Kinematics of Particles Chapter 11 Kinematics of Particles (Part 1)

Introduction Dynamics includes: Kinematics: study of the geometry of motion. Kinematics is used to relate displacement, velocity, acceleration, and time without reference to the cause of motion. Kinetics: study of the relations existing between the forces acting on a body, the mass of the body, and the motion of the body. Kinetics is used to predict the motion caused by given forces or to determine the forces required to produce a given motion. Rectilinear motion: position, velocity, and acceleration of a particle as it moves along a straight line. Curvilinear motion: position, velocity, and acceleration of a particle as it moves along a curved line in two or three dimensions.

RECTILINEAR MOTION OF PARTICLES 11.2 Position, Velocity and Acceleration Particle moving along a straight line is said to be in rectilinear motion. Position coordinate of a particle is defined by positive or negative distance of particle from a fixed origin on the line. The motion of a particle is known if the position coordinate for particle is known for every value of time t. Motion of the particle may be expressed in the form of a function, e.g., or in the form of a graph x vs. t.

Consider particle which occupies position P at time t and P’ at t+Dt, Average velocity Instantaneous velocity Instantaneous velocity may be positive or negative. Magnitude of velocity is referred to as particle speed. From the definition of a derivative, e.g.,

Consider particle with velocity v at time t and v’ at t+Dt, Instantaneous acceleration Instantaneous acceleration may be: positive: increasing positive velocity or decreasing negative velocity negative: decreasing positive velocity or increasing negative velocity. From the definition of a derivative,

Consider particle with motion given by at t = 0, x = 0, v = 0, a = 12 m/s2 at t = 2 s, x = 16 m, v = vmax = 12 m/s, a = 0 at t = 4 s, x = xmax = 32 m, v = 0, a = -12 m/s2 at t = 6 s, x = 0, v = -36 m/s, a = 24 m/s2

11.3 Determination of the motion of a particle Recall, motion of a particle is known if position is known for all time t. Typically, conditions of motion are specified by the type of acceleration experienced by the particle. Determination of velocity and position requires two successive integrations. Three classes of motion may be defined for: acceleration given as a function of time, a = f(t) - acceleration given as a function of position, a = f(x) - acceleration given as a function of velocity, a = f(v)

11.3 Determination of the motion of a particle Acceleration given as a function of time, a = f(t): Acceleration given as a function of position, a = f(x):

Acceleration given as a function of velocity, a = f(v):

Sample problem 11.1 The position of a particle which moves along a straight line is defined by the relation , where x is expected in meter and t in seconds. Determine the time at which the velocity will be zero the position and distance traveled by the particle at that time the acceleration of the particle at that time the distance traveled by the particle from t= 4s to t= 6s

Solution x (m) v (m/s) a (m/s ) 2 t(s)

Solution x (m) v (m/s) a (m/s ) 2 t(s)

Sample problem 11.2 A ball is tossed with a velocity of 10 m/s directed vertically upward from a window located 20m above the ground. Knowing that the acceleration of the ball is constant and equal to 9.81 m/s downward, Determine the velocity v and elevation y of the ball above the ground at any time t, the highest elevation reached by the ball and the corresponding value of t, the time when the ball will hit the ground and the corresponding velocity. Draw the v-t and y-t curves. 2

Integrate twice to find v(t) and y(t). Solution a.Velocity and Elevation Integrate twice to find v(t) and y(t).

b. Highest Elevation Solve for t at which velocity equals zero and evaluate corresponding altitude. Solve for t at which altitude equals zero and evaluate corresponding velocity.

c. Balls Hits the Ground Solve for t at which altitude equals zero and evaluate corresponding velocity.

Sample problem 11.3 The brake mechanism used to reduce recoil in certain types of guns consists essentially of piston attached to the barrel and moving in a fixed cylinder filled with oil. As the barrel recoils with an initial velocity v0, the piston moves and oil is forced through orifices in the piston, causing the piston and the barrel to decelerate at a rate proportional to their velocity. that is, a = -kv Determine v(t), x(t), and v(x). Draw the corresponding motion curves.

Solution Integrate a = dv/dt = -kv to find v(t). Integrate v(t) = dx/dt to find x(t).

Solution Integrate a = v dv/dx = -kv to find v(x). Alternatively, with and then

11.4 Uniform Rectilinear Motion For particle in uniform rectilinear motion, the acceleration is zero and the velocity is constant.

11.5 Uniformly Acceleration Rectilinear Motion For particle in uniformly accelerated rectilinear motion, the acceleration of the particle is constant.

11.6 Motion of Several Particles For particles moving along the same line, time should be recorded from the same starting instant and displacements should be measured from the same origin in the same direction. relative position of B with respect to A relative velocity of B with respect to A relative acceleration of B with respect to A

constant (one degree of freedom) Dependent Motions Position of a particle may depend on position of one or more other particles. Position of block B depends on position of block A. Since rope is of constant length, it follows that sum of lengths of segments must be constant. constant (one degree of freedom) Positions of three blocks are dependent. constant (two degrees of freedom) For linearly related positions, similar relations hold between velocities and accelerations.

Sample Problem 11.4 A ball is thrown vertically upward from the 12 m level in an elevator shaft with an initial velocity of 18 m/s. At the same instant, an open-platform elevator passes the 5m level, moving upward with a constant velocity at 2 m/s. Determine when and where the ball will hits the elevator the relative velocity of ball with respect to the elevator when the ball hits the elevator.

Substitute initial position and velocity and constant acceleration of ball into general equations for uniformly accelerated rectilinear motion. (1) (2) Substitute initial position and constant velocity of elevator into equation for uniform rectilinear motion. (3) (4)

Ball hits elevator We first note that the same time t and the same origin O were used in writing the equations of motion of both the ball and the elevator. We see from the figure that when the ball hits the elevator, (5) Substituting for yE and yB from (2) and (4) into (5), we have and Only the root t=3.65s corresponds to a time after the motion has begun, Substituting this value into (4), we have Elevation fro ground = 12.30m

The relative velocity of the ball with respect to the elevator is When the ball hits the elevator at time t= 3.65 s , we have The negative sign means that the ball is observed from the elevator to be moving in the negative sense (downward)

Sample Problem 11.5 200 mm Collar A and block B are connected by a cable passing over three pulleys C,D, and E as shown. Pulleys C and E are fixed, while D is attached to a collar which is pulled downward with a constant velocity of 75mm/s. At t=0, collar A starts moving downward from position K with constant acceleration and no initial velocity. Knowing that the velocity of collar A is 300 mm/s as it passes through point L, determine the change in elevation, the velocity and the acceleration of block B when collar A passes through L.

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