Engineering Mechanics: Dynamics, Twelfth Edition. Russell C. Hibbeler. Page 2. Engineering Mechanics: Dynamics, Twelfth Edition. Russell C. Hibbeler. ENGINEERING MECHANICS DYNAMICS TWELFTH EDITION R. C. HIBBELER PRENTICE HALL Upper Saddle River, NJ SOLUTION 1. Access Engineering Mechanics Dynamics SI 12th Edition solutions now. Our solutions are written by Chegg experts so you can be assured of the highest.

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Engineering Mechanics: Dynamics, Hibbeler, 12th Edition, Solution Manual 1. ayofoto.info, L.G. Kraige Engineering Mechanics Dynamics 7th edition ENGINEERING MECHANICS DYNAMICS TWELFTH EDITION R. C. HIBBELER Upper Saddle River, NJ P R E NTICE HALL Library of Congress. Engineering Mechanics: Dynamics, Twelfth Edition website is included inside the The Dynamics Study Pack and MasteringEngineering resources are.

Scott Disanno Production Editor: Also, the results can be checked in part by inspection. A particle travels along the circular path from A to B in 1 s. Looking for the textbook? The dragster starts from rest and has an graph for the time interval 0:

Dynamics is considered to be more involved than statics since both the forces applied to a body and its motion must be taken into account. Also, many applications require using calculus, rather than just algebra and trigonometry. In any case, the most effective way of learning the principles of dynamics is to solve problems.

To be successful at this, it is necessary to present the work in a logical and orderly manner as suggested by the following sequence of steps: Read the problem carefully and try to correlate the actual physical situation with the theory you have studied.

Draw any necessary diagrams and tabulate the problem data. Establish a coordinate system and apply the relevant principles, generally in mathematical form.

Solve the necessary equations algebraically as far as practical; then, use a consistent set of units and complete the solution numerically.

Report the answer with no more significant figures than the accuracy of the given data. Study the answer using technical judgment and common sense to determine whether or not it seems reasonable. Once the solution has been completed, review the problem. Try to think of other ways of obtaining the same solution. In applying this general procedure, do the work as neatly as possible.

Being neat generally stimulates clear and orderly thinking, and vice versa. Recall that a particle has a mass but negligible size and shape. Therefore we must limit application to those objects that have dimensions that are of no consequence in the analysis of the motion. In most problems, we will be interested in bodies of finite size, such as rockets, projectiles, or vehicles. Each of these objects can be considered as a particle, as long as the motion is characterized by the motion of its mass center and any rotation of the body is neglected.

Rectilinea r Kinematics. The kinematics of a particle is characterized by specifying, at any given instant, the particle 's position, velocity, and acceleration. The straight-line path of a particle will be defined using a single coordinate axis s, Fig. The origin 0 on the path is a fixed s point, and from this point the position coordinate s is used to specify the s location of the particle at any given instant.

The magnitude of s is the Position distance from 0 to the particle, usually measured in meters m or feet ft , and the sense of direction is defined by the algebraic sign on s. Likewise, it is negative if the particle is located to the left of O. Realize that position is a vector quantity since it has both magnitude and direction. Here, however, it is being represented by the algebraic scalar s since the direction always remains along the coordinate axis.

The displacement of the particle is defined as the change in its position. For example, if the particle moves from one point to another, Fig. Likewise, if the final position were to the f b S left of its initial position, Lls would be negative.

Specifically, the distance traveled is a positive scalar that represents the total length of path over which the particle travels. For example, if the particle is moving to the right, Fig. This is emphasized here by the arrow written at the left of Eq. Velocity Occasionally, the term "average speed" is used. The average speed is always a positive scalar and is defined as the total distance traveled by a particle, ST, divided by the elapsed time Llt; i.

Provided the velocity of the particle is known at two points, the average acceleration of the particle during the time interval! The instantaneous acceleration at time t is a vector that is found by taking smaller and smaller values of! In particular, when the particle is slowing down, or its speed o is decreasing, the particle is said to be decelerating.

In this case, v' in a Fig. Also, note that when the velocity is constant, the acceleration is zero since! Deceleration f Finally, an important differential relation involving the displacement, velocity, and acceleration along the path may be obtained by eliminating the time differential dt between Eqs. Velocity as a Function of Time. Either solve for t in Eq. A typical example of constant accelerated motion occurs when a body falls freely toward the earth.

If air resistance is neglected and the distance of fall is short, then the downward acceleration of the body when it is close to the earth is constant and approximately 9. The proof of this is given in Example This is different from the average velocity, which is the displacement divided by the time. Since motion is along a straight line, the vector quantities position, velocity, and acceleration can be represented as algebraic scalars. Kinematic Equations. Remember that Eqs. The position coordinate extends from the fixed origin 0 to the car, positive to the right.

Since v! The formulas for constant acceleration cannot be used to solve this problem, because the acceleration is a function of time. Using the condition that at t 0, s 0, then C 0. Determine the projectile's velocity and position 4 s after it is fired. Since the motion is downward, the position coordinate is positive downward, with origin located at 0, Fig.

Determine the maximum height SB reached by the rocket and its speed just before it hits the ground. While in motion the rocket is subjected to a constant downward acceleration of 9. Neglect the effect of air resistance. The origin 0 for the position coordinate s is taken at ground level with positive upward, Fig.

Since ac is constant the rocket's position may be related to its velocity at the two points A and B on the path by using Eq. To obtain the velocity of the rocket just before it hits the ground, we can apply Eq. S 10 The negative root was chosen since the rocket is moving downward. Similarly, Eq. It should be realized that the rocket is subjected to a deceleration from A to B of 9.

As shown in Fig. The positive root is chosen since the particle is traveling downward, i. The formulas for constant acceleration cannot be used here because the acceleration changes with position, i. If it is initially located at the origin 0, determine the distance traveled in 3. Here positive motion is to the right, measured from the origin 0, Fig.

If we consider a graph of the velocity function, Fig. Hence, the distance traveled in 3. Determine the Fl Initially, the car travels along a straight road with a FS. If the brakes are applied and the speed of the car is reduced to 10 mls in 15 s, determine the constant time when the velocity of the particle is zero, and the total deceleration of the car.

A ball is thrown vertically upward with a speed of Fl A particle travels along a straight line with a velocity F A particle travels along a straight line with a speed FS. A car starts from rest and with constant Afterwards it maintains this and the time required.

Determine the distance traveled by car A when they pass A train starts from rest at a station and travels with a each other. A car is traveling at 15 mi s, when the traffic light 50 m ahead turns yellow. Determine the required constant ft deceleration of the car and the time needed to stop the car at the light. Determine the velocity and the position of the particle as a Determine Also, find the velocity of the ball when it strikes the bottom of the shaft.

What is the displacement of the car during the How much time is needed to stop the car? A particle travels along a straight-line path such Va Determine the particle's average velocity and function of time t.

Tests reveal that a normal driver takes about 0. Two particles A and B start from rest at the origin takes about 3 s for a driver having 0. If such drivers are traveling on a straight road seconds. If you must drink, please don't drive!

One second later k is a constant and v is the velocity of the particle. Determine the height from the ground where the two balls pass each other.

If It then travels with constant velocity for 60 seconds. A particle moves along a straight line with an meters. Use Simpson's rule to of a parking garage, which is 48 ft above the ground. If the evaluate the integral. If seconds. Plot the path to determine the total distance is maximum. Ball A is thrown vertically upwards with a velocity of B all B is thrown upwards from the same point with Vo The acceleration of a particle as it moves along a At so that it does not fall back to the earth.

Determine the height from the ground and the time at which they pass. Accounting for the vanatlOn of gravitational acceleration a with respect to altitude y see Prob. Afterwards it maintains With what velocity does the particle strike the earth if it is this speed. Use the road is traveling toward the motorcycle at a constant speed numerical data in Prob.

Determine the time and the distance traveled by the motorcycle when they pass each other. Initially the particle falls How long does the particle take to reach this position if from rest. A particle is moving along a straight line such that Determine the total distance traveled seconds, determine its velocity, after it has traveled 10 m.

Also, find the average How much time does this take? E rratic M otion When a particle has erratic or changing motion then its position, velocity, and acceleration cannot be described by a single continuous mathematical function along the entire path.

Instead, a series of functions will be required to specify the motion at different intervals. For this reason, it is convenient to represent the motion as a graph. Several situations occur frequently.

The s-t, v-t, a n d a-t Graphs. To construct the v-t graph given the s-t graph, Fig. The v-t graph can be v constructed by plotting this and other values at each instant.

The a-t graph can be constructed from the v-t graph in a similar manner, Fig. Since differentiation reduces a polynomial of degree n to that of degree n - 1, then if the s-t graph is parabolic a second-degree curve , the v-t graph will be a sloping line a first-degree curve , and the b a-t graph will be a constant or a horizontal line a zero-degree curve. Notice b that an algebraic addition of the area increments of the a-t graph is necessary, since areas lying above the t axis correspond to an increase in Fig.

Similarly, if the v-t graph is given, Fig. If segments of the a-t graph can be described by a series of equations, then each of these equations can be integrated to yield equations So describing the corresponding segments of the v-t graph. In a similar manner, the s-t graph can be obtained by integrating the equations which describe the segments of the v-t graph.

Construct the v-t and a-t graphs for We can also obtain specific values of v by measuring the slope of the s-t graph at a given instant.

This yields 2 f How far has the car traveled? This yields, Fig. Using this initial condition, 10 s: The graphical results can be checked in part by calculating slopes. Also, the results can be checked in part by inspection. The particle travels along a straight track such that F Construct the v- that its position is described by the graph.

Construct the v- graph for the same time interval. The dragster starts from rest and has an graph for the time interval 0: S ', where ' is the time Fl A van travels along a straight road with a velocity acceleration described by the graph. Construct the v- described by the graph. Construct the s-t and a- graphs during the same period. A bicycle travels along a straight road where its F The dragster starts from rest and has a velocity velocity is described by the v-s graph.

Construct the a-s described by the graph. Construct the s- graph during the graph for the same time interval. Also, determine the total v distance traveled during this time interval. The speed of a train during the first minute has A train starts from station A and for the first been recorded as follows: Then, for the next two kilometers, it travels with a uniform speed.

Plot the v-t graph, approximating the curve as straight-line segments between the given points. Determine the total The particle travels along a straight line with the distance traveled.

Construct the a-s graph. A two-stage missile is fired vertically from rest with the acceleration shown. In 15 s the first stage A burns out and the second stage B ignites. Construct the v-s graph. A freight train starts from rest and travels with a 2 constant acceleration of 0. Determine the time t' and draw the v-t graph for the motion. A car travels up a hill with the speed shown.

Plot the a-t graph. Draw Determine the total distance the car travels until it stops the s - t, v - t, and a - t graphs for the particle for o:: A truck is traveling along the straight line with a velocity described by the graph.

Construct the a-s graph for The snowmobile moves along a straight course according to the v-t graph. Construct the s-t and a-t graphs for the same s time interval. A car starts from rest and travels along a straight road with a velocity described by the graph. Determine the IL Construct the s-t 30 50 and a-t graphs. If he then maintains this speed, determine the time needed for him to reach a point located ft in front of the truck.

Draw the v-t and s-t graphs for the motorcycle during this time.

An airplane traveling at 70 mls lands on a straight The dragster starts from rest and travels along a runway and has a deceleration described by the graph. Construct the v-s graph for 0: Construct the v-t and s-t graphs for determine the distance s' traveled before the dragster again this time interval, 0: The position of a cyclist traveling along a straight A sports car travels along a straight road with an road is described by the graph.

Construct the v-t and a-t acceleration-deceleration described by the graph. If the car graphs. A missile starting from rest travels along a straight The v-t graph of a car while traveling along a road track and for 10 s has an acceleration as shown.

Draw the is shown. Draw the s-t and a-t graphs for the motion. A motorcyclist starting from rest travels along a The boat travels in a straight line with the straight road and for 10 s has an acceleration as shown. If it starts from rest, Draw the v-t graph that describes the motion and find the construct the v-s graph and determine the boat's maximum distance traveled in 10 s. What distance S' does it travel before it stops?

The rocket has a n acceleration described by the The acceleration of the speed boat starting from graph. If it starts from rest, construct the v- and s- graphs for the motion for the time interval 0: The jet bike is moving along a straight road with the speed described by the v- s graph.

Construct the s- and a-s graphs. The s-t graph for a train has been determined The airplane travels along a straight runway with experimentally.

From the data, construct the v-t and a-t an acceleration described by the graph. If it starts from rest graphs for the motion. Construct the v-t and s-t graphs. The a-t graph of the bullet train is shown. If the and has a deceleration described by the graph. Determine train starts from rest, determine the elapsed time t ' before it the distance s' traveled before its speed is decreased to again comes to rest.

Draw the s-t graph. Since this path is often described in three dimensions, vector analysis will be used to formulate the particle's position, velocity, and acceleration. Consider a particle located at a point on a space curve o s defined by the path function set , Fig.

Notice that both the magnitude and direction of this vector will change as the particle moves along the curve.

Position Displacement. Suppose that during a small time interval! The displacement Llr represents the change in the particle's position and is determined by vector subtraction; i. During the time! The magnitude of v, which is called the speed, is obtained by realizing that the length of the straight line segment Llr in o s Fig. To study this time rate of change, the two velocity vectors in Fig. To clarify this point, realize that Llv and consequently a must account for the change made in both the magnitude and direction of the velocity v as the particle moves from one point to the next along a the path, Fig.

However, in order for the particle to follow any curved path, the directional change always "swings" the velocity vector Acceleration path toward the "inside" or "concave side" of the path, and therefore a g cannot remain tangent to the path.

In summary, v is always tangent to the path and a is always tangent to the hodograph. Recta n g u l a r Components Occasionally the motion of a particle can best be described along a path that can be expressed in terms of its x, y, z coordinates. The first time derivative of r yields the velocity of the particle.

For example, the derivative of the i component of r is - Xl dx. As discussed in Sec. The acceleration of the particle is obtained by taking the first time derivative of Eq. Since a represents the time rate of change in both the magnitude and direction of the velocity, in general a will not be tangent to the path, Fig.

Coordinate System. Kinematic Quantities. A review of this concept is given in Appendix C. B-3, and their coordinate direction angles from the components of their unit vectors, Eqs.

B-4 and B-S. See Appendix A for a full explanation.

The direction is tangent to the path, Fig. Q - V x S Acceleration. The relationship between the acceleration components b is determined using the chain rule. See Appendix C.

Using the chain rule, the time derivative of Eq. These results are shown in Fig. To illustrate the kinematic analysis, consider a projectile launched at point xo , Yo , with an initial velocity of vo, having components vo x and vo y , Fig.

The red ball The first and last equations indicate that the horizontal component of falls from rest, whereas the yellow ball is given a horizontal velocity when released.

Both balls accelerate downward at the same rate, and so they remain at the same Vertica l M otio n. Since the positive y axis is directed upward, then elevation at any instant.

Applying Eqs. Recall that the last equation can be formulated on the basis of eliminating the time t from the first two equations, and therefore only two of the above three equations are independent of one another. Once Vx and Vy are obtained, the resultant velocity v, which is always tangent to the path, can be determined by the vector sum as shown in Fig. Establish the fixed x, y coordinate axes and sketch the trajectory of the particle.

In all cases the acceleration of gravity acts downward and equals 9. The particle's initial and final velocities should be represented in terms of their x and y components.

H o rizontal M otion. The velocity in the horizontal or direction is constant, i. Rectangular coordinates are used for the analysis since the acceleration is only in the vertical the first and third of these equations will not be useful.

If the height of the ramp is 6 m from the floor, y determine the time needed for the sack to strike the floor and the range R where sacks begin to pile up. The origin of coordinates is established at the beginning of the path, point A, Fig. Horizontal Motion.

Since tAB has been calculated, R is determined as follows: The calculation for tAB also indicates that if a sack were released from rest at A, it would take the same amount of time to strike the floor at C, Fig. Since we do not need to determine VA y ' we have Horizontal Motion.

During a race it was observed that the rider shown in Fig. Determine the speed at which he was traveling off the ramp, the horizontal distance he travels before striking the ground, and the maXImum height he attains. Neglect the size of the bike and rider. Since the time of flight and the vertical distance between the ends of the path are known, we can determine vA.

The range R can now be determined. In order to find the maximum height h we will consider the path AC, Fig. Here the three unknowns are the time of flight tAc , the horizontal distance from A to C, and the height h. Substitute for g and 0.

Since the horizontal velocity is always constant,. Substitute Therefore, the magnitude of ball velocity is. Substitute 2. Substitute for and for g. Therefore, the tangential component of the acceleration of the ball is. Therefore, the normal component of the acceleration of the ball is. Chegg Solution Manuals are written by vetted Chegg Classical Mechanics experts, and rated by students - so you know you're getting high quality answers.

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