Leaving Certificate Honours Applied Mathematicsc2014 Stephen Easley-Walsh

Dynamics Further Pulley, Block & Wedge Systems

1. A one kg block (the upper block) is placed on a one kg block which is on a horizontal table (the lower block). The lowerone kg block has a pulley attached to it with a string passing over it. The string has a hanging weight of one kg on it andthe other end of the string is attached to the upper block as so to cause it to move. All surfaces are smooth. What isthe acceleration of the lower block?

2. A one kg block (the upper block) is placed on a one kg block which is on a horizontal table (the lower block). The lowerone kg block has a pulley attached to it with a string passing over it. The string has a hanging weight of one kg on it andthe other end of the string is attached to the upper block as so to cause it to move. The coefficent of friction betweenall surfaces is 0.1. What is the acceleration of the lower block?

3. * Two particles of mass m kg and 2m kg lie at rest on horizontal rough tables. The coefficient of friction between eachparticle and the table it lies on is < 23 . The particles are connected by a light inextensible string which passes undera smooth movable pulley of mass 4m kg. The system is released from rest.

(a) Find, in terms of m and , the tension in the string.

(b) If the acceleration of the m kg mass is f , find the acceleration of the 2m kg mass in terms of f .

4. * A mass m1 kg is at rest on a smooth horizontal table. It is attached to a light inextensible string. The string, afterpassing over a small fixed pulley at the edge of the table, passes under a small moveable pulley C, of mass m2 kg. Thestring then passes over a smooth fixed pulley and supports a mass of 1 kg. The system is released from rest.

(a) Find, in terms of m1 and m2 , the tension in the string.

(b) The pulley C will remain at rest if 2m2 1m1 = k. Find the value of k.5. * Particles of mass 2m and m are connected by a light inextensible string which passes over a smooth pulley at the

vertex of a wedge-shaped block, one particle resting on each of the smooth faces. The mass of the wedge is 4m andthe inclination of each face to the horizontal is 30. The wedge rests on a smooth horizontal surface and the system isreleased from rest.

(a) Show, on separate diagrams, the forces acting on the wedge and on the particles.

(b) Find the acceleration of the wedge.

6. A smooth wedge of mass 10 kg rests on a smooth horizontal tabletop. The wedge is a right-angle shape (the right-angle on the right) that makes an angle of 45 with the tabletop. At the top of the wedge is a smooth light pulley. A lightinextensible string passes over the pulley, connecting smooth blocks of mass 6 kg and 2 kg. The 6 kg mass is on theinclined surface of the wedge (the hypotenuse). When the system is released from rest the 6 kg block moves down theinclined surface and the wedge moves to the right. What is the acceleration of the 10 kg wedge?

7. A wedge of mass 20 kg rests on a smooth horizontal table (no friction between the wedge and table). The wedge is aright-angle shape (the right-angle on the right) that makes an angle of tan1

(34

)with the table. At the top of the wedge

is a smooth light pulley. A light inextensible string passes over the pulley, connecting a rough block of mass 12 kg and asmooth block of mass 4 kg. The 12 kg mass is on the inclined surface of the wedge (the hypotenuse) and experience afriction force given by = 0.1. When the system is released from rest the 12 kg block moves down the inclined surfaceand the wedge moves to the right. What is the acceleration of the 20 kg wedge?

8. A three kg wedge is placed on a horizontal table. The two inclined surfaces are at angles of 45 to the horizontal. Twomasses of mass one and two kg are places on the inclined surfaces and are connected by a light inextensible stringwhich is placed over a pulley on the vertex. All surfaces are smooth. What is the acceleration of the wedge?

9. A three kg wedge is placed on a horizontal table. The two inclined surfaces are at angles of 45 to the horizontal. Twomasses of mass one and two kg are places on the inclined surfaces and are connected by a light inextensible stringwhich is placed over a pulley on the vertex. All surfaces are rough with coefficient of friction of 12 . What is theacceleration of the wedge?

10. A smooth wedge of mass 8 kg rests on a smooth horizontal tabletop. The wedge is a right-angle shape (the right-angleon the right) that makes an angle of tan1

(34

)with the tabletop. At the top of the wedge is a smooth light pulley. A

light inextensible string passes over the pulley, connecting smooth blocks of mass 5 kg and 1 kg. The 1 kg mass is onthe inclined surface of the wedge (the hypotenuse). When the system is released from rest the 5 kg block moves upthe inclined surface, the wedge moves to the left and the 5 kg mass moves out from the vertical surface of the wedge,making a small angle to the wedge surface. Assuming the angle is small (sin and cos 1 in radians) find thevalue of the angle to the nearest half of a degree.

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Leaving Certificate Honours Applied Mathematicsc2014 Stephen Easley-Walsh

Source(s):

http://www. Maths Grinds .ie/ http://www.examinations.ie/

Further Information:

Questions marked with an asterisk * are past Leaving Cert Exam questions.

Answers:

1. 3.27 m/s2

2. 0.10 m/s2

3. (a) 45mg (1 + )(b) 3f 2g

4. (a) 3m1m2g4m1+m2+m1m2(b) k = 1

5. (a) N/A

(b)3

19 g

6. 0.74 m/s2

7. 0.68 m/s2

8. 4945 m/s2

9. 2009265 m/s2

10. 3 and 12 degrees

c 2008-2014. Stephen Easley-Walsh. All rights reserved. email: Stephen@MathsGrinds.ie homepage: http://www.MathsGrinds.ie

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