DepartamentodeEngenhariaMecânicaÁreaCientíficadeMecânicadosMeiosSólidos

RMM–2015/16

MecânicaAplicada

Cap.131-ImpulsoequantidadedemovimentoProblema13.140

O último segmento da prova de salto triplo do atletismo é o saltofinalemqueoatletaaterranumacaixadeareia.Considerandoqueavelocidade do atleta de84 𝑘𝑔 justamente antes de aterrar é de9,14 𝑚/𝑠com um ângulo de35°com a horizontal, e que o atletapára por completo 0,22 𝑠 após a aterragem, determine acomponentehorizontaldaforçaimpulsivamédiaexercidasobreosseuspésduranteaaterragem.

Problema13.146

Umamala𝐴de135 𝑁foi encostada contra uma extremidade de um carrinho de bagagem𝐵de360 𝑁eestáimpedidadeescorregarparabaixoporoutrabagagem.Quandoabagagemédescarregadaeoultimobaupesadoétiradodocarrinho,amalaficalivreparaescorregar,fazendocomqueocarrinhosemovaparaaesquerdacomvelocidade𝒗!deintensidade0,75 𝑚/𝑠.Desprezandooatrito,determine:

(a) a velocidade𝒗! !damala em relação ao carrinho enquanto ela semovimenta sobreopisodocarrinho;

(b) avelocidadedocarrinhoapósamalaatingiroladodireitodocarrinhosemvoltarparatrás;(c) aenergiaperdidanoimpactodamalasobreopisodocarrinho.

Problema13.171

Ocoeficientederestituiçãoéde0,9entreasduasbolasdebilhar𝐴e𝐵, de60 𝑚𝑚de diâmetro. A bola𝐴move-se na direçãomostrada nafigura,comvelocidadede1 𝑚/𝑠,quandobatenabola𝐵,queestáemrepouso.Sabendoque,apósoimpacto,abola𝐵move-senadireção𝑥,determine:

(a) oângulo𝜃;(b) avelocidadede𝐵apósoimpacto.

Problema13.178

Um bloco𝐵de11,25 𝑁move-se com velocidade𝑣!de intensidade𝑣! = 1,8 𝑚/𝑠 quando bate na esfera𝐴 de6,75 𝑁𝑚 que está emrepouso e pendurada por uma corda presa em𝑂 . Sabendo que𝜇! = 0,6entreoblocoeasuperfíciehorizontaleque𝑒 = 0,8entreoblocoeaesfera,determineapósoimpacto:

(a) aalturamáximaℎalcançadapelaesfera;(b) adistância𝑥percorridapelobloco.

1Osproblemasapresentadosreferem-seaolivro“MecânicaVetorialparaEngenheiros–Dinâmica,FerdinandP.Beer,E.RussellJohnstonJr.,WilliamE.Clausen,7ªEdMcGraw-Hill”

817Problems 13.141 The last segment of the triple jump track-and-field event is the jump, in which the athlete makes a final leap, landing in a sand-filled pit. Assuming that the velocity of a 185-lb athlete just before landing is 30 ft/s at an angle of 35° with the horizontal and that the athlete comes to a complete stop in 0.22 s after landing, deter-mine the horizontal component of the average impulsive force exerted on his feet during landing.

13.142 An estimate of the expected load on over-the-shoulder seat belts is to be made before designing prototype belts that will be evalu-ated in automobile crash tests. Assuming that an automobile travel-ing at 45 mi/h is brought to a stop in 110 ms, determine (a) the average impulsive force exerted by a 200-lb man on the belt, (b) the maximum force Fm exerted on the belt if the force-time diagram has the shape shown.

13.143 A 46-g golf ball is hit with a golf club and leaves it with a velo c-ity of 50 m/s. We assume that for 0 # t # t0, where t0 is the duration of the impact, the magnitude F of the force exerted on the ball can be expressed as F 5 Fm sin (pt/t0). Knowing that t0 5 0.5 ms, determine the maximum value Fm of the force exerted on the ball.

13.144 The design for a new cementless hip implant is to be studied using an instrumented implant and a fixed simulated femur. Assuming the punch applies an average force of 2 kN over a time of 2 ms to the 200 g implant, determine (a) the velocity of the implant imme-diately after impact, (b) the average resistance of the implant to penetration if the implant moves 1 mm before coming to rest.

13.145 A 20-Mg railroad car moving at 4 km/h is to be coupled to a 40-Mg car which is at rest with locked wheels (mk 5 0.30). Determine (a) the velocity of both cars after the coupling is completed, (b) the time it takes for both cars to come to rest.

30 ft/s 35°

Landing pit

Fig. P13.141

1100

F (lb)

Fm

t (ms)

Fig. P13.142

Fig. P13.144

40 Mg

20 Mg

4 km/h

Fig. P13.145

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837Problems 13.166 Two identical hockey pucks are moving on a hockey rink at the same speed of 3 m/s and in parallel and opposite directions when they strike each other as shown. Assuming a coefficient of restitu-tion e 5 1, determine the magnitude and direction of the velocity of each puck after impact.

13.167 Two identical pool balls of 2.37-in.-diameter, may move freely on a pool table. Ball B is at rest and ball A has an initial velocity v 5 v0i. (a) Knowing that b 5 2 in. and e 5 0.7, determine the velocity of each ball after impact. (b) Show that if e 5 1, the final velocities of the balls form a right angle for all values of b.

20°

A

B

vA

vB

Fig. P13.166

x

y

v

bA

B

Fig. P13.167

13.168 The coefficient of restitution is 0.9 between the two 2.37-in. diameter billiard balls A and B. Ball A is moving in the direction shown with a velocity of 3 ft/s when it strikes ball B, which is at rest. Knowing that after impact B is moving in the x direction, determine (a) the angle u, (b) the velocity of B after impact.

A

B

x

y

vA

vB'

6 in.

10 in.

q

Fig. P13.168

13.169 A boy located at point A halfway between the center O of a semi-circular wall and the wall itself throws a ball at the wall in a direc-tion forming an angle of 45° with OA. Knowing that after hitting the wall the ball rebounds in a direction parallel to OA, determine the coefficient of restitution between the ball and the wall.

R2

R

OA

B

v

v!

45°

Fig. P13.169

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839Problems 13.174 A 1-kg block B is moving with a velocity v0 of magnitude v0 5 2 m/s as it hits the 0.5-kg sphere A, which is at rest and hanging from a cord attached at O. Knowing that mk 5 0.6 between the block and the horizontal surface and e 5 0.8 between the block and the sphere, determine after impact (a) the maximum height h reached by the sphere, (b) the distance x traveled by the block.

13.175 A 1.5-kg block B is attached to an undeformed spring of constant k 5 80 N/m and is resting on a horizontal frictionless surface when it is struck by an identical block A moving at a speed of 5 m/s. Con-sidering successively the cases when the coefficient of restitution between the two blocks is (1) e 5 1, (2) e 5 0, determine (a) the maximum deflection of the spring, (b) the final velocity of block A.

13.176 Block A is released from rest and slides down the frictionless sur-face of B until it hits a bumper on the right end of B. Block A has a mass of 10 kg and object B has a mass of 30 kg and B can roll freely on the ground. Determine the velocities of A and B imme-diately after impact when (a) e 5 0, (b) e 5 0.7.

AB

O

v0h

x

Fig. P13.174

B A

5 m/s

k = 80 N/m

Fig. P13.175

Fig. P13.176

0.2 m

B

A

13.177 A 90-g ball thrown with a horizontal velocity v0 strikes a 720-g plate attached to a vertical wall at a height of 900 mm above the ground. It is observed that after rebounding, the ball hits the ground at a distance of 480 mm from the wall when the plate is rigidly attached to the wall (Fig. 1) and at a distance of 220 mm when a foam-rubber mat is placed between the plate and the wall (Fig. 2). Determine (a) the coefficient of restitution e between the ball and the plate, (b) the initial velocity v0 of the ball.

v0

900 mm

720 g

90 g

v0

90 g

220 mm480 mm

(1) (2)

720 g

Fig. P13.177

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250 𝑚𝑚

150 𝑚𝑚

9,14 𝑚/𝑠