Particle Kinetics Homework - Purdue Engineeringjfrhoads/me274/H4_fall_2016_pwl.pdf · Given: At the...

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Chapter 4: Particle Kinetics Homework Chapter 4 Particle Kinetics Homework Freeform c 2016 4-1

Transcript of Particle Kinetics Homework - Purdue Engineeringjfrhoads/me274/H4_fall_2016_pwl.pdf · Given: At the...

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Chapter 4: Particle Kinetics Homework

Chapter 4

Particle Kinetics Homework

Freeform c©2016 4-1

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Chapter 4: Particle Kinetics Homework

4-2 Freeform c©2016

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Chapter 4: Particle Kinetics Homework

Homework 4.A

Given: At the instant shown, a projectile (of mass m) is moving along a path with the tangentto the path at an angle of θ with respect to the horizontal. In addition to the weight force, a dragforce FD = bv2 acts on the particle that opposes its motion.

Find: Determine the rate of change of speed of the particle and the radius of curvature of its pathat the instant shown.

g

FD

v!

F

!v

smooth fixed peg

R

HORIZONTAL PLANE

P

eR

e!

30°

Use the following parameters in your analysis: m = 5 kg, v = 40 m/s, b = 0.2 Ns2/m2 andθ = 36.87◦.

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Chapter 4: Particle Kinetics Homework

Homework 4.B

Given: Particle P (of mass m) moves through a smooth, fixed slot in a horizontal plane with aconstant speed of v. The shape of the slot is given in Cartesian coordinates as y = x2/2, where xand y are in meters.

Find: Determine the reaction force on P due to the smooth slot at x = 0 m.

g

B

m

m

µk

smooth pulley

A

x v

P

y = x2 / 2

HORIZONTAL PLANE

y

Use the following parameters in your analysis: m = 4 kg and v = 30 m/s.

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Chapter 4: Particle Kinetics Homework

Homework 4.C

Given: The mechanism, shown below, consisting of a rotating disk, frictionless slot, cable, andblock of mass m, is driven at a constant speed ω in the horizontal plane by a motor attached atpoint O.

Find: Determine:(a) The tension in the cable;(b) The normal force exerted on the block.

Particle Kinetics Homework Problems ME 274

Problem IV-5 The mechanism, shown below, consisting of a rotating disk, frictionless slot, cable, and block of mass m, is driven at a constant speed ! in the horizontal plane by a motor attached at point O. Determine:

(a) the tension in the cable (b) the normal force exerted on the block

Use the following: ! = 10 rad / sec , h = 0.3m , d = 0.4 m, and m = 4 kg.

HORIZONTAL PLANE

!

O

r

dh

m

!

O

r

dh

m

Use the following parameters in your analysis: ω = 12 rad/s, h = 0.4 m, d = 0.6 m and m = 10 kg.

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Chapter 4: Particle Kinetics Homework

Homework 4.D

Given: An arm rotates about a vertical axis passing through O at a rate of ω with this rotationchanging at a rate of ω. Block C, having a mass of m, slides smoothly over the arm at a rate of rwith this sliding motion changing at a rate of r.

Find: At this instant, determine:(a) The tension force in the cord;(b) The normal force of the arm on block C;(c) Which side of the arm (A or B) that the block makes contact.

𝑟

A

B

stringO

C

Use the following parameters in your analysis: m = 7 kg, ω = 4 rad/s, ω = -1 rad/s2, r = 0.25 m,r = -0.5 m/s and r = 0 m/s2.

4-6 Freeform c©2016

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Chapter 4: Particle Kinetics Homework

Homework 4.E

Given: Particle P of mass m slides in the direction shown within a tube with a speed of vrelrelative to the tube as the tube rotates in the CCW sense with an angular speed of ω and angularacceleration α.

Find: For this problem:

(a) Determine the acceleration of P;

(b) Determine the friction force acting on P.

Particle Kinetics Homework Problems ME 274

Problem IV-56 Given: Particle P of mass m slides in the direction shown within a tube with a speed of

vrel relative to the tube as the tube rotates in the CCW sense with an angular speed of ! and angular acceleration ! .

Find: For this problem:

a) Determine the acceleration of P.

b) Determine the friction force acting on P. Use the following parameters in your analysis: m = 10 kg , ! = 5rad / sec ,

! = 2rad / sec2 , vrel = 4m / sec , µs = 0.6 , µs = 0.3 , L = 0.4meters and

h = 0.2meters .

HORIZONTAL PLANE

µk ,µs vrel

O !,"

L

P

y

x

h

Use the following parameters in your analysis: m = 10 kg, ω = 5 rad/s, α = 2 rad/s2, vrel = 4m/s, µs = 0.6, µk = 0.3, L = 0.4 m and h = 0.2 m.

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Chapter 4: Particle Kinetics Homework

Homework 4.F

Given: The system shown is released from rest.

Find: On release, determine:(a) The acceleration of each block;(b) The tension in the cable.

Am

Bm

, ks

Use the following parameters in your analysis: µs = 0.30, µk = 0.2, mA = 3 kg, mB = 4 kg, andθ = 36.87◦.

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Chapter 4: Particle Kinetics Homework

Homework 4.G

Given: The system shown is released from rest. Assume all surfaces to be smooth and that themass of the pulleys is negligible.

Find: Determine the speed of both masses after B has moved a distance of d.

AmBm

O

Use the following parameters in your analysis: θ = 20◦, mA = 6 kg, mB = 14 kg and d = 0.2 m.

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Chapter 4: Particle Kinetics Homework

Homework 4.H

Given: A block with mass m is attached to a pulley system. The other end of the cable is attachedto a spring with stiffness k, as shown in the diagram. The spring is initially unstretched when theblock is at a distance h from the ground. The system is released from rest. Assume all pulleys aremassless.

Find: Determine the velocity of the block when it hits the ground.

Particle Kinetics Homework Problems ME 274

Problem IV-35 A block with mass m is attached to a pulley system. The other end of the cable is attached to a spring with stiffness k, as shown in the diagram. The spring is initially unstretched when the block is at a distance h from the ground. If the system is released from rest, determine the velocity of the block when it hits the ground. Assume all pulleys are massless. Use the following parameters for your analysis: m = 150 kg, k = 4 N/m, h = 1 m.

m

h

g

k

Use the following parameters in your analysis: m = 200 kg, k = 3 N/m and h = 0.75 m.

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Chapter 4: Particle Kinetics Homework

Homework 4.I

Given: Particles A and B (having masses of m and 2m, respectively) are connected by the cable-pulley system shown. Particle A is constrained to move along a horizontal guide. A constant forceF acts to the right on particle A. When A is at a position of sA1 = 0 m, it is given an initialspeed of vA1 to the right. Assume that the cable remains taut at all times and that all surfaces aresmooth.

Find: Determine the speeds of A and B when A is at the position sA2.

sA

sB

h B

m

A

g

2m

F

Use the following parameters in your analysis: m = 5 kg, F = 50 N, h = 2 m, sA2 = 1.5 m andvA1 = 10 m/s.

Freeform c©2016 4-11

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Chapter 4: Particle Kinetics Homework

Homework 4.J

Given: Particles A and B (having masses of m and 2m, respectively) are attached to a lightweightrigid bar as shown in the figure. A constant horizontal force F acts on particle A. At the initialstate of θ = 0◦, the system is at rest.

Find: Determine the angular speed of the bar as a function of θ and in terms of the parameters ofthe problem.

g B

A

F

L / 2

L / 2

O

!

initial position

2m

m

4-12 Freeform c©2016

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Chapter 4: Particle Kinetics Homework

Homework 4.K

Given: Pellets A and B (having masses m and 3m, respectively) are placed within a smooth tubetrapping a small compartment of fuel. At a time when the pellets are initially at rest, the fuel isignited. The combustion occurs over a short time ∆t, and over this time the combustion appliesequal and opposite forces on the pellets with this force idealized by the force time history F (t)shown below.

Find: Determine the speed of each pellet at time t2, where t2 > ∆t.

Particle Kinetics Homework Problems ME 274

Problem IV-11 Pellets A and B (having masses m and 3m, respectively) are placed within a smooth tube trapping a small compartment of fuel. At a time when the pellets are initially at rest, the fuel is ignited. The combustion occurs over a short time !t, and over this time the combustion applies equal and opposite forces on the pellets with this force idealized by the force time history F(t) shown below.

a) Draw individual free body diagrams (FBD’s) for pellets A and B. From these FBD’s, determine the speed of each pellet at a time t2, where t2 > !t.

b) Draw an FBD for pellets A and B together (i.e., treat A and B as a single system). From this FBD, provide an argument that linear momentum for the system of A and B together should be conserved. Use your results from a) above to verify this argument.

Use the following parameters in your analysis: F0 = 2000 N , ! t = 0.006 s and m = 0.5 kg .

0 !t t

F(t)

F0 A B

m 3m

smooth smooth

combusting fuel

Use the following parameters in your analysis: F0 = 3000 N, ∆t = 0.005 s and m = 0.75 kg.

Freeform c©2016 4-13

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Chapter 4: Particle Kinetics Homework

Homework 4.L

Given: Block B (of mass 2m) is able to slide along a smooth horizontal surface. Block A (of massm) is able to slide along the rough top surface of block B, as shown in the figure. Initially, A istraveling to the right with a speed of vA1, and block B is traveling to the left with a speed of vB1.

Find: Determine the velocity of block B when block A has to come rest relative to block B.

Particle Kinetics Homework Problems ME 274

Problem IV-55 Given: Block B (of mass 2m) is able to slide along a smooth horizontal surface. Block

A (of mass m) is able to slide along the rough top surface of block B, as shown in the figure. Initially, A is traveling to the right with a speed of vA1 , and block

B is traveling to the left with a speed of vB1 .

Find: What is the velocity of block B when block A has to come rest relative to block

B? Write your answer as a vector. Use the following parameters in your analysis: m = 25kg , vA1 = 26meters / sec and vB1 = 10meters / sec .

HINT: Consider applying the linear impulse momentum equation to the system

made up of A and B together.

smooth

m vA1 A

B 2m vB1 rough

4-14 Freeform c©2016

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Chapter 4: Particle Kinetics Homework

Homework 4.M

Given: A smooth, circular slot is cut into block A with block A being constrained to move alonga smooth, horizontal surface. The slot is vertical at the top surface of the block with the slot beinghorizontal at the right edge of the block, as shown in the figures below. In Position 1, block A isstationary, and a particle B is released from rest into the upper opening of the slot. At Position 2shown below, particle B is exiting the slot at the right edge of the block. The masses of A and Bare M and m, respectively. The radius of the circular slot is r.

Find: Determine the velocity of block A and the velocity of particle B at position 2.

Particle Kinetics Homework Problems ME 274

Problem IV-50 Given: A smooth, circular slot is cut into block A with block A being constrained to

move along a smooth, horizontal surface. The slot is vertical at the top surface of the block with the slot being horizontal at the right edge of the block, as shown in the figures below. In Position 1, block A is stationary, and a particle B is released from rest into the upper opening of the slot. At Position 2 shown below, particle B is exiting the slot at the right edge of the block. The masses of A and B are M = 40 kg and m = 20 kg, respectively. The radius of the circular slot is given by r = 2 meters.

Find: Determine the velocity of block A and the velocity of particle B at position 2. You must provide an accurate free body diagram of the system used in your solution in order to receive full credit for your work. Consider using both the linear impulse-momentum and work-energy equations in solving this problem.

r

B

A B

A VB2

VA2

g

POSITION 1 POSITION 2

M

m

Use the following parameters in your analysis: m = 20 kg, M = 40 kg and r = 2 m.

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Chapter 4: Particle Kinetics Homework

Homework 4.N

Given: At the instant the cable fails, the crate of mass m is traveling up the slope with a speedv0. The coefficient of kinetic friction between the crate and the plane is µk.

Find: Determine the speed of the crate, v, at a time of ∆t after the cable fails.

0v

B

k

Use the following parameters in your analysis: m = 100 kg, v0 = 16 m/s, µk = 0.25, θ = 30◦ and∆t = 0.5 s.

4-16 Freeform c©2016

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Chapter 4: Particle Kinetics Homework

Homework 4.O

Given: A particle of mass m is projected horizontally to the right at a height h above a smooth,horizontal floor with a speed of v0. The particle strikes the floor at a horizontal distance d fromwhere it was initially projected. The coefficient of restitution of the impact of the particle with thefloor is e.

Find: For this problem:(a) Determine the angle θ1 that the velocity of the particle has with the horizontal immediately

before impact;(b) Determine the angle θ2 that the velocity of the particle has with the horizontal immediately

after impact.

Particle Kinetics Homework Problems ME 274

Problem IV-15 A particle of mass m is projected horizontally to the right at a height h above a smooth, horizontal floor with a speed of v0. The particle strikes the floor at a horizontal distance d from where it was initially projected. The coefficient of restitution of the impact of the particle with the floor is e.

a) Determine the angle !1 that the velocity of the particle has with the horizontal before impact. Use conservation of energy prior to impact.

b) Determine the angle !2 that the velocity of the particle has with the horizontal immediately after impact. Use conservation of linear momentum in the horizontal direction along with the coefficient of restitution equation during impact.

Use the following parameters in your analysis: e = 0.6 , h = 125 ft and v0 = 150 ft / sec .

v0

v1

!1

v2

!2

BEFORE impact immediately AFTER impact

h

d

Use the following parameters in your analysis: e = 0.5, h = 150 ft and v0 = 160 ft/s.

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Chapter 4: Particle Kinetics Homework

Homework 4.P

Given: Spheres A and B (having masses of 2m and m, respectively) are able to move on a smoothHORIZONTAL surface. Sphere A is given an initial velocity of vA1, as shown below. Sphere Aimpacts sphere B at exactly the same instant that A also impacts a smooth bumper, with the lineof impact of A and B being parallel to the bumper. The coefficient of restitution between A andthe bumper is e = 0.4, and the coefficient of restitution between spheres A and B is e = 1.

Find: Determine the velocities of spheres A and B after the impacts described above.

Particle Kinetics Homework Problems ME 274

Problem IV-43 Given: Spheres A and B (having masses of 2m and m, respectively) are able to move

on a smooth HORIZONTAL surface. Sphere A is given an initial velocity of vA1 = 10 m/sec, as shown below. Sphere A impacts sphere B at exactly the same instant that A also impacts a smooth bumper, with the line of impact of A and B being parallel to the bumper. The coefficient of restitution between A and the bumper is e = 0.4, and the coefficient of restitution between spheres A and B is e = 1.

Find: The velocities of spheres A and B after the impacts described above. Include

in your solution a free body diagram of each sphere individually, as wheel as a free body diagram of the two spheres together.

Write your final answers as vectors.

v A1

A

B

!A1

e = 0.4

e = 1

m

2m

Use the following parameters in your analysis: vA1 = 10 m/s and θA1 = 36.87◦.

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Chapter 4: Particle Kinetics Homework

Homework 4.Q

Given: A bullet strikes the lower particle of a stationary pendulum with a speed of v. Afterimpact, the bullet sticks to the particle.

Find: Determine:(a) The angular speed of the pendulum immediately after impact;(b) The maximum rotation angle through which the pendulum swings after impact.

20m

M

Mv

400 mm

200 mm

O

Use the following parameters in your analysis: v = 100 m/s, m = 50 g and M = 4 kg.

Freeform c©2016 4-19

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Chapter 4: Particle Kinetics Homework

Homework 4.R

Given: The rod shown below is of negligible mass and is subjected to a couple moment of M =(30t2) Nm. The car’s engine provide a traction force of F = 15t N. The car starts from rest andthe total mass of the car and rider is m.

Find: Determine the speed of the car at time t .

4m

2(30 ) N mM t

(15 ) NF t

Use the following parameters in your analysis: m = 175 kg and t = 8 s.

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Chapter 4: Particle Kinetics Homework

Homework 4.S

Given: Particles A and B (each having a mass of m) are connected by rod AB, with rod AB beingpinned to a vertical shaft. With the shaft rotating with a rate of ω1 and θ = θ1, and with A andB being stationary respect to the shaft, bar AB is released. Consider the mass of rod AB to benegligible.

Find: Determine the angular speed of the shaft when θ = θ2.

O

Bm

smooth bearings m

L

2L

θ

ωsmooth bearings

g

A

Use the following parameters in your analysis: m = 10 kg, L = 0.25 m, ω1 = 8 rad/s, θ1 = 30◦ andθ2 = 90◦.

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Chapter 4: Particle Kinetics Homework

Homework 4.T

Given: Particle P, having a mass of m, is able to slide on the smooth, horizontal top of a table. Aflexible cable is attached to P, with the cable being fed through a hole in the table at O. A constantforce F acts on the other end of the cable. The system is released with P being at a radial distanceR = R1 from O, and with P having a velocity perpendicular to OP with a speed of vP1.

Find: Determine the numerical values for R and φ when P has moved to a position for whichR = R2.

O

R

P vP1

F

φm

Use the following parameters in your analysis: m = 2 kg, R1 = 1.5 m, R2 = 0.5 m, vP1 = 5 m/sand F = 236 N.

4-22 Freeform c©2016