MODULE 2. DYNAMICS AND PRINCIPLES OF NEWTON'S LAW

This module tackled the curvilinear motion of a particle which describes the position, velocity, and acceleration of a particle using vectors for the motion has direction in 2D or 3D. Thus, the principle of Newton's Second Law and its applications were also discussed, concerning forces acting on particles that led to a state of equilibrium.

position, velocity, and acceleration vectors

Curvilinear motion occurs when a particle moves along a curved path. Since the path is often described in three dimensions, vector analysis is used to formulate the particle's position, velocity, and acceleration

Consider a point P occupied by a particle on a space curve, for example. The vector r is characterized by magnitude r (position vector) at time t and r' defining position P' occupied by some particle at time t + Δt. The vector Δr is called the displacement vector.

The average velocity is defined as the quotient of Δr and Δt and instantaneous velocity is obtained by taking the limit as time interval Δt approaches zero. Thus, the limit of the quotient Δr/Δt is referred to as the derivative of the function r(t).

The magnitude v of vector v is called the speed of the particle. It can also be obtained by finding the length of arc and differentiating with respect to t.

The average acceleration is defined as the quotient of Δv and Δt, and the instantaneous acceleration is obtained by choosing increasingly smaller values for Δt and Δv. Thus, the limit of quotient Δv/Δt is referred to as the derivative of v with respect to t.

derivatives of vector functions

The acceleration a of the particle can be represented using the derivative of the vector function v(t). The following are the rules governing the differentiation of sums and products of vector functions.

Sum of two vector functions

Scalar and Vector Product of two vector functions

The properties shown above can also be used to determine the rectangular components of derivative of vector function. The rate of change of a vector is the same with respect to a fixed frame and with respect to a frame in translation

rectangular components of velocity and acceleration

When the vector P is a function of time t, the derivative dP/dt represents the rate of change of P. Since the position of particle P is defined by rectangular coordinates x,y, and z, the velocity v and acceleration a can be resolved into rectangular components.

motion relative to a frame in translation

Consider two particles A and B moving in space. The vectors rA and rB define their positions at any given moment with respect to the fixed frame of reference. The vector rB/A joining A and B defines the position of B relative to A. Differentiating the relative position with respect to t will yield relative velocity. Likewise, by differentiating relative velocity, the relative acceleration is obtained.

The motion of particle along a path can be analyzed without using rectangular components. These include the tangential and normal components, which is based on the path of particle, and radial distance and the angular displacement of the particle.

tangential and normal components of acceleration

It is sometimes convenient to resolve the acceleration into components that are normal and tangent to the path of the particle. The reference frame is referred to as the path coordinates. which includes tangential and normal coordinates.

The tangential component of the acceleration is equal to the rate of change of the speed of the vehicle, and the normal component is equal to the square of the speed divided by the radius of curvature of the path at P.

radial and transverse components of acceleration

There are some situations wherein the position of the particle P is defined by polar coordinates r and θ. Therefore, it will be convenient to use the system of radial and transverse components. The unit er defines the radial direction and the unit vector eθ defines the transverse direction. Derivation of unit vectors er and eθ with respect to time t leads to the following relations shown in the left figure.

The position of a particle can also be defined using cylindrical coordinates (R,θ, and z).Resolving the position vector r of particle P yields to the following relations shown in the right figure.

newton's second law of motion

Kinetics is a branch of dynamics that deals with the relationship between the change in motion of a body and the forces that cause this change. The basis for kinetics is Newton's Second Law, which states that when an unbalanced force acts on a particle, the particle will accelerate in the direction of the force with a magnitude that is proportional to the force.

When a particle of mass m is acted upon by a force F, the force F and the acceleration must satisfy the relation shown in the left. If the resultant ∑ F of the forces acting on the particle is zero, the acceleration of the particle is also zero. If the particle is initially at rest (vo = 0) with respect to the Newtonian frame of reference used, it will remain at rest (v=0). If originally moving with velocity v0 , the particle will maintain a constant velocity v=v0, it will move with constant speed in a straight line.

linear momentum of a particle and its rate of change

The resultant of the forces acting on the particle is also equal to the rate of change in linear momentum of the particle, denoted by the equation in the right. The conservation of linear momentum states that if the resultant force acting on a particle is zero, the linear momentum of the particle remains constant in both magnitude and direction.

systems of units

In order to satisfy the equation F = ma, two systems of consistent kinetic units are used. These are SI units and U.S Customary Units. The conversion factors obtained for the units of length, force and mass are shown in the left.

EQUATIONS OF MOTION

In order to model dynamic systems and apply appropriate equations of motion, a free body diagram shown in the left figure best illustrates the steps in drawing an FBD and kinetic diagram for solving dynamics problems.

Also, it is convenient to resolve equivalent equations involving vector quantities into its components, depending on the type of problem that needs to be solved.

Rectangular Components

Tangential and Normal Components

Radial and Transverse Components

ASSIGNMENT/s in module 2

Assignment#2 Part 1_BARTOLOME.pdf

Assignment #2_Part 2_BARTOLOME.pdf

Assignment #2_Part 3_BARTOLOME.pdf

COURSEWORK #2

Coursework#2_BARTOLOME.pdf

REFLECTION:

Unlike rectilinear motion where the inclination of displacement vector does not change, the concepts of velocity and acceleration in curvilinear motion must be extended because the displacement vector changes in both magnitude and inclination. But there are some cases wherein using rectangular components are not viable enough to analyze the motion. I have learned that a particle moving in a curvilinear path can be solved using rectangular or non-rectangular components. Problems that uses rectangular components of acceleration and velocity generally involves projectile motion. On the other hand, using non-rectangular components introduces the path coordinates (tangential and normal components), and the radial and transverse components. The equation used for radial distance and angular displacement seemed new to me because I usually solve kinematic problems in our physics class involving curvilinear path using only the rectangular components and path coordinates. I haven't encountered solving kinematics problem yet expressed in terms of polar coordinates. Solving problems involving curvilinear motion seems difficult to solve at first, but as I analyze further the motion of a particle both in 2-D and 3-D, and expressing the velocity and acceleration whether in rectangular or non-rectangular components, I was able to simplify the derivation of expressions used for each problem, and that helped me to get the answer correctly.

The second part of the module tackled kinetics, which relates the force acting on the body to its mass and acceleration (Newton's Second Law). The principles of kinematics are applied to determine the velocity or acceleration of a moving particle at any instant, and by means of relations that were developed in kinematics, I have learned that it can be used to determine the force or forces required that satisfies the motion of a particle. Also, I found it convenient to follow the steps on how to solve problems applying Newton's Second Law of Motion, particularly the steps on how to construct a free body diagram and kinetic diagram. Similar to Statics of rigid bodies, I found out that I can also assume a possible motion when solving kinetic problems, then to check it later if the assumption was correct. Those steps helped me to improve my skills in solving both kinematic and kinetic problems.

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