Particle Kinetics Homework - Purdue University · Chapter 4: Particle Kinetics Homework Chapter 4...

Chapter 4: Particle Kinetics Homework

Chapter 4

Particle Kinetics Homework

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Homework H.4.A

Given: Particle P (of mass m) moves along a rough surface (with a coefficient of kinetic frictionof µk between P and the surface). At one instant when P is at a position on the path that hasa radius of curvature of ρ, P has a speed of v with the velocity vector for P being oriented at aclockwise angle of θ from the vertical.

Find: For this position,(a) Determine the normal component of force of the path acting on particle P.(b) Determine the rate of change of speed of particle P.

xTOPview

HORIZONTALPLANE

y

P

v

µk

θ g

Use the following parameters in your analysis: µk = 0.2, m = 4 kg, ρ = 0.5 m, θ = 36.87 and v= 30 m/s.



Homework H.4.B

Given: Particle P (of mass m) is attached to an inextensible cable, with the opposite end of thecable being attached to a fixed pin O. P is also constrained to move along the rough surface of aright circular cone, with P following a circular path of radius r. The coefficient of kinetic frictionbetween P and the cone is known to be µk. The angle of cable OP with respect to the vertical isknown to be θ. At a particular instant in time, P is moving with a speed of v.

Find: For this instant in time,(a) Determine the normal component of force of the cone acting on particle P.(b) Determine the rate of change of speed of particle P.

FRONTview

P

O

L

θ

r g

r

µk

v

P

TOPviewofthepathofP

A A

Use the following parameters in your analysis: µk = 0.3, m = 10 kg, θ = 36.87, r = 0.5 m and v= 1.5 m/s.



Homework H.4.C

Given: A slotted arm is rotating about end O with a constant rate of Ω. Particle P (of mass m)is attached to link AP, with the mass of AP assumed to be negligible compared to the mass of P.P is constrained to move within a smooth, straight slot. At the position of interest, AP is at anangle of θ measured counterclockwise from the slotted arm. Note that AP is a two-force member.The mechanism moves in a horizontal plane.

Find: For this position,(a) Determine the velocity and acceleration of P. Write your answers as vectors, in terms of their

xy coordinates. It is suggested that you use the moving reference frame equations in youranalysis for the velocity and acceleration of P. Use an observer attached to the slotted arm.

(b) Determine the normal force acting on P and the force acting on P by link AP.

d

O

P

L

A

Ω

y

x

θ negligible mass

m

HORIZONTALplane

Use the following parameters in your analysis: m = 15 kg, θ = 36.87, d = 0.5 m, L = 1.5 m andΩ = 4 rad/s.



Homework H.4.D

Given: Blocks A and B (having masses of m and 2m, respectively) are constrained to move alonga smooth inclined surface. Cable (1) is connected to fixed ground at D and to the center of pulleyC, as shown, with cable (1) being wrapped around a pulley connected to block A. A second cable(2) is connected between the fixed ground at D and block B. The pulleys are to be assumed to beof negligible mass, and the cables are assumed to be inextensible and not allowed to go slack. Thesections of the cables not wrapped around pulleys are parallel to the incline on which blocks A andB move. A force F acts along the direction of the incline on block A.

Find: For this problem, determine the accelerations of blocks A and B.

θ

sA

C sB

sC

F

B

A

D

(1)

(2)

smooth g

m

2m

smooth



Homework H.4.E

Given: Blocks A and B (having masses of m and 2m, respectively) are constrained to move alongthe smooth surfaces shown in the figure below. Member AB, in the shape of a quarter-circle arc,connects blocks A and B, with AB having a mass that is negligible compared to the masses of Aand B. At the position shown, when the center O of the circular arc AB is directly below block B,the system is released from rest.

Find: For this position, determine the acceleration of blocks A and B on release.

A

B

R

O

g

smooth

2m

m

smooth

negligible mass

HINT: Note that AB is a two-force member. What does this say about the direction of the reactionforces on blocks A and B due to member AB?



Homework H.4.F

Given: Particle P (having a mass of m) is constrained to move around the wall of a horizontalcircular cavity, with the path of P in the cavity being a circle of radius R. The horizontal surfaceon which P moves is smooth, with the wall of the cavity along which P moves is rough having acoefficient of kinetic friction between the wall and P of µk. When at position A, P is known to havea speed of vA.

Find: For this problem,(a) Show that the speed of P as it moves around the cavity is governed by the differential equation:

dv/ds = −µkv/R, where s is the distance traveled by P.(b) Using the result of (a) above, determine the speed v of P as a function of s as it moves around

the cavity wall. (HINT: Integrate the differential equation found in (a).) Leave your answer interms of, at most, vA, µk, R and s.

(c) How far does P travel before it comes to rest?

R

smooth

µk

TOPview SIDEview

R

s P

v

P

O O

A

mo/onofPintheHORIZONTALplane



Homework H.4.G

Given: Particle P, having a mass of m, is pressed against a spring (having a stiffness of k) withthe spring being compressed by an amount of ∆. Upon release from rest, the particle travels alonga rough horizontal surface for which the kinetic coefficient of friction is known to be µk.

Find: Determine the total distance d traveled by P before coming to rest. Assume that d > ∆.Leave your answer in symbolic form.

Particle Kinetics Homework Problems ME 274

Problem IV-49 Given: Particle P, having a mass of m, is pressed against a spring (having a stiffness

of k) with the spring being compressed by an amount of ! . Upon release from rest, the particle travels along a rough horizontal surface for which the kinetic coefficient of friction is known to be µK.

Find: Determine the total distance d traveled by P before coming to rest. Assume

that d > !. Leave your answer in symbolic form.

P

rough, µK

P

d

!



Homework H.4.H

Given: Blocks A and B (having masses of 2m and m, respectively) are connected by rigid, masslessrod AB of length L. Block A is constrained to move along a smooth vertical guide, and B movesalong a smooth, horizontal surface. The system is released from rest when A is at a height of h1ABOVE the path of B.

Find: Determine the speed of block A when A has dropped to a position that is at a distance ofh2 BELOW the path of B.


Problem IV-53 Given: Blocks A and B (having masses of 2m and m, respectively) are connected by

rigid, massless rod AB of length L. Block A is constrained to move along a smooth vertical guide, and B moves along a smooth, horizontal surface. The system is released from rest when A is at a height of h1 ABOVE the path of B.

Find: For this problem,

a) Draw a free body diagram of the system made up of block A, block B and rod AB.

b) Determine the speed of block A when A has dropped to a position that is at a distance of h2 BELOW the path of B.

In your analysis, use the following parameter values: m = 10 kg, k = 1000 N/m, L = 0.5 meters, h1 = 0.4 meters and h2 = 0.3 meters.

A

B

L h1 g

m

2m

Use the following parameters in your analysis: m = 10 kg, L = 0.5 m, h1 = 0.4 m and h2 = 0.3 m.



Homework H.4.I

Given: Particle P (of mass m) is constrained to move along a smooth circular guide (of radiusR). A spring of stiffness k is attached between P and the fixed point O, where O is on the samehorizontal line as the bottom of the circular guide, as shown in the figure. At position 1, P is atrest and is on the same horizontal line as the center of the guide C. At position 2, P is on the samehorizontal line as O and the spring is unstretched.

Find: Determine the speed of particle P at position 2.

k

P

R

position 1

position 2

P

O

g

smooth guide

d

C

Use the following parameters in your analysis: m = 50 kg, R = 0.6 m, d = 0.8 m and k = 500N/m.



Homework H.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



Homework H.4.K

Given: A cannonball P of mass m is fired toward a steel barrier on a stationary cart. At some timeafter rebounding from the barrier, the cannonball is observed to have a speed of vP and is moving inthe direction shown below in the figure. Let M be the combined mass of the cannon/cart. Assumethat the cart is able to move without friction along the horizontal surface and ignore the influenceof air resistance.

Find: For this problem:(a) Determine the velocity vector of the cart after the cannonball bounces off the steel barrier at

the instant shown below;(a) If ∆t represents the elapsed time between the firing of the cannonball and the instant shown

below, determine the average value of the horizontal force acting on the combined cannon/cartover the time period of 0 < t < ∆t.


Problem IV-14 A cannonball P of mass m is fired toward a steel barrier on a stationary cart. At some time after rebounding from the barrier, the cannonball is observed to have a speed of vP and moving in a direction shown below in the figure. Let M be the combined mass of the cannon and cart. Assume that the cart is able to move without friction along the horizontal surface and ignore the influence of air resistance.

a) Determine the velocity (both magnitude AND direction) of the cart after the cannonball bounces off the steel barrier at the instant shown below.

b) Let !t represents the elapsed time between the firing of the cannonball and the instant shown below. Determine the average value of the horizontal force acting on the combined cannon and cart over the time period of 0 < t < !t.

Use the following parameters in your analysis: mg = 50 lb , Mg = 250 lb , ! t = 0.4 s , ! = 25° and vP = 100 ft / s .

!

vP P

cannon

cart

M

m

Use the following parameters in your analysis: mg = 60 lb, Mg = 200 lb, ∆t = 0.2 s, θ = 20 andvP = 120 ft/s.



Homework H.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.


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



Homework H.4.M

Given: Particle B (having a mass of m) is constrained to move within a circular slot (of radius R)that is cut into block A (having a mass of M). The system is released from rest with particle B ona horizontal line passing through the circle’s center O. Consider all surfaces to be smooth.

Find: For this problem:(a) Determine the velocities of A and B when B has moved position 2 where B is directly below O

(write your answers as vectors);(b) Determine the work done on block A in moving from position 1 to position 2.


Problem IV-47 Given: Particle B (having a mass of m) is constrained to move within a circular slot (of

radius R) that is cut into block A (having a mass of M). The system is released from rest with particle B on a horizontal line passing through the circle’s center O. Consider all surfaces to be smooth.

Find: For this problem:

a) Determine the velocities of A and B when B has moved position 2 where B is directly below O. Write your answers as vectors. Consider using the work/energy and linear impulse/momentum equations for the system of A+B.

b) Determine the work done on block A in moving from position 1 to position 2. Clearly indicate the sign of your answer (positive or negative). Consider using the work/energy equation for the system of A alone.

Use the following parameters in your analysis: m = 20kg , M = 40kg and R = 0.5 meters .

B

A A

g B

R

O O

!"#$%$"&'(' !"#$%$"&')'

Use the following parameters in your analysis: m = 20 kg, M = 40 kg and R = 0.5 m.



Homework H.4.N

Given: Blocks A and C (having masses m and 2m, respectively) are connected by an inextensiblecable. This cable has been fed through a third block B (having a mass of 2m), as shown in thefigure below. The system is released from rest with A being located at a distance of D below B.In the subsequent motion, block A contacts block B and sticks after a short time ∆t followingtheir initial contact. Assume that the cable does not break and that the mass of the two pulleys isnegligible.

Find: For this problem:(a) Determine the maximum height attained by block B after its contact with and sticking to block

A.(b) Determine the average tension force in the cable during the during of the impact time ∆t

between A and B.

A

BC

2m

m

2m

g

d

Use the following parameters in your analysis: m = 20 kg, d = 2 m and ∆t = 0.003 s.



Homework H.4.O

Given: Spheres A, B and C (having masses of 3m, 2m and m, respectively, are able to slide ona smooth horizontal surface. Spheres B and C moving with speeds of vB1 and vC1, respectively,strike the stationary sphere A at exactly the same time. The coefficient of restitution is e for allimpacts.

Find: Determine the direction of travel for A after the impact.

m

2m 3m

AB

C

vC1

vB1

Use the following parameters in your analysis: m = 2 kg, vB1 = 10 m/s, vC1 = 4 m/s and e = 0.6.



Homework H.4.P

Given: Particles A and B impact each other with initial velocities shown in the figure. Thecoefficient of restitution for this impact is e.

Find: For this problem:

(a) Determine the velocities of A and B after the impact. Write your answers as vectors.

(b) Determine the energy loss during the impact as a percentage of the initial energy of the system.

vA1 B

vB1

m 2m

smooth

vA1

B

HORIZONTAL PLANE

m 2m

! vB1

Use the following parameters in your analysis: m = 2 kg, vA1 = 10 m/s, vB1 = 10 m/s, θ = 30

and e = 0.



Homework H.4.Q

Given: Seven particles, A through G, move within a single plane. The mass of each particle isshown in the figure below, along with the velocity and position of each particle.

Find: Determine the total angular momentum about the fixed point O for the entire system ofseven particles.

A Ox

y

B

C

D

E

(2,2) ft(−2,2) ft

(−2,−2) ft (2,−2) ft(1.5,−2) ft

6 ft / s

6 ft / s

8 ft / s

8 ft / s

10 ft / s mmm

m

m

(−2,−1) ftF

1 ft / s

4 ft / s

G

m

2m

Use the following parameter in your analysis: m = 10 slugs.



Homework H.4.R

Given: A particle P, having a mass of m, is free to slide on a smooth, lightweight bar. The baris free to rotate in a horizontal plane about a vertical shaft passing through O. A spring, having astiffness k and unstretched length R0, is connected between P and O. The spring is compressed tohalf of its unstretched length and released when the bar has a rotational speed of ω1, as shown inthe figure below left.

Find: For the position when R = R0 (shown in the figure below right):(a) Determine the rotation rate ω2 of the bar;(b) Determine the value of R.


Problem IV-51 Given: A particle P, having a mass of m = 10 kg, is free to slide on a smooth,

lightweight bar. The bar is free to rotate in a horizontal plane about a vertical shaft passing through O. A spring, having a stiffness k = 400 N/m and unstretched length R0 = 2 meters, is connected between P and O. The spring is compressed to half of its unstretched length and released when the bar has a rotational speed of !1 = 5 rad/sec, as shown in the figure below left.

Find: For the position when R = R0 (shown in the figure below right):

a) Determine the rotation rate !2 of the bar.

b) Determine the value of

dRdt

.

P

O

R = R0/2

HORIZONTAL PLANE

k

P

O R = R0

k !2

!1

Use the following parameters in your analysis: m = 10 kg, k = 400 N/m, R0 = 2 m and ω1 = 5rad/s.



Homework H.4.S

Given: Particle A (having a mass of m) is attached to rod OA, with end O being pinned to avertical shaft. Particle A is also attached to particle B (having a mass of m) through rod AB, withparticle B being constrained to slide along the rotation axis of the shaft. At a time when the shaftis rotating with a rate of ω1 and with θ = θ1, and while particles A and B are stationary withrespect to the shaft, the particles are released. Assume all surfaces to be smooth and the mass ofrods OA and AB to be negligible.

Find: Determine the speed of particle B when θ = θ2.

O

B

m

smooth bearings

L

θ

ω

smooth bearings

g

m

A

L

Use the following parameters in your analysis: m = 20 kg, L = 0.5 m, ω1 = 6 rad/s, θ1 = 30 andθ2 = 90.



Homework H.4.T

Given: Rigid arm OA (having length L and having negligible mass) is pinned to ground at endO. A particle of mass M is attached to end A of OA. At instant “1”, a pellet P (having a mass ofm) strikes the stationary particle A with a speed of vP1 in the direction shown below in the figure.At the end of a short time interval impact, P sticks to A.

Find: Determine the angular speed of arm OA immediately after P sticks to A.


Problem IV-19 Rigid arm OA (having length L and having negligible mass) is pinned to ground at end O. A particle of mass M is attached to end A of OA. At instant “1” a pellet P (having a mass of m) strikes the stationary particle A with a speed of vP1 in the direction shown below in the figure. At the end of a short time interval impact, P sticks to A.

a) Draw a free body diagram (FBD) of the system made up of particles A and P during impact. Recall that P and A are particles (i.e., the physical dimensions of these two bodies are negligible). Based on your FBD, provide argument that supports the claim that the angular momentum for system AP of about O is conserved during impact.

b) Determine the angular speed of arm OA !2 immediately after P sticks to A. Consider using the angular momentum equation.

Use the following parameters in your analysis: ! = 36.87° , L = 5 ft ,mg = 2 lb , Mg = 10 lb and vP1 = 120 ft / s .

O

P

L

!

g

VERTICAL PLANE

A M

m

vP1

Use the following parameters in your analysis: φ = 30, L = 4 ft, mg = 4 lb, Mg = 8 lb andvP1 = 150 ft/s.


Particle Kinetics Homework - Purdue University · Chapter 4: Particle Kinetics Homework Chapter 4...

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