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engineering
engineering mechanics dynamics

Engineering Mechanics - Dynamics 11th Edition R. C. Hibbeler - Solutions

The smooth block B, having mass M, is attached to the vertex A of the right circular cone using a light cord. The cone is rotating at a constant angular rate about the z axis such that the block attains speed v. At this speed, determine the tension in the cord and the reaction which the cone exerts
A ball having a mass M and negligible size moves within a smooth vertical circular slot. If it is released from rest at θ1, determine the force of the slot on the ball when the ball arrives at points A and B. Given: M = 2 kg 8 = 90 deg 01 = 10 deg r = 0.8 m g = 9.81 m S
The box of mass M has a speed v0 when it is at A on the smooth ramp. If the surface is in the shape of a parabola, determine the normal force on the box at the instant x = x1. Also, what is the rate of increase in its speed at this instant? Given: M = 35 kg VO 2 m S = 3 m a = 4 m b 1 1 9 m
The man has mass M and sits a distance d from the center of the rotating platform. Due to the rotation his speed is increased from rest by the rate v'. If the coefficient of static friction between his clothes and the platform is μs, determine the time required to cause him to slip. Given: M = 80
The collar A, having a mass M, is attached to a spring having a stiffness k. When rod BC rotates about the vertical axis, the collar slides outward along the smooth rod DE. If the spring is unstretched when x = 0, determine the constant speed of the collar in order that x = x1. Also, what is the
The block has weight W and it is free to move along the smooth slot in the rotating disk. The spring has stiffness k and an unstretched length δ. Determine the force of the spring on the block and the tangential component of force which the slot exerts on the side of the block, when the block is
A particle having mass M moves along a path defined by the equations r = a + bt, θ = ct2 + d and z = e + ft3. Determine the r, θ, and z components of force which the path exerts on the particle when t = t1. Given: M = 1.5 kg c = 1 rad 2 f = -1 3 S a = 4 m b = 3 d = 2 rad t₁ = 2 s t1 EI m S e =
The boy has weight W and hangs uniformly from the bar. Determine the force in each of his arms at time t = t1 if the bar is moving upward with(a) A constant velocity v0(b) A speed v = bt2 Given: W = 80 lb t1 || 2 s VO = 3 b = 4 ft S ft 3 S
The path of motion of a particle of weight W in the horizontal plane is described in terms of polar coordinates as r = at + b and θ = ct2 + dt. Determine the magnitude of the unbalanced force acting on the particle when t = t1. Given: W = 5 lb d = -1 rad S a = 2 t1 = ft - S 2 s b = 1 ft 8 =
A satellite S travels in a circular orbit around the earth. A rocket is located at the apogee of its elliptical orbit for which the eccentricity is e. Determine the sudden change in speed that must occur at A so that the rocket can enter the satellite’s orbit while in free flight along the blue
The rocket is traveling in free flight along an elliptical trajectory A'A .The planet has no atmosphere, and its mass is k times that of the earth’s. The rocket has an apoapsis and periapsis as shown in the figure. If the rocket is to land on the surface of the planet, determine the required
An asteroid is in an elliptical orbit about the sun such that its perihelion distance is d. If the eccentricity of the orbit is e, determine the aphelion distance of the orbit. Given: d = 9.30 x 10 km e = 0.073
A rocket is in free-flight elliptical orbit around the planet Venus. Knowing that the periapsis and apoapsis of the orbit are rp and ap, respectively, determine(a) The speed of the rocket at point A',(b) The required speed it must attain at A just after braking so that it undergoes a free-flight
The rocket is traveling in a free-flight elliptical orbit about the earth such that the eccentricity is e and its perigee is a distanced d as shown. Determine its speed when it is at point B. Also determine the sudden decrease in speed the rocket must experience at A in order to travel in a
A satellite is in an elliptical orbit around the earth with eccentricity e. If its perigee is hp, determine its velocity at this point and also the distance OB when it is at point B, located at angle θ from perigee as 238 shown. Units Used: Mm Given: e = 0.156 0 = 135 deg hp = 5 Mm G = 66.73 x
The rocket is traveling in free flight along an elliptical trajectory A'A. The planet has no atmosphere, and its mass is k times that of the earth’s. If the rocket has an apoapsis and periapsis as shown in the figure, determine the speed of the rocket when it is at point A. Units used: Mm = 10³
The speed of a satellite launched into a circular orbit about the earth is given by Eq. 13-25. Determine the speed of a satellite launched parallel to the surface of the earth so that it travels in a circular orbit a distance d from the earth’s surface.Equation 13-25 Vc = GMe d
A communications satellite is to be placed into an equatorial circular orbit around the earth so that it always remains directly over a point on the earth’s surface. If this requires the period T (approximately), determine the radius of the orbit and the satellite’s velocity. Units
An elliptical path of a satellite has an eccentricity e. If it has speed vp when it is at perigee, P, determine its speed when it arrives at apogee, A. Also, how far is it from the earth's surface when it is at A? Units Used: Mm = 10 km Given: e = 0.130 Vp 15 = G = 6.673 ×
The rocket is in circular orbit about the earth at altitude h. Determine the minimum increment in speed it must have in order to escape the earth's gravitational field. Given: h = 4106 m G = 66.73 x 10 12 m Me = Re = 6378 km 5.976 5.976 × 1024 kg 3 2 kg.s
Using air pressure, the ball of mass M is forced to move through the tube lying in the vertical plane and having the shape of a logarithmic spiral. If the tangential force exerted on the ball due to the air is F, determine the rate of increase in the ball’s speed at the instant θ = θ1. What
The arm is rotating at the rate θ' when the angular acceleration is θ'' and the angle is θ0. Determine the normal force it must exert on the particle of mass M if the particle is confined to move along the slotted path defined by the horizontal hyperbolic spiral rθ = b. Given: 8 = 5 Ꮎ 8' Ꮎ'
The pilot of an airplane executes a vertical loop which in part follows the path of a “four-leaved rose,” r = a cos 2θ . If his speed at A is a constant vp, determine the vertical reaction the seat of the plane exerts on the pilot when the plane is at A. His weight is W. Given: a = -600 ft Vp
The collar has mass M and travels along the smooth horizontal rod defined by the equiangular spiral r = aeθ. Determine the tangential force F and the normal force NC acting on the collar when θ = θ1 if the force F maintains a constant angular motion θ'. Given: M = 2 kg a = 1 m 01 = 90
Using a forked rod, a smooth cylinder C having a mass M is forced to move along the vertical slotted path r = aθ. If the angular position of the arm is θ = bt2, determine the force of the rod on the cylinder and the normal force of the slot on the cylinder at the instant t. The cylinder is in
The ball has mass M and a negligible size. It is originally traveling around the horizontal circular path of radius r0 such that the angular rate of rotation is θ'0. If the attached cord ABC is drawn down through the hole at constant speed v, determine the tension the cord exerts on the ball at
The forked rod is used to move the smooth particle of weight W around the horizontal path in the shape of a limacon r = a + bcosθ. If θ = ct2, determine the force which the rod exerts on the particle at the instant t = t1. The fork and path contact the particle on only one side. Given: W = 2 lb a
The smooth particle has mass M. It is attached to an elastic cord extending from O to P and due to the slotted arm guide moves along the horizontal circular path r = b sin θ. If the cord has stiffness k and unstretched length δ determine the force of the guide on the particle when θ = θ1. The
Determine the normal and frictional driving forces that the partial spiral track exerts on the motorcycle of mass M at the instant θ, θ', and θ''. Neglect the size of the motorcycle. Units Used: KN = 10³ N Given: M = 200 kg b = 5 m 0 = Ө 5 π 3 8 = 0.4 Ꮎ' = 0.8 rad rad S rad 2 S
The smooth particle has mass M. It is attached to an elastic cord extending from O to P and due to the slotted arm guide moves along the horizontal circular path r = b sin θ. If the cord has stiffness k and unstretched length δ determine the force of the guide on the particle when θ = θ1. The
The collar of weight W slides along the smooth vertical spiral rod r = bθ, where θ is in radians. If its angular rate of rotation θ' is constant, determine the tangential force P needed to cause the motion and the normal force that the rod exerts on the spool at the instant θ = θ1. Given: W =
The collar of weight W slides along the smooth horizontal spiral rod r = bθ, where θ is in radians. If its angular rate of rotation θ' is constant, determine the tangential force P needed to cause the motion and the normal force that the rod exerts on the spool at the instant θ = θ1. Given: W
A smooth can C, having a mass M, is lifted from a feed at A to a ramp at B by a rotating rod. If the rod maintains a constant angular velocity of θ', determine the force which the rod exerts on the can at the instant θ = θ1. Neglect the effects of friction in the calculation and the size of the
The particle has mass M and is confined to move along the smooth horizontal slot due to the rotation of the arm OA. Determine the force of the rod on the particle and the normal force of the slot on the particle when θ = θ1. The rod is rotating with a constant angular velocity θ'. Assume the
The particle has mass M and is confined to move along the smooth horizontal slot due to the rotation of the arm OA. Determine the force of the rod on the particle and the normal force of the slot on the particle when θ = θ1. The rod is rotating with angular velocity θ' and angular acceleration
The particle of weight W is guided along the circular path using the slotted arm guide. If the arm has angular velocity θ' and angular acceleration θ'' at the instant θ = θ1, determine the force of the guide on the particle. Motion occurs in the horizontal plane. Given: W = 0.5 lb rad 8 = 4 ' =
The spring-held follower AB has weight W and moves back and forth as its end rolls on the contoured surface of the cam, where the radius is r and z = asin(2θ). If the cam is rotating at a constant rate θ', determine the force at the end A of the follower when θ = θ1. In this position the spring
The spring-held follower AB has weight W and moves back and forth as its end rolls on the contoured surface of the cam, where the radius is r and z = a sin(2θ). If the cam is rotating at a constant rate of θ', determine the maximum and minimum force the follower exerts on the cam if the spring is
The spool of mass M slides along the rotating rod. At the instant shown, the angular rate of rotation of the rod is θ', which is increasing at θ''. At this same instant, the spool is moving outward along the rod at r' which is increasing at r'' at r. Determine the radial frictional force and the
The girl has a mass M. She is seated on the horse of the merry-go-round which undergoes constant rotational motion θ'. If the path of the horse is defined by r = r0, z = b sin(θ), determine the maximum and minimum force Fz the horse exerts on her during the motion. Given: M = 50 kg Ꮎ =
Ball A is released from rest at height h1 at the same time that a second ball B is thrown upward from a distance h2 above the ground. If the balls pass one another at a height h3 determine the speed at which ball B was thrown upward. Given: h1 h2 = 5 ft h3 = 20 ft Do = 40 ft = 32.2 ft 2 S
A car starts from rest and moves along a straight line with an acceleration a = k s−1/3. Determine the car’s velocity and position at t = t1.Given: $9 = [₁ 3 S 6 m E = Y
The acceleration of a rocket traveling upward is given by ap = b + c sp. Determine the rocket’s velocity when sp = sp1 and the time needed to reach this altitude. Initially, vp = 0 and sp = 0 when t = 0. Given: b = 6 m 2 S C = 0.02 2 S Spl = 2000 m
A particle has an initial speed v0. If it experiences a deceleration a = bt, determine its velocity when it travels a distance s1. How much time does this take? Given: vo Vo = 27 S b = -6 m 3 S $1 = 10 m
A stone A is dropped from rest down a well, and at time t1 another stone B is dropped from rest. Determine the time interval between the instant A strikes the water and the instant B strikes the water. Also, at what speed do they strike the water? Given: d = 80 ft t1 = 1 s 8 = 32.2 2 S
A particle is moving with velocity v0 when s = 0 and t = 0. If it is subjected to a deceleration of a −kv3= , where k is a constant, determine its velocity and position as functions of time.
A truck traveling along a straight road at speed v1, increases its speed to v2 in time t. If its acceleration is constant, determine the distance traveled. Given: VI = 20 km hr V2 = 120 km hr t = 15 s
The position of a particle along a straight line is given by sp = at3 + bt2 + ct. Determine its maximum acceleration and maximum velocity during the time interval t0 ≤ t ≤ tf. Given: a = 1 ft b = -9 c = 15 ft S to = 0 s tf = 10 s
A baseball is thrown downward from a tower of height h with an initial speed v0. Determine the speed at which it hits the ground and the time of travel. Given: h = 50 ft 8 = 32.2 2 S Vo ft = 18. S
Starting from rest, a particle moving in a straight line has an acceleration of a = (bt + c). What is the particle’s velocity at t1 and what is its position at t2? Given: b = 2 w|B 3 S c = -6 2 S t₁ = 6 s t2 = 11 s
The acceleration of a particle as it moves along a straight line is given by a = bt + c. If s = s0 and v = v0 when t = 0, determine the particle’s velocity and position when t = t1. Also, determine the total distance the particle travels during this time period. Given: b = 2 m 3 S c = -1 m 2 S S0
A car starts from rest and reaches a speed v after traveling a distance d along a straight road. Determine its constant acceleration and the time of travel. Given: V = 80 ft S d = 500 ft
A particle moves along a straight line such that its position is defined by sp = at3 + bt2 + c. Determine the average velocity, the average speed, and the acceleration of the particle at time t1. Given: 11 1 m b = -3 5.3 S c = 2 m to = 0 s t₁ = 4 s
The velocity of a particle traveling in a straight line is given v = bt + ct2. If s = 0 when t = 0, determine the particle’s deceleration and position when t = t1. How far has the particle traveled during the time t1, and what is its average speed? Given: m b = 6- 2 S c = -3 m 3 S to = 0 s t1 = 3
A particle is moving along a straight line such that its acceleration is defined as a = −kv. If v = v0 when d = 0 and t = 0, determine the particle’s velocity as a function of position and the distance the particle moves before it stops. Given: k = 2 S Vo = 20 EI m S
Traveling with an initial speed v0 a car accelerates at rate a along a straight road. How long will it take to reach a speed vf ? Also, through what distance does the car travel during this time? VO = 70 km hr a 6000 km 2 hr vf = 120 km hr
The acceleration of a rocket traveling upward is given by ap = b + c sp. Determine the time needed for the rocket to reach an altitute sp1. Initially, vp = 0 and sp = 0 when t = 0. Given: b = 6 m 2 S C = 0.02 -| 2 S Spl = 100 m
The elevator starts from rest at the first floor of the building. It can accelerate at rate a1 and then decelerate at rate a2. Determine the shortest time it takes to reach a floor a distance d above the ground. The elevator starts from rest and then stops. Draw the a-t, v-t, and s-t graphs for the
A freight train travels at v = v0 (1− e−bt) where t is the elapsed time. Determine the distance traveled in time t1, and the acceleration at this time. Given: VO = 60 = S t1 = 3 s S
From approximately what floor of a building must a car be dropped from an at-rest position so that it reaches a speed vf when it hits the ground? Each floor is a distance h higher than the one below it. Given: vf = 55 mph h = 12 ft 8 = 32.2 2 S
A car is to be hoisted by elevator to the fourth floor of a parking garage, which is at a height h above the ground. If the elevator can accelerate at a1, decelerate at a2, and reach a maximum speed v, determine the shortest time to make the lift, starting from rest and ending at rest. Given: h =
A particle moves along a straight line such that its position is defined by s = bt2 + ct + d. Determine the average velocity, the average speed, and the acceleration of the particlewhen t = t1. Given: m b = 1 m c = -6 - S d = 5 m to = 0 s t1 = 6 s
A particle is moving along a straight line such that when it is at the origin it has a velocity v0. If it begins to decelerate at the rate a = bv1/2 determine the particle’s position and velocity when t = t1. Given: + = 0₁ m S b = -1.5 m 3 S t1 = 2 s = b√v a(v) =
A particle, initially at the origin, moves along a straight line through a fluid medium such that its velocity is defined as v = b(1− e−ct). Determine the displacement of the particle during the time 0 1. Given: m b = 1.8- 1 S С 0.3 S t1 = 3 s
A stone A is dropped from rest down a well, and at time t1 another stone B is dropped from rest. Determine the distance between the stones at a later time t2. Given: d = 80 ft t₁ = 1 s t2 = 2 s 8 = 32.2 2 S
A particle travels to the right along a straight line with a velocity vp = a / (b + sp). Determine its deceleration when sp = sp1. Given: a = 5 m S b = 4 m Spl = 2 m
Two particles A and B start from rest at the origin s = 0 and move along a straight line such that aA = (at − b) and aB = (ct2 − d), where t is in seconds. Determine the distance between them at t and the total distance each has traveled in time t. Given: a=6 ft - 3 b = 3 ft c = 12 d = 8 ft t =
A car can have an acceleration and a deceleration a. If it starts from rest, and can have a maximum speed v, determine the shortest time it can travel a distance d at which point it stops. Given: a = :5 m 2 S V = 60 m S d = 1200 m
An airplane starts from rest, travels a distance d down a runway, and after uniform acceleration, takes off with a speed vr It then climbs in a straight line with a uniform acceleration aa until it reaches a constant speed va. Draw the s-t, v-t, and a-t graphs that describe the motion. Given: d =
A particle has an initial speed v0. If it experiences a deceleration a = bt, determine the distance traveled before it stops. Given: v0 = 27 Vo m S b = -6 m 3 S
The acceleration of a particle along a straight line is defined by ap = b t + c. At t = 0, sp = sp0 and vp = vp0. When t = t1 determine(a) The particle's position,(b) The total distance traveled,(c) The velocity. Given: m b = 2. c = -9 m Sp0 = 1 m Vp0 10 = ES m t1 = 9 s
The v-t graph for a particle moving through an electric field from one plate to another has the shape shown in the figure, where tf and vmax are given. Draw the s-t and a-t graphs for the particle. When t = tf /2 the particle is at s = sc. Given: tf If = 0.2 s m S Vmax = 10- Sc = 0.5 m
A particle is moving along a straight line such that its acceleration is defined as a = k s2. If v = v0 when s = sp0 and t = 0, determine the particle’s velocity as a function of position. Given: k = 4 -|~ 2 ms VO = -100 m S Sp0 = 10 m
If the position of a particle is defined as s = bt + ct2, construct the s–t, v–t, and a–t graphs for 0 ≤ t ≤ T. Given: b = 5 ft c = -3 ft T = 10 s t = 0,0.017.. T
Determine the time required for a car to travel a distance d along a road if the car starts from rest, reaches a maximum speed at some intermediate point, and then stops at the end of the road. The car can accelerate at a1 and decelerate at a2. Given:Let t1 be the time at which it stops
The v-t graph for a particle moving through an electric field from one plate to another has the shape shown in the figure. The acceleration and deceleration that occur are constant and both have a magnitude a. If the plates are spaced smax apart, determine the maximum velocity vmax and the time
When two cars A and B are next to one another, they are traveling in the same direction with speeds vA0 and vB0 respectively. If B maintains its constant speed, while A begins to decelerate at the rate aA, determine the distance d between the cars at the instant A stops. A B
A car travels along a straight road with the speed shown by the v–t graph. Determine the total distance the car travels until it stops at t2. Also plot the s–t and a–t graphs. Given: t₁ = 30 s t1 12 = 48 s VO = 6 m S
If the effects of atmospheric resistance are accounted for, a freely falling body has an acceleration defined by the equation a = g(1− cv2), where the positive direction is downward. If the body is released from rest at a very high altitude, determine(a) The velocity at time t1(b) The body’s
A motorcycle starts from rest at s = 0 and travels along a straight road with the speed shown by the v-t graph. Determine the motorcycle's acceleration and position when t = t4 and t = t5. Given: v0 = 5 m S t1 = 4 s 12 = 10 s 13 = 15 s 14 = 8 s 15 = 12 s
From experimental data, the motion of a jet plane while traveling along a runway is defined by the v–t graph shown. Construct the s-t and a-t graphs for the motion. Given: VI = 80 El t1 = 10 s 12 = 40 s S
A car starting from rest moves along a straight track with an acceleration as shown. Determine the time t for the car to reach speed v. Given: v = 50 a1 = 8 a] m S m 2 S t1 = 10 s
If the position of a particle is defined by sp = b sin(ct) + d, construct the s-t, v-t, and a-t graphs for 0 ≤ t ≤ T. Given: b = 2 m 1 5 S C =- d = 4 m T = 10 s t = 0,0.017..T
Accounting for the variation of gravitational acceleration a with respect to altitude y (see Prob. 12-34), derive an equation that relates the velocity of a freely falling particle to its altitude. Assume that the particle is released from rest at an altitude y0 from the earth’s surface. With
As a body is projected to a high altitude above the earth ’s surface, the variation of the acceleration of gravity with respect to altitude y must be taken into account. Neglecting air resistance, this acceleration is determined from the formula a = −g[R2/(R+y)2], where g is the constant
When a particle falls through the air, its initial acceleration a = g diminishes until it is zero, and thereafter it falls at a constant or terminal velocity vf. If this variation of the acceleration can be expressed as a = (g/vf2)(v2f − v2), determine the time needed for the velocity to become v
The v–t graph for the motion of a train as it moves from station A to station B is shown. Draw the a–t graph and determine the average speed and the distance between the stations. Given: t1 = 30 s 12 = 90 s 13 = 120 s V1 = 40 ft S
The a–t graph for a motorcycle traveling along a straight road has been estimated as shown. Determine the time needed for the motorcycle to reach a maximum speed vmax and the distance traveled in this time. Draw the v–t and s–t graphs. The motorcycle starts from rest at s = 0. Given: t1 = 10
The v-t graph for the motion of a car as if moves along a straight road is shown. Draw the a-t graph and determine the maximum acceleration during the time interval 0 2. The car starts from rest at s = 0. Given: t1 = 10 s 12 = 30 s V1 = 40 VI V2 = 60 S ft S
Two cars start from rest side by side and travel along a straight road. Car A accelerates at the rate aA for a time t1, and then maintains a constant speed. Car B accelerates at the rate aB until reaching a constant speed vB and then maintains this speed. Construct the a-t, v-t, and s-t graphs for
The a–s graph for a boat moving along a straight path is given. If the boat starts at s = 0 when v = 0, determine its speed when it is at s = s2, and s3, respectively. Use Simpson’s rule with n to evaluate v at s = s3. Given: a] = 5 a2 = 6 ft 2 S ft ` 2 S $1 = 100 ft $2 = 75 ft $3 = 125 ft b =
The s–t graph for a train has been experimentally determined. From the data, construct the v–t and a–t graphs for the motion; 0 ≤ t ≤ t2. For 0 ≤ t ≤ t1, the curve is a parabola, and then it becomes straight for t ≥ t1. Given: t₁ = 30 s 12 = 40 s $1 = 360 m : 600 m $2 =
A two-stage rocket is fired vertically from rest at s = 0 with an acceleration as shown. After time t1 the first stage A burns out and the second stage B ignites. Plot the v-t and s-t graphs which describe the motion of the second stage for 0 2. Given: t1 = 30 s t2 = 60s al = 9 m 2 S a2 = 15 2 S
The v-t graph for the motion of a car as it moves along a straight road is shown. Draw the s-t graph and determine the average speed and the distance traveled for the time interval 0 2. The car starts from rest at s = 0. Given: t1 = 10 s 12 = = 30 s VI = 40 ft S ft S v2 = 60-
A motorcyclist at A is traveling at speed v1 when he wishes to pass the truck T which is traveling at a constant speed v2. To do so the motorcyclist accelerates at rate a until reaching a maximum speed v3. If he then maintains this speed, determine the time needed for him to reach a point located a
The v-s graph for a go-cart traveling on a straight road is shown. Determine the acceleration of the go-cart at s3 and s4. Draw the a-s graph. Given: VI = 8 m S $1 = 100 m $2 = 200 m $3 = 50 m $4 = 150 m
The a–t graph for a car is shown. Construct the v–t and s–t graphs if the car starts from rest at t = 0. At what time t' does the car stop? Given: aj || a2 = จ Ele t] = 10 s
The a-s graph for a train traveling along a straight track is given for 0 ≤ s ≤ s2. Plot the v-s graph. v = 0 at s = 0.Given: $1 = 200 m $2 = 400 m a₁ = 2 m 2 S
The v-s graph for an airplane traveling on a straight runway is shown. Determine the acceleration of the plane at s = s3 and s = s4. Draw the a-s graph. Given: s] = 100 m s4 = 150 m $2 = 200 m vj $3 = 50 m m 40- S V1 = 40 m S v2 V2 = 50-
The jet plane starts from rest at s = 0 and is subjected to the acceleration shown. Determine the speed of the plane when it has traveled a distance d. Also, how much time is required for it to travel the distance d? Given: d = 200 ft a0 = 75 ao ft 2 S S] = 500 ft

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