dynamics

1 Rectilinear Motion with Constant Acceleration 1. On a certain stretch of track, trains run at 60 mph. How far back of a stopped train should a warning torpedo be placed to signal an oncoming train? Assume that the brakes are applied at once and retard the train at the uniform rate of 2 ft per sec2. 2. A stone is thrown vertically upward and returns to earth in l0sec. What was its initial velocity and how high did it go? 3. A ball is dropped from the top of a tower 80 ft high at the same instant that a second ball is thrown upward from the ground with an initial velocity of 40 ft per see. When and where do they pass, and with what relative velocity? 4. A stone is dropped down a well and 5 sec later the sound of the splash is heard. If the velocity of sound is 1120 ft per sec, what is the depth of the well? 5. A stone is dropped from a captive balloon at an elevation of 1000 ft. Two seconds later another stone is projected vertically upward from the ground with a velocity of 248 ft per sec. If g is 32 ft per sec2, when and where will the stones pass each other? 6. A stone is thrown vertically upward from the ground with a velocity of 48.3 ft per sec. One second later another stone is thrown vertically upward with a velocity of 96.6 ft per sec. How far above the ground will the stones be at the same level? 7. A ball is shot vertically into the air at a velocity of 193.2 ft per sec. After 4 sec, another

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ball is shot vertically into the air. What initial velocity must the second ball have in order to meet the first ball 386.4 ft from the ground?

decelerate at the constant rate of 10 ft per sec . In what time will A overtake B, and how far will each car have traveled?

8. A stone is thrown vertically up from the ground with a velocity of 300 ft per sec. How long must one wait before dropping a second stone from the top of a 600-ft tower if the two stones are to pass each other 200 ft from the top of the tower?

14. An automobile moving at a constant velocity of 45 ft per sec passes a gasoline station. Two seconds later, another automobile leaves the gasoline station and accelerates at the constant rate of 6 ft per sec2. How soon will the second automobile overtake the first?

9. A ship being launched slides down the ways with a constant acceleration. She takes 8 sec to slide the first foot. How long will she take to slide down the ways if their length is 625 ft?

15. A balloon rises from the ground with a constant acceleration of 4 ft per sec2. Five seconds later, a stone is thrown vertically up from the launching site. What must be the minimum initial velocity of the stone for it to just touch the balloon? Note that the balloon and stone have the same velocity at contact.

10. A train moving with constant acceleration travels 24 ft during the 10th sec of its motion and 18 ft during the 12th sec of its motion. Find its initial velocity and its constant acceleration. 11. An automobile starting from rest speeds up to 40 ft per sec with a constant acceleration of 4 ft per sec2, runs at this speed for a time, and finally comes to rest with a deceleration of 5 ft per sec2. If the total distance traveled is 1000 ft, find the total time required. 12. A train travels between two stations 1- mile apart in a minimum time of 41 sec. If the train accelerates and decelerates at 8 ft per sec2, starting from rest at the first station and coming to a stop at the second station, what is its maximum speed in mph? How long does it travel at this top speed? 13. Two cars A and B have a velocity of 60 mph in the same direction. A is 250 ft behind B when the brakes are applied to car B, causing it to

Rectilinear Motion with Variable Acceleration Using Equations of Motions 1. The motion of a particle is given by the 1 equation s  2t 4  t 3  2t 2 , where s is in feet and 6 t in seconds. Compute the values of v and a when t = 2 sec. 2. A particle moves in a straight line according to the law s = t3 - 40 t where s is in feet and t in seconds. (a) When t = 5 sec, compute the velocity. (b) Find the average velocity during the fourth second. (c) When the particle again comes to rest, what is its acceleration?

2 3. A ladder of length L moves with its ends in contact with a vertical wall and a horizontal floor. If the ladder starts from a vertical position and its lower end A moves along the floor with a constant velocity vA, show that the velocity of its upper end B is vB = -vAtan θ where θ is the angle between the ladder and the wall. 4. The rectilinear motion of a particle is given by s = v2 - 9 where s is in feet and v in feet per second. When t = 0, s = 0 and v = 3 ft per sec. Find the s-t, v-t, and a-t relations.

t is in seconds. When t is zero, v = 2 ft per sec and s = 4 ft. Find the values of v and s when t = 2 sec.

Rectilinear Motion with Variable Acceleration Using Motion Curves 1. From the v-t curve shown, determine the distance traveled in 4 sec and also in 6 sec.

4. A particle having an initial velocity of 200 ft per sec decelerates according to the a-t curve shown. Compute the change in displacement in the time interval of 30 sec.

5. The velocity of a particle moving along the X axis is defined by v = x3 - 4x2 + 6 x where v is in feet per second and x is in feet. Compute the value of the acceleration when x = 2 ft. 6. The motion of a particle is defined by the relation a = 4 t, where a is in feet per second2 and t in seconds. It is known that s = 1 ft and v = 2 ft per sec when t = 1 sec. Determine the relations between v and t, s and t, v and s.

2. The motion of a particle starting from rest is governed by the a-t curve shown. Determine the displacement at t = 9 sec.

7. The motion of a particle is governed by the 8 equation a   2 , where a is in feet per second2 s and s is in feet. When t = 1 sec, s = 4 ft and v = 2 ft per Sec. Determine the relations between v and t, s and t, v and s. 1

8. The motion of a particle is given by a  6v 2 , where a is in feet per second2 and v in feet per second. When t is zero, s = 6 ft and v = 0. Find the relations between v and t, s and t, v and s. 9. The motion of a particle is governed by the relation a = 4t2, where a is n feet per second2 and

5. A car moving at 60 ft per sec is brought to rest in 12 sec with a deceleration which varies uniformly with time from 2 ft per sec2 to a maximum deceleration. Compute the distance traveled in stopping. 6. A car starts from rest and reaches a speed of 48 ft per sec in 15 sec. The acceleration increases from zero uniformly with time for the first 6 sec after which it remains constant. Compute the distance traveled in 15 sec.

3. The curved portions of the v-t curve shown are second degree parabolas with horizontal slope at t = 0 and t = 12 sec. Determine the value of s when t = 18 sec

7. A car accelerates for t sec from rest to a velocity of 48 ft per sec, the acceleration increasing uniformly from zero to 12 ft per sec2. During the next 4 sec, the car decelerates at a constant rate to a velocity of 4 ft per sec. Sketch the a-t, v-t, and s-t curves.

3 8. An object attains a velocity of 16 ft per sec by moving in a straight line with an acceleration which varies uniformly from zero to 8 ft per sec2 in 6 sec. Compute its initial velocity and the change in displacement during the 6-sec interval. 9. The acceleration of an object decreases uniformly from 8 ft per sec2 to zero in 6 sec at which time its velocity is 10 ft per see. Find the initial velocity and the change in displacement during the 6-sec interval.

Kinetics of Rectilinear Motion, Analysis as a Particle 1. An elevator weighing 3220 lb starts from rest and acquires an upward velocity of 600 ft per min in a distance of 20 ft. If the acceleration is constant what is the tension in the elevator cable? 2. A man weighing 161 lb is in an elevator moving upward with an acceleration of 8 ft per sec2. (a) What pressure does he exert on the floor of the elevator? (b) What will the pressure be if the elevator is descending with the same acceleration? 3. The block shown reaches a velocity of 40 ft per sec in 100 ft, starting from rest. Compute the coefficient of kinetic friction between the block and the ground.

4. Determine the force P that will give the body shown an acceleration of 6 ft per sec2. The coefficient of kinetic friction is 0.20.

5. A magnetic particle weighing 3.6 grams is pulled through a solenoid with an acceleration of 6 meters per sec2. Compute the force in pounds acting on the particle. 6. When a 644-lb boat is moving at 10 ft per sec, the motor conks out. How much farther will the boat glide, assuming its resistance to motion is 2v lb where v is in feet per second? 7. A bullet weighing 1 lb is fired vertically upward with a muzzle velocity of 3000 ft per sec. If the velocity is 2950 ft per sec after 1 sec, what is the average air resistance on the bullet? What maximum height will the bullet reach, assuming that the air resistance remains constant? 8. Two blocks A and B are released from rest on a 30° incline when they are 50 ft apart. The coefficient of friction under the upper block A is 0.2 and that under the lower block B is 0.4. Compute the elapsed time until the blocks touch. 9. Determine the acceleration of the bodies shown if the fixed drum is smooth and A is heavier than B.

10. Referring to Prob. 9, assume A weighs 200 lb and B weighs 100 lb. Determine the acceleration of the bodies if the coefficient of kinetic friction is 0.10 between the cable and the fixed drum. 11. Two bodies A and B are separated by a spring. Their motion down the incline is resisted by a force P = 200 lb. The coefficient of kinetic friction is 0.30 under A and 0.10 under B. Determine the force in the spring.

12. If the pulleys shown are weightless and frictionless, find the acceleration of body A.

4 13. Determine the acceleration of body B shown, assuming the pulleys to be weightless and frictionless.

17. Compute the acceleration of body B and the tension in the cord attached to it. 20. Determine the acceleration of each weight shown, assuming the pulleys to be weightless and frictionless.

14. The coefficient of kinetic friction under block A is 0.30 and under block B it is 0.20. Find the acceleration of the system and the tension in each cord.

18. Compute the time required for the 100-lb body to move 10 ft starting from rest.

15. 1068. Determine the magnitude of W so that the 200-lb body will have acceleration up the plane of 4.025 ft per sec2. 21. Determine the tension in the cord supporting body C. The pulleys are frictionless and of negligible weight. 19. If the pulleys are weightless and frictionless, determine the acceleration of each weight in fps2. 16. Compute the acceleration of body B and the tension in the cord supporting body A.

5 4. The uniform block shown weighs 200 lb. It is pulled up the incline by the force P = 300 lb. Determine the maximum and minimum values of d so that the block does not tip over as it slides up the incline. The coefficient of friction is 0.20. Ans. Max. d = 2.32 ft; mm. d = 1.253 ft

22. In the system of connected blocks shown, the coefficient of kinetic friction is 0.20 under bodies B and C. Determine the acceleration of each body and the tension in the cord.

23. Repeat Prob. 22, but change the weight of A to 600 lb, of B to 1000 lb and of C to 500 lb. 24. Two blocks A and B, each weighing 96.6 lb and connected by a rigid bar of negligible weight, move along the smooth surfaces shown. They start from rest at the given position. Determine the acceleration of B at this instant.

Kinetics of Rectilinear Motion, Analysis as a Rigid Body 1. A juggler places the lower end of a vertical rod upon his finger. As it starts to tip, explain how he keeps the rod in balance by moving his finger horizontally back and forth. 2. The cable of a cargo crane can support a maximum load of 2 tons. While the crane is lowering a 1610-lb weight at uniform speed, the brake on the winch is applied too rapidly, thereby causing a sudden deceleration of the weight equal to 100 fps2. The cable snaps and the weight falls, badly injuring a workman. For the purpose of establishing liability in this accident, is it likely that failure of the cable was due to its being weaker than its test strength of 2 tons? Why? 3. A uniform box is 2 ft square and 6ft high. It stands on end upon a truck with its sides parallel to the truck’s motion. If the box weighs 240 lb and the coefficient of friction between the box and the truck is 0.30, show whether the block will slide or tip first as the acceleration of the truck is increased.

5. The 240-lb body shown is supported by wheels at B which roll freely without friction and by a skid at A under which the coefficient of friction is 0.40. Compute the value of P to cause an acceleration of ⅓g. 6. If the value of P in Prob. 5 is 100 Ib, compute the acceleration. If this value of P were applied at a higher position on the body, would the acceleration be changed in any way? 7. An auto with a rear wheel drive has a wheelbase of 10 ft. The c.g. is 3 ft above the pavement and 4 ft ahead of the rear wheels. The coefficient of friction is 0.60 between the tires and the pavement. Determine the maximum acceleration the auto could have when moving along a level road. 8. An auto, equipped with only front wheel brakes, has a wheelbase of 120 in. with its c.g. located 60 in. ahead of the rear wheels and 36 in.

6 above the pavement. 1f μ = 0.80 at the tires, compute the minimum distance in which the auto can be brought to rest from a speed of 60 mph if the driver’s reaction time before applying the brakes is ¾ sec. 9. A car with a four-wheel drive weighs 3000 lb and has a wheelbase of 10 ft. The c.g. is 3 ft above the pavement and 4 ft ahead of the rear wheels. Compute the tractive force acting at the rear wheels when the car accelerates at ⅓g fps2. Assume the coefficient of friction is equal at all four wheels. 10. The coefficient of kinetic friction under the sliding supports t A and B is 0.30. What force P will give the 600-lb door a leftward acceleration of 8.05 fps2? What will be the normal pressures at A and B?

13. A bar weighing 2 lb per ft is bent at right angles into segments 26 in. and 13 in. long. It takes the position shown when the frame F to which it is pinned at A is accelerated horizontally. Determine this acceleration and the components of the reaction at A.

15. Determine the value of W in Prob. 14 if the 200-lb crate is on the verge of tipping forward as it slides up the incline. Assume d = 3.32 ft. 16. The frame of a machine is accelerated leftwards at 3 g fps2. As shown, it carries a 5

uniform angle ABC weighing 80 lb which is braced by the uniform strut CD weighing 60 lb. Determine the components of the pin pressure at C upon CD.

11. Repeat Prob. 10 if μ = 0.30 at A and μ = 0.20 at B. 12. From the figure, find the angle θ at which a uniform bar will be maintained inside the smooth surface of a cylindrical drum accelerating leftward at

3 g fps2. 5

14. The uniform crate shown weighs 200 lb. It is pulled up the incline by a counterweight W of 400 lb. Find the maximum and minimum values of d so that the crate does not tip over as it slides up the incline.

17. The uniform bar AB weighing 240 lb is mounted as shown upon a carriage weighing 480 lb. The center of gravity of the carriage is at C midway between the wheels. If P = 180 lb and there is no frictional resistance at the wheels, find

7 the wheel reactions RA and RB, and also the horizontal and vertical components of the pin pressure at A.

1. A stone is thrown from a hill at an angle of 60° to the horizontal with an initial velocity of 100 ft per sec. After hitting level ground at the base of the hill, the stone has covered a horizontal distance of 500 ft. How high is the hill? 2. A shell leaves a mortar with a muzzle velocity of 500 ft per sec directed upward at 600 with the horizontal. Determine the position of the shell and its resultant velocity 20 sec after firing. How high will it rise?

18. Two bodies A and B, each weighing 96.6 lb, are connected by a rigid bar of negligible weight attached to them at their gravity centers. The coefficient of friction at the wall and floor is 0.268. If the bodies start from rest at the given position, determine the acceleration of B at this instant.

3. A projectile is fired with an initial velocity of 193.2 ft per sec upward at an angle of 300 to the horizontal from a point 257.6 ft above a level plain. What horizontal distance will it cover before it strikes the level plain? 4. A projectile is fired with an initial velocity of v0 ft per sec upward at an angle of θ with the horizontal. Find the horizontal distance covered before the projectile returns to its original level. Also determine the maximum height attained.

ground and at a angle of 60° to the horizontal, what was the initial velocity of the ball? 7. Determine the distance s at which a ball thrown with a velocity v of 100 fps at an angle 3   tan 1 0 will strike the incline shown. 4

8. In Prob. 7, a ball thrown down the incline strikes it at a distance s = 254.5 ft. If the ball rises to a maximum height h = 14.4 ft above the point of release, compute its initial velocity, vo and inclination θ. 9. Refer to the figure shown and find α to cause the projectile to hit point B in exactly 4 sec. What is the distance x?

5. The car shown is just to clear the water-filled gap. Find the take-off velocity v0.

Curvilinear Motion: Projectile Motion

6. A ball is thrown so that it just clears a 10-ft fence 60 ft away. If it left the hand 5 ft above the

10. Boat A moves with a constant velocity of 20 fps, starting from the position shown. Find θ in order for the projectile to hit the boat 5 sec after starting, under the conditions given. How high is the hill above the water?

8 also the resultant acceleration in magnitude and inclination.

11. A stone has an initial velocity of 100 ft per sec up to the right at 30° with the horizontal. The components of acceleration are constant at ax = -4 fps2 and ay = -20 fps2. Compute the horizontal distance covered until the stone reaches a point 60 ft below its original elevation. 12. A particle has such a curvilinear motion that its x- coordinate is defined by x = 5 t3 - 105 t where x is in inches and t in seconds. When t = 2 sec, the total acceleration is 75 in. per sec2. If the y component of acceleration is constant and the particle starts from rest at the origin when t = 0, determine its total velocity when t = 4 sec.

6. A particle is moving along a curved path. At a certain instant when the slope of the path is 0.75, ax= 6 ft fps2 and ay = 10 fps2. Compute the values of at and an at this instant and sketch how the path curves. 2. A particle moves in such a manner that ax = -6 fps2 and ay = -30 fps2. If its initial velocity is 100 fps directed at a slope of 4 to 3 as shown, compute the radius of curvature of the path 2 sec later.

1. Point A moves in a circular path of 20 ft radius so that its arc distance from an initial position B is given by the relation s. = 6t3 - 4 t, where s is in feet and t in seconds. Determine the tangential and normal components of acceleration of the point for the instant when t = 2 sec. Find

7. A stone is thrown with an initial velocity of 100 fps upward at 60 to the horizontal. Compute the radius of curvature of its path at the point where it is 50 ft horizontally from its initial position. 8. A stone has an initial velocity of 200 fps up to the right at a slope of 4 to 3. The components of acceleration are constant at ax = -12 fps2 and ay = -20 fps2. Compute the radius of curvature at the start and at the top of the path.

13. A projectile is fired from the top of a cliff 300 ft high with a velocity of 1414 ft per sec directed at 45° above the horizontal. Find the range on a horizontal plane through the base of the cliff.

Curvilinear Motion: Two Components of Acceleration

5. A particle moves on a circular path of 20 ft radius so that its arc distance from a fixed point on the path is given by s = 4t3 - 10t where a is in fps2 and t in seconds. Compute the total acceleration at the end of 2 sec.

3. The normal acceleration of a particle on the rim of a pulley 10 ft in diameter is constant at 8000 fps2. Determine the speed of the pulley in rpm. 4. At the bottom of a loop the speed of an airplane is 400 mph. This causes a normal acceleration of 9g fps2. Determine the radius of the loop.

9. A particle moves counterclockwise on a circular path of 400 ft radius. It starts from a fixed point which is horizontally to the right of the center of the path and moves so that s = 10t2 + 20t, where s is the arc distance in feet and t is the time in seconds. Compute the x and y components of acceleration at the end of 3 sec. 10. A particle moves on a circle in accordance with the equation s = t4 - 8 t, where s is the displacement in feet measured along the circular path and t is in seconds. Two seconds after starting from rest, the total acceleration of the

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particle is 48 2 fps . Compute the radius of the circle. Kinetics of Curvilinear Motion, Dynamic Equilibrium 1. As shown in the figure, a bob of weight W is moving with a constant velocity v in a horizontal plane at the end of a cord of length L. Because the string generates a cone while in motion, the system is called a conical pendulum. It is required to determine the tension in the supporting cord, its inclination with the vertical, and the period or time required to complete one revolution.

5. A body of weight W rests on the smooth inclined surface of the frame shown. A peg attached to the frame forces the body to rotate with it about the vertical axis. Determine the speed in rpm at which the tension in the cord is equal to the weight of the body.

2. A rod 4 ft long rotates in a horizontal plane about a vertical axis through its center. At each end of the rod is fastened a cord 3 ft long. Each cord supports a weight W. Compute the speed of rotation n in rpm to incline each cord at 30° with the vertical. 3. A weight concentrated at the end of a cord forms a conical pendulum for which the period is 1 sec. Determine the velocity v of the weight if the cord rotates inclined at 30° with the vertical. 4. In the figure, the 20-lb ball is forced to rotate around the smooth inside surface of a conical shell at the rate of one revolution in π/4 sec. If g = 32 fps2, find the tension in the cord and the force on the conical shell. At what speed in rpm will the force on the shell become 0?

6. The hammer of an impact testing machine weighs 64.4 lb. It is attached to the end of a light rod 4 ft long which is pivoted to a horizontal axis at A as shown. (a) What is the bearing reaction on the pivot an instant after being released from the given position? (b) What is the bearing reaction just before impact at B if the velocity of the hammer is then 5.9 ft per sec?

7. To check the radius of a railroad curve, the effect of a 20-lb weight is observed to be 20.7 lb on a spring scale suspended from the roof of an experimental ear rounding the curve at 40 mph. What is the radius of the curve? 8. Figure shown represents a schematic diagram of a Porter governor. Each fly ball weighs 16.1 lb and the central weight D is 40 lb. Determine the rotational speed in rpm about the vertical axis AD at which the weight D begins to rise. 9. What counterweight W will maintain the Corliss engine governor in the position shown at a rotational speed n = 120 rpm. Each fly ball weighs 16.1 lb. Neglect the weight of the other links.

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10. The side rod of the engine is 8 ft long and weighs 100 lb. The cranks AD and BC are of length r = 18 in. and rotate at 300 rpm. Determine the maximum bending moment M in the rod if M = WL/8 where W is the total distributed load and L is the length of the rod.

11. The segment of road passing over the crest of 4x x 2 a hill is defined by the parabola y  .A  10 100 car weighing 3220 lb travels along the road at a constant speed of 30 ft per sec. What is the pressure on the wheels of the car when it is at the crest of the hill where y = 4 ft? At what speed will the road pressure be zero?