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engineering
engineering mechanics dynamics

Questions and Answers of Engineering Mechanics Dynamics

A spacecraft moving in a west-to-east equatorial orbit is observed by a tracking station located on the equator. If the spacecraft has a perigee altitude H = 150 km and velocity v when directly over
The perigee and apogee altitudes above the surface of the earth of an artificial satellite are hp and ha, respectively. Derive the expression for the radius of curvature ρp of the orbit at the
A satellite is placed in a circular polar orbit a distance H above the earth. As the satellite goes over the north pole at A, its retro-rocket is activated to produce a burst of negative thrust which
Determine the necessary amount Δv by which the circular-orbit velocity of satellite C should be reduced if the catch-up maneuver of Prob. 3/288 is to be accomplished with not one but two periods in
The 175,000-lb space-shuttle orbiter is in a circular orbit of altitude 200 miles. The two orbital-maneuvering-system (OMS) engines, each of which has a thrust of 6000 lb, are fi red in retrothrust
Compare the orbital period of the moon calculated with the assumption of a fixed earth with the period calculated without this assumption.
Two satellites B and C are in the same circular orbit of altitude 500 miles. Satellite B is 1000 mi ahead of satellite C as indicated. Show that C can catch up to B by “putting on the brakes.”
Determine the unstretched spring length which would cause the 3-lb collar of the previous problem to have no speed as it arrives at position B. All other conditions of the previous problem remain the
All conditions of the previous problem remain the same, except now the applied couple varies with time according to M = 2t, where t is in seconds and M is in newton-meters. Determine the angular
The 0.4-lb ball and its supporting cord are revolving about the vertical axis on the fixed smooth conical surface with an angular velocity of 4 rad /sec. The ball is held in the position b = 14 in.
The assembly of two 5-kg spheres is rotating freely about the vertical axis at 40 rev/min with θ = 90°. If the force F which maintains the given position is increased to raise the base collar and
As a check of the basketball before the start of a game, the referee releases the ball from the overhead position shown, and the ball rebounds to about waist level. Determine the coefficient of
Compute the final velocities v1′ and v2′ after collision of the two cylinders which slide on the smooth horizontal shaft. The coefficient of restitution is e = 0.8. v1 = 7 m/s U2 = 1 m/s %3D m1 =
Car B is initially stationary and is struck by car A, which is moving with speed v. The mass of car B is pm, where m is the mass of car A and p is a positive constant. If the coefficient of
The sphere of mass m1 travels with an initial velocity v1 directed as shown and strikes the stationary sphere of mass m2. For a given coefficient of restitution e, what condition on the mass ratio
A tennis ball is projected toward a smooth surface with speed v as shown. Determine the rebound angle θ′ and the final speed v′. The coefficient of restitution is 0.6. 45°
Determine the coefficient of restitution e for a steel ball dropped from rest at a height h above a heavy horizontal steel plate if the height of the second rebound is h2. h h2
Determine the value of the coefficient of restitution e for which the outgoing angle is one-half of the incoming angle θ as shown. Evaluate your general expression for θ = 40°. v' 2
To pass inspection, steel balls designed for use in ball bearings must clear the fixed bar A at the top of their rebound when dropped from rest through the vertical distance H = 36 in. onto the heavy
The cart of mass m1 = 3 kg is moving to the right with a speed v1 = 6 m/s when it collides with the initially stationary barrier of mass m2 = 5 kg. The coefficient of restitution for this collision
If the center of the ping-pong ball is to clear the net as shown, at what height h should the ball be horizontally served? Also determine h2. The coefficient of restitution for the impacts between
Two steel balls of the same diameter are connected by a rigid bar of negligible mass as shown and are dropped in the horizontal position from a height of 150 mm above the heavy steel and brass base
Freight car A of mass mA is rolling to the right when it collides with freight car B of mass mB initially at rest. If the two cars are coupled together at impact, show that the fractional loss of
A small ball is projected horizontally toward an incline as shown. Determine the slant range R. The initial speed is v0 = 16 m /s, and the coefficient of restitution for the impact at A is e = 0.6.
A miniature-golf shot from position A to the hole D is to be accomplished by “banking off” the 45° wall. Using the theory of this article, determine the location x for which the shot can be
The pendulum is released from the 60° position and then strikes the initially stationary cylinder of mass m2 when OA is vertical. Determine the maximum spring compression δ. Use the values m1 = 3
In a pool game the cue ball A must strike the eight ball in the position shown in order to send it to the pocket P with a velocity v2′. The cue ball has a velocity v1 before impact and a velocity
Determine the coefficient of restitution e which will allow the ball to bounce down the steps as shown. The tread and riser dimensions, d and h, respectively, are the same for every step, and the
Sphere A has a mass of 23 kg and a radius of 75 mm, while sphere B has a mass of 4 kg and a radius of 50 mm. If the spheres are traveling initially along the parallel paths with the speeds shown,
During a pregame warmup period, two basketballs collide above the hoop when in the positions shown. Just before impact, ball 1 has a velocity v1 which makes a 30° angle with the horizontal. If
Two identical hockey pucks moving with initial velocities vA and vB collide as shown. If the coefficient of restitution is e = 0.75, determine the velocity (magnitude and direction θ with
The 3000-kg anvil A of the drop forge is mounted on a nest of heavy coil springs having a combined stiffness of 2.8(106) N/m. The 600-kg hammer B falls 500 mm from rest and strikes the anvil, which
The 0.5-kg cylinder A is released from rest from the position shown and drops the distance h1 = 0.6 m. It then collides with the 0.4-kg block B; the coefficient of restitution is e = 0.8. Determine
The elements of a device designed to measure the coefficient of restitution of bat–baseball collisions are shown. The 1-lb “bat” A is a short length of wood or aluminum which is projected to
A child throws a ball from point A with a speed of 50 ft/sec. It strikes the wall at point B and then returns exactly to point A. Determine the necessary angle α if the coefficient of restitution in
The 2-kg sphere is projected horizontally with a velocity of 10 m /s against the 10-kg carriage which is backed up by the spring with stiffness of 1600 N/m. The carriage is initially at rest with the
A small ball is projected horizontally as shown and bounces at point A. Determine the range of initial speed v0 for which the ball will ultimately land on the horizontal surface at B. The coefficient
Calculate the velocity of a spacecraft which orbits the moon in a circular path of 80-km altitude. 80 km
What velocity v must the space shuttle have in order to release the Hubble space telescope in a circular earth orbit 590 km above the surface of the earth? 590 km
Show that the path of the moon is concave toward the sun at the position shown. Assume that the sun, earth, and moon are in the same line. Sunlight Earth Moon
A satellite is in a circular polar orbit of altitude 300 km. Determine the separation d at the equator between the ground tracks (shown dashed) associated with two successive overhead passes of the
Determine the apparent velocity vrel of a satellite moving in a circular equatorial orbit 200 mi above the earth as measured by an observer on the equator (a) For a west-to-east orbit (b)
A spacecraft is in an initial circular orbit with an altitude of 350 km. As it passes point P, onboard thrusters give it a velocity boost of 25 m/s. Determine the resulting altitude gain Δh at point
If the perigee altitude of an earth satellite is 240 km and the apogee altitude is 400 km, compute the eccentricity e of the orbit and the period τ of one complete orbit in space.
Determine the energy difference ΔE between an 80 000-kg space-shuttle orbiter on the launch pad in Cape Canaveral (latitude 28.5°) and the same orbiter in a circular orbit of altitude h = 300 km.
The Mars orbiter for the Viking mission was designed to make one complete trip around the planet in exactly the same time that it takes Mars to revolve once about its own axis. This time is 24 h, 37
A “drag-free” satellite is one which carries a small mass inside a chamber as shown. If the satellite speed decreases because of drag, the mass speed will not, and so the mass moves relative to
Determine the speed v required of an earth satellite at point A for (a) A circular orbit, (b) An elliptical orbit of eccentricity e = 0.1, (c) An elliptical orbit of eccentricity e =
Initially in the 240-km circular orbit, the spacecraft S receives a velocity boost at P which will take it to r →∞ with no speed at that point. Determine the required velocity increment Δv at
If the earth were suddenly deprived of its orbital velocity around the sun, find the time t which it would take for the earth to “fall” to the location of the center of the sun.
Just after launch from the earth, the space-shuttle orbiter is in the 37 × 137-mi orbit shown. The first time that the orbiter passes the apogee A, its two orbital-maneuvering-system (OMS) engines
A spacecraft is in a circular orbit of radius 3R around the moon. At point A, the spacecraft ejects a probe which is designed to arrive at the surface of the moon at point B. Determine the necessary
A projectile is launched from B with a speed of 2000 m/s at an angle α of 30° with the horizontal as shown. Determine the maximum altitude hmax. D UB = 2000 m/s 'max B
Compute the magnitude of the necessary launch velocity at B if the projectile trajectory is to intersect the earth’s surface so that the angle β equals 90°. The altitude at the highest point of
Compute the necessary launch angle α at point B for the trajectory prescribed in Prob. 3/286.Data from problem 286 R/2 B
A particle is launched with a horizontal velocity v0 = 0.55 m/s from the 30° position shown and then slides without friction along the funnel-like surface. Determine the angle θ which its velocity
The central attractive force F on an earth satellite can have no moment about the center O of the earth. For the particular elliptical orbit with major and minor axes as shown, a satellite will have
A pendulum consists of two 3.2-kg concentrated masses positioned as shown on a light but rigid bar. The pendulum is swinging through the vertical position with a clockwise angular velocity ω = 6 rad
A particle moves on the inside surface of a smooth conical shell and is given an initial velocity v0 tangent to the horizontal rim of the surface at A. As the particle slides past point B, a distance
At the point A of closest approach to the sun, a comet has a velocity vA = 188,500 ft /sec. Determine the radial and transverse components of its velocity vB at point B, where the radial distance
A particle of mass m is released from rest in position A and then slides down the smooth vertical-plane track. Determine its angular momentum about both points A and D (a) As it passes position
A wad of clay of mass m1 with an initial horizontal velocity v1 hits and adheres to the massless rigid bar which supports the body of mass m2, which can be assumed to be a particle. The pendulum
The particle of mass m is launched from point O with a horizontal velocity u at time t = 0. Determine its angular momentum HO relative to point O as a function of time. m y L--x
The small spheres, which have the masses and initial velocities shown in the figure, strike and become attached to the spiked ends of the rod, which is freely pivoted at O and is initially at rest.
A particle of mass m moves with negligible friction on a horizontal surface and is connected to a light spring fastened at O. At position A the particle has the velocity vA = 4 m/s. Determine the
The two spheres of equal mass m are able to slide along the horizontal rotating rod. If they are initially latched in position a distance r from the rotating axis with the assembly rotating freely
A 0.7-lb particle is located at the position r1 = 2i + 3j + k ft and has the velocity v1 = i + j + 2k ft/sec at time t = 0. If the particle is acted upon by a single force which has the moment MO =
The 6-kg sphere and 4-kg block (shown in section) are secured to the arm of negligible mass which rotates in the vertical plane about a horizontal axis at O. The 2-kg plug is released from rest at A
A particle with a mass of 4 kg has a position vector in meters given by r = 3t2i − 2tj − 3k, where t is the time in seconds. For t = 5 s determine the angular momentum of the particle and the
The small particle of mass m and its restraining cord are spinning with an angular velocity ω on the horizontal surface of a smooth disk, shown in section. As the force F is slightly relaxed, r
The rigid assembly which consists of light rods and two 1.2-kg spheres rotates freely about a vertical axis. The assembly is initially at rest and then a constant couple M = 2 N∙ m is applied for 5
Just after launch from the earth, the space-shuttle orbiter is in the 37 × 137–mi orbit shown. At the apogee point A, its speed is 17,290 mi/hr. If nothing were done to modify the orbit, what
The assembly starts from rest and reaches an angular speed of 150 rev/min under the action of a 20-N force T applied to the string for t seconds. Determine t. Neglect friction and all masses except
The particle of mass m is gently nudged from the equilibrium position A and subsequently slides along the smooth elliptical path which lies in a vertical plane. Determine the magnitude of its angular
The 3-kg sphere moves in the x-y plane and has the indicated velocity at a particular instant. Determine its (a) Linear momentum, (b) Angular momentum about point O, (c) Kinetic
At a certain instant, the particle of mass m has the position and velocity shown in the figure, and it is acted upon by the force F. Determine its angular momentum about point O and the time rate of
Determine the magnitude HO of the angular momentum of the 2-kg sphere about point O (a) By using the vector definition of angular momentum (b) By using an equivalent scalar approach.
Two barges, each with a displacement (mass) of 500 Mg, are loosely moored in calm water. A stunt driver starts his 1500-kg car from rest at A, drives along the deck, and leaves the end of the 15°
The two mine cars of equal mass are connected by a rope which is initially slack. Car A is given a shove which imparts to it a velocity of 4 ft/sec with car B initially at rest. When the slack is
The wad of clay A is projected as shown at the same instant that cylinder B is released. The two bodies collide and stick together at C and then ultimately strike the horizontal surface at D.
The 80-lb boy has taken a running jump from the upper surface and lands on his 10-lb skateboard with a velocity of 16 ft/sec in the plane of the figure as shown. If his impact with the skateboard has
A tennis player strikes the tennis ball with her racket while the ball is still rising. The ball speed before impact with the racket is v1 = 15 m/s and after impact its speed is v2 = 22 m/s, with
The ballistic pendulum is a simple device to measure projectile velocity v by observing the maximum angle θ to which the box of sand with embedded projectile swings. Calculate the angle θ if the
The 1.2-lb sphere is moving in the horizontal x-y plane with a velocity of 10 ft /sec in the direction shown and encounters a steady flow of air in the x-direction. If the air stream exerts an
The slider of mass m1 = 0.4 kg moves along the smooth support surface with velocity v1 = 5 m/s when in the position shown. After negotiating the curved portion, it moves onto the inclined face of an
A spacecraft in deep space is programmed to increase its speed by a desired amount Δv by burning its engine for a specified time duration t. Twenty-five percent of the way through the burn, the
The baseball is traveling with a horizontal velocity of 85 mi/hr just before impact with the bat. Just after the impact, the velocity of the 5 1/8 -oz ball is 130 mi/hr directed at 35° to the
The ice-hockey puck with a mass of 0.20 kg has a velocity of 12 m/s before being struck by the hockey stick. After the impact the puck moves in the new direction shown with a velocity of 18 m/s. If
The 1.62-oz golf ball is struck by the fi ve-iron and acquires the velocity shown in a time period of 0.001 sec. Determine the magnitude R of the average force exerted by the club on the ball. What
The 2.4-kg particle moves in the horizontal x-y plane and has the velocity shown at time t = 0. If the force F = 2 + 3t2/4 newtons, where t is time in seconds, is applied to the particle in the
Car B is initially stationary and is struck by car A moving with initial speed v1 = 20 mi/hr. The cars become entangled and move together with speed v′ after the collision. If the time duration of
The 100-lb block is stationary at time t = 0, and then it is subjected to the force P shown. Note that the force is zero for all times beyond t = 15 sec. Determine the velocity v of the block at time
The hydraulic braking system for the truck and trailer is set to produce equal braking forces for the two units. If the brakes are applied uniformly for 5 seconds to bring the rig to a stop from a
The space shuttle launches an 800-kg satellite by ejecting it from the cargo bay as shown. The ejection mechanism is activated and is in contact with the satellite for 4 s to give it a velocity of
The pilot of a 90,000-lb airplane which is originally flying horizontally at a speed of 400 mi/hr cuts off all engine power and enters a 5° glide path as shown. After 120 seconds the airspeed is 360
The spring of modulus k = 200 N/m is compressed a distance of 300 mm and suddenly released with the system at rest. Determine the absolute velocities of both masses when the spring is unstretched.
All elements of the previous problem remain unchanged, except that the force P is now held at a constant 30° angle relative to the horizontal. Determine the time ts at which the initially stationary
The initially stationary 20-kg block is subjected to the time-varying horizontal force whose magnitude P is shown in the plot. Note that the force is zero for all times greater than 3 s. Determine
The third and fourth stages of a rocket are coasting in space with a velocity of 18 000 km/h when a small explosive charge between the stages separates them. Immediately after separation the fourth
The 140-g projectile is fired with a velocity of 600 m /s and picks up three washers, each with a mass of 100 g. Find the common velocity v of the projectile and washers. Determine also the loss ΔE|

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