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Quantities and types of movement Vectors For non-vector quantities, adding or subtracting them is a simple algebraic process, but in the case of vector quantities, to add two or more vectors, the direction must be taken into account and there are two ways to add vector...

Quantities and types of movement Vectors For non-vector quantities, adding or subtracting them is a simple algebraic process, but in the case of vector quantities, to add two or more vectors, the direction must be taken into account and there are two ways to add vectors: - 1. Drawing method: If two forces F1 and F2 affect a body at a point, to graph the resultant of these forces, the line OA represents the force F1 as a magnitude and direction and the line OB represents the force F2 as magnitude and direction. By completing the parallelogram OAOB, the OC represents the magnitude and direction of the resultant 1 (R) for the two forces F1 and F2 (See Figure 1). On the other hand, the force F2 can be represented by magnitude and direction by the AC line, and the resultant R is also represented by the OC line, that is, if any two forces can be represented by a magnitude and direction by two sides in a triangle, then the third side represents the resultant of these two forces. In a similar way, two vectors P, G can be subtracted, for example, we assume that R is the vector resulting from the subtraction of the two vectors, i.e: P-G = P+ (-G) = R So, if we change the direction of G and then add it to P, we get our answer. 2 2. Analysis method: If a set of forces F1, F2, F3.... affect a body, the sum of these forces can be determined by analyzing these forces in two perpendicular directions: the X-axis and the Y-axis. Therefore, the components in the direction of the X- axis RX are given by RX = F1x ، F2x ، F3x + …….. = Fx In the direction of the Y-axis Ry, it is given by Ry = F1y ، F2y ، F3y +........ = Fy The magnitude of the resultant R is: The resultant R is made at angle θ with the X-axis where: 3  B C y F2 R F1 F2 F3 3θ θ2 1θ O F1 A x (Fig. 1) (Fig. 2) 4 Newton's Laws of Motion Newton's First Law of Motion Each body continues in its state of rest or uniform motion in a straight line unless it is forced to change that state by the forces acting on it. To simplify this law, we will reformulate it as follows: A body at rest continues at rest unless acted upon by a net force different from zero. A moving body continues to move in a straight line at a constant speed unless 5 acted upon by a net force other than zero. Newton's Second Law of Motion If a net force different from zero acts on an body, this force causes the body to move in the direction of the force, the magnitude of the acceleration is directly proportional to the magnitude of the net force and inversely to the amount of substance in the body. For example, if a person wants to push a child in a stroller, it is clear to everyone that the greater the thrust of the stroller, the greater its acceleration Power === (Direct proportionality with) === accelaration and the force required for the motion of a body also depends on the mass of the body. 6 Each body has a distinctive property called: Inertia: All bodies tend to continue in a state of rest unless affected by an unbalanced force and also all moving bodies tend to continue to move, so it is said that the body has an inertia and the inertia of the body is associated with its fullness, i.e. inertia is associated with mass. The greater the mass of a particular body.... The inertia of that body increased. Inertia is simply the resistance of the body to change in its state in which it is, that is, the static body resists movement according to its mass, the moving body resists stopping, and the greater the mass, the greater the inertia of that body. 7 Newton's Third Law of Motion Every action always has an equal and opposite reaction in the direction or the reciprocal actions between two bodies against each other are always equal in magnitude and opposite in direction. This is a simplification of the third law to say: If a body exerts a force on a second body, the second body acts on the first with a force equal in magnitude and opposite in direction. One of these forces is called the action force and the other is called the reaction force. 8 Types of movement 1. Linear motion If a physical body moves on a straight line under the action of a constant force, we call this motion linear motion. If v1 and v2 represent the velocity of the body at t1 and t2 respectively and t = t2 – t1, then the equations of linear motion can be deduced as follows: 1. V2 = V1 + a t 2. S = V1 t + ½ a t2 3. where S represents the displacement of the body, a represents its acceleration. These equations are also used to study the motion of body in a curved trajectory and in particular to study the motion of ballistics. 9 1. Rotational motion B The motion of body θ ZA 0 rotating on a fixed axis is called rotational motion or angular motion. The velocity of the body is the angular w = θ/t , where θ is the angle that the body rotates in t-time. Angular velocity units are estimated in radial units per second (radios/sec). To find the relationship between angular velocity and linearity, suppose that oz represents the distance of a fixed point z in the moving body from the location of 10 the 0 axis around which the body rotates. During the rotation of the body, the point z will move on the circumference of a radius oz = r and travel the distance in the time of the power of t so that , and the relationship between the linear velocity V and the angular velocity W is: That is, linear velocity = angular velocity × radius It can also be proved that linear acceleration = angular acceleration × radius 11 2. Circular motion It is said that a body moves at a constant speed if both the magnitude and direction of the speed are constant, and therefore the speed of the body is variable if its magnitude and direction change or both together, and the rate of this change gives us the value of the acceleration by which the body moves, and it is necessary for the events of this change to have a force that affects the movement of the body. The uniform motion of a body on a circular trajectory is a type of motion that changes with a constant numerical value of velocity while the direction of motion changes continuously under the influence of a force called the centripetal force. 12 If we consider the motion of a small sphere of mass m on the circumference of a radius r at a uniform speed of v, then to calculate the centrifugal force, we first begin by assigning the acceleration by which this body moves in magnitude and direction. The figure shows the magnitude of the displacement of the sphere from position P to position in time t and is Note that the magnitude of the angle θ rotated by the body around the center 0 is equal to 13 the angle between the velocity vectors at P, and is produced from the two triangles : ‫ ان‬M N ، O P If the change in the speed of the body ∆v over the time period t is equal to t , the value of the acceleration by which the body moves is given by the equation: 14 where v = w r. When the point is too close to P, it can be shown that the direction of the acceleration by which the body moves is always towards the center of the circle o. Using Newton's law to find the centripetal force F that causes the uniform motion of the body on a circular path, we find that the magnitude of the force: Its direction is also in the direction of the radius towards the center of the circle. It should be noted that at the same time that there is a gravitational force in the direction of the center, there is also, according to Newton's third law, a 15 reaction force acting in the opposite direction to the outside and is called the centrifugal force. Fc = M L T-2 dyne is Whereas: Fc cetrifigal force m body mass V speed r radius Using the equations w=2n or The equation can be developed as: Fc = m (2  n)² r follows: Fc = ms r The above equations can be applied in many advanced devices and examples of these devices: 16  1- Colloidal and liquied separator 2- Milk and honey sorters 3- Butter dryer 4- Moisture equivelant estimator 5- Extrusion pump 6- Steam control device 4. Vibrational motion There is another F=0 type of motion that F differs from the previous types in that F x the acting force is not x o constant, but changes according to the change in the 17 positions of the body. For example, a small ball of mass m is fixed at the end of a spring wire and placed on a smooth horizontal surface as in the figure. If the ball is pulled to the right by a distance of x and then left to oscillate around the equilibrium position o, the flexible wire exerts a recovery force F that always acts in the direction of the equilibrium 18 point and is given by the equation: F = - k X ………… (1) where k is the constant of proportion, which is called the force constant. A negative sign means that the direction of the force is opposite to the direction in which the displacement increases x. Applying Newton's second law where F = m a , we find that the acceleration by which the ball moves is a= ……….. (2) A negative signal for the accelaration means the same as for the recovery force. Equations (1) and (2) describe the simple harmonic motion, whose oscillation amplitude is defined as the maximum displacement reached by the body during its oscillation around the equilibrium position, and the periodic time T for this 19 movement is defined as the time required to make its full oscillation and is given from the equation: T= It is noted that the periodic time does not depend on the amplitude of the oscillation contrary to what is expected. Simple pendulum A simple pendulum consists of a small ball T L suspended at the end θ of a string. Flexible and light weight and fixes the other end to a hanging point. If the string is left to dwell, it takes an 20 anchored position in equilibrium, and if the ball is slightly tightened to the right and then left to oscillate around the equilibrium position, it is easy to prove that the movement of the ball is a simple harmonic movement as follows: The diagram represents a simple pendulum of mass m and length L in a position inclined at an angle θ on the vertical equilibrium position. The forces acting on the pendulum ball are the gravitational force and the tensile force of the string T. 21  Since the ball does not change its distance from the suspension point during its movement, the tensile T is equivalent to the gravitational force component m g Cos θ and the recovery force acting in the direction of the circular path towards the equilibrium position is: F = - m g sin θ We can see that this force is not proportional to the magnitude of the angular displacement θ but instead is proportional to the sine of θ. So, you can't say that the motion in this case is simple harmonic motion. This can be overcome if the angle θ is small and then we can say that the sine of angle θ is approximately equal to the value of the angle itself in radians. In this case: F=-mgθ=-mg =- x 22 where the magnitude of the displacement x on the circular path = θ L and applying the second Newton, we find that: That is, acceleration a is equal to This is the condition of simple harmonic motion where ، The periodic time T in the case of small displacement is: Note that the periodic time in this case does not depend on the amplitude of the 23 oscillation nor on the mass of the simple pendulum. Example: Calculate the acceleration due to gravity at a place where a simple pendulum has a string length of 150 meters and runs 100 oscillations in a time of 245 seconds? Solution: 24