Particle Kinetics

Particle Kinetics Homework Problems ME 274 – Spring 2009

Problem 3-I The truck shown below is travelling along I-65 when a deer runs out onto the highway. If the truck is initially travelling at 70 mph and decelerates at a constant rate, find the minimum distance s over which the truck can stop to ensure that its load does not shift (and the deer is safely avoided). Assume the load’s weight is 1200 lb and that µS = 0.35.


Problem 3-II The system shown below is released from rest in the configuration shown. If the masses A and B are 50 kg and 75 kg, respectively, µk = 0.25, µS = 0.35, and θ = 45°. Determine:

(a) the acceleration of block A (b) the acceleration of block B (c) the tension T in the cable

g

µs, µk

mAmB

!

µs, µk

mAmB

!


Problem 3-III The racecar, shown below, travels around a banked curve at Daytona International Speedway. Given that the curve has slope θ = 20°, radius of curvature ρ = 1000 ft, and static coefficient of friction µS = 1.0, determine:

(a) the speed at which the racecar can circumvent the curve without the assistance of a lateral friction force

(b) the maximum speed the racecar can circumvent the curve with the assistance of a lateral friction force

θ

ρ


Problem 3-IV The mechanism, shown below, consisting of a rotating arm, frictionless slot, and block of mass m, is driven at a constant speed ω by a motor attached at point O. Determine:

(a) the tension in the cable and normal force exerted on the block at the instant shown corresponding to ω = 2 rad/s

(b) the minimum angular velocity ω necessary to keep the cable taut at this instant Use r = 0.15 m, m = 0.1 kg and θ = 30° in your analysis.

!

O

r

"

m

!

O

r

"

mg


Problem 3-V The mechanism, shown below, consisting of a rotating disk, frictionless slot, cable, and block of mass m, is driven at a constant speed (ω = 5 rad/s) in the horizontal plane by a motor attached at point O. Using h = 0.15 m, d = 0.20 m, r = 0.5 m, and m = 0.2 kg, determine:

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

HORIZONTAL PLANE

!

O

r

d

h

m

!

O

r

d

h

m


Problem 3-VI The drum of an industrial polisher, shown below, rotates about point O with a constant speed ω = 15 rad/s. A small cube of aluminum is placed on the inner surface of the drum. It is observed that this piece of metal has no motion relative to the polisher’s surface as it passes through θ = 0. For r = 0.1 m, determine the relationship between µS (the static coefficient of friction between the block and the drum’s inner surface) and θcrit (the angle of orientation at which the block first slips). Using your favorite software, plot µS vs. θcrit for θcrit values between 0 and 180°. Comment on the plot.

g !

Or

"

µs

m

!

Or

"

µs

m


Problem 3-VII The gantry crane, shown below, is used to move raw materials, of mass m, around a local factory. To ensure safety, the angular deflection of the mass is constrained to be less than θ = 5°. Determine the maximum speed at which the crane can move, such that when it comes to an abrupt halt, the aforementioned constraint is not violated. Assume that the mass hangs vertically while the crane is in motion and that L = 6 m.

m

!

v

L

m

!

v

L

g


Problem 3-VIII The collar, shown below, of mass m = 0.05 kg, starts from rest at point A. A constant force F = 12 N is applied to the collar in the direction shown. Assuming that r = 0.1 m, determine the speed of the collar when it reaches point B. Note that the mechanism lies in the vertical plane.

A

B

F

m

r

A

B

F

m

rg


Problem 3-IX The collar, shown below, of mass m = 2.5 kg is pushed against a spring with stiffness k = 1500 N/m until it is compressed 0.1 m, and then released. Assuming that there is a kinetic coefficient of friction µk = 0.30 between the collar and rod, θ = 45°, and that the collar fails to launch clear of the rod, determine the maximum height obtained by the collar, as measured from the point of release.

m

k

µk

!

m

k

µk

!

g


Problem 3-X The mechanism shown below, consisting of two blocks A and B (m = 0.1 kg) and a massless bar, is released from rest in the configuration shown (θ = 45°). Given that L = 0.20 m, determine the speed of block A when it reaches the bottom of the slot.

m

m

!

A

B

L

m

m

!

A

B

Lg


Problem 3-XI 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 = 750 N , ! t = 0.004 s and m = 0.05 kg .

0 ∆t t

F(t)

F0 A B

m 3m

smooth smooth

combusting fuel


Problem 3-XII A particle of mass m is dropped from rest when at a height h1 above a rigid floor. The particle impacts the floor with a speed of v1. This impact of the particle with the floor lasts for a short duration of time ∆t, and after the impact is complete, the particle rebounds upward with a speed of v2. The particle continues upward reaching a maximum height of h2.

a) Draw a free body diagram (FBD) of the particle for the time period during its impact with the floor.

b) Based on your FBD in a), determine the average force acting on the particle by the floor during impact. Recall that the average value of a force F(t) acting over a

period of time ∆t is given by: Fave =1

! tF t( ) dt

0

! t

! .

c) Repeat your calculations in b) above but now ignore the weight force during the time of impact. Did your answer for the average impact force change much?

d) Determine the value of h2 / h1 . Use the following parameters in your analysis:! t = 0.002 s , m = 4 kg , v1 = 30 meters / sec and v2 = 20 meters / sec .

h1

v = v1

v = 0

h2 v = v2

v = 0


Problem 3-XIII Automobile A (having a mass of m) is parked on a long, stationary barge B (having a mass of M) that is floating in a body of water. The driver of the automobile starts driving to the right to a point where the speedometer of the car registers a speed of vS after an elapsed time of ∆t. In the analysis that follows, ignore the drag effects on the barge by the water as it moves, and treat the automobile and barge as particles.

a) Draw a free body diagram (FBD) of the system made up of the automobile and barge. From this FBD, determine the true velocity of the automobile and the velocity of the barge at t = ∆t.

b) Draw an FBD of the automobile alone. From this FBD, determine the average traction force acting on the automobile over the time period of 0 < t < ∆t.

Use the following parameters in your analysis:! t = 10 s , m = 1800 kg , M = 9000 kg and vS = 18 meters / sec .

A

B M m


Problem 3-XIV 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:m = 10 kg , M = 200 kg , ! t = 0.5 s and vP = 40 meters / s .

θ

vP P

cannon

cart

M

m


Problem 3-XV 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 θ1 that the velocity of the particle has with the horizontal before impact. Use conservation of energy prior to impact.

b) Determine the angle θ2 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 = 0.5 , h = 60 meters and v0 = 100 meters / sec .

v0

v1

θ1

v2

θ2

BEFORE impact immediately AFTER impact

h

d


Problem 3-XVI Disks A and B have masses of 2m and m, respectively. Disk B is traveling in the direction shown with a speed of vB1 when it strikes the stationary disk A (vA1 = 0). Let e represent the coefficient of restitution of impact between A and B. Find the velocity of disk A after impact. Write your answer as a vector in terms of its “n” and “t” components.

Use the following parameters in your analysis: e = 0.8 , ! = 36.87°and vB1 = 20 ft / s .

A

B θ

t n

m

2m

vB1


Problem 3-XVII A satellite P of mass m is in orbit around a planet whose center is at E. At position 1 shown below, the satellite is at a distance R1 from E and moving with a speed of v1. At position 2 shown, P is at a distance of R2 from E and moving with a speed of v2 with the velocity of P oriented as shown below. Assume that the only force acting on the satellite is a gravitational force directed toward E, and that the planet is not accelerating.

a) Determine the angular momentum of P about E at positions 1 and 2. b) Provide an argument to support the claim that the angular momentum about E is

constant.

c) Based on b) above, determine the speed of P at position 2, v2, in terms of R1, R2, θ2 and v1.

d) Explain why your result in c) does not depend on the gravitational force acting on the satellite.

eR R1

eθ

eθ eR

R2 v1

v2 θ2

P

P

E


Problem 3-XVIII Arm OA, having negligible mass, rotates about a smooth vertical shaft passing through end O of the arm. Particle P is able to slide along arm OA as the arm rotates. A spring of stiffness k and unstretched length R0 is connected between P and the fixed point O. At instant “1”, P is stationary with respect to the arm ( !R = 0 ), the spring is compressed by an amount of ∆R and arm OA is rotating counterclockwise with a rotational speed of ! =!1 . At instant “2”, the spring is stretched by an amount of ∆R. Assume that the rod is smooth (i.e., neglect friction).

a) Draw a free body diagram (FBD) of P. Note that since arm OA is of negligible mass the normal force on P by the arm is zero. Based on your FBD, provide arguments that the angular momentum of P about O is conserved AND energy for P is conserved.

b) Determine the angular speed of arm OA !2 at instant 2. (Note that this result is not influenced by the spring.) Consider using the angular momentum equation.

c) Determine the speed of P at instant 2. Consider using the work-energy equation.

d) Determine the value !R of at instant 2. Consider the polar form of the velocity equation.

Use the following parameters in your analysis: !1 = 3 rad / s , R1 = 0.5 meters , ! R = 0.6 meters , k = 400 N / meter and m = 2 kg .

O P

eR

R

ω

k

eθ

HORIZONTAL PLANE

A


Problem 3-XIX 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 = 8 meters ,m = 1 kg , M = 9 kg and vP1 = 50 meters / s .

O

P

L

φ

g

VERTICAL PLANE

A M

m

vP1


Problem 3-XX Particles A and B each have a mass of m. A is constrained to move on a horizontal arm OC, and B is constrained to move on the vertical shaft about which arm OC rotates. A taut cable connects A and B as shown in the figure below. At an instant when the shaft is rotating with a rate of ! =!1 and when R = R1 , A and B are released from rest relative to the rotating arm and shaft. Consider all surfaces to be smooth, assume the masses of the arm, shaft and pulley to be negligible, and assume the radius of the pulley to be small. When A has moved outward to a position of R = R2 ,

a) determine the angular speed of the arm !2 . Consider using the angular momentum equation.

b) determine the speed of B. Consider using the work-energy equation along with appropriate kinematics relating the motions of A and B. Be careful – the velocity of A has two components at position 2 (one due to the rotation of the arm and one due to its sliding along the arm).

Use the following parameters in your analysis: m = 10kg , R1 = 0.2 ft , R2 = 0.5 ft and !1 = 20rad / s .

A

B

O

R

g

ω

C

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Particle Kinetics

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