 /kg.s^2 in SI units or approximately 34. x 10 -^9 ft^4 /lb.s4 in U.S. customary units. Gravitational forces exist between any pair of bodies, but their effect is appreciable only when one of the bodies has a very large mass. The effect of gravitational forces is apparent in the cases of the motion of a planet about the sun, of satellites orbiting about the earth, or of bodies falling on the surface of the earth. Since the force exerted by the earth on a body of mass m located on or near its surface is defined as the weight W of the body, we can substitute the magnitude W = mg of the weight for F , and the earth’s radius R for r. We obtain where M is the mass of the earth. Sample Problems

  1. A satellite is launched in a direction parallel to the surface of the earth with a velocity of 18, mi/h from an altitude of 240 mi. Determine the velocity of the satellite as it reaches its maximum altitude of 2340 mi. Recall that the earth’s radius is 3960 mi.

SOLUTION WILL BE PRESENTED DURING CLASS

DISCUSSION

  1. A space tug travels a circular orbit with a 6000-mi radius around the earth. In order to transfer it to a larger orbit with a 24,000-mi radius, the tug is first placed on an elliptical path AB by firing its engines as it passes through A , thus increasing its velocity by 3810 mi/h. Determine how much the tug’s velocity should be increased as it reaches B to insert it into the larger circular orbit.

SOLUTION WILL BE PRESENTED DURING CLASS

DISCUSSION