## Applied Mathematics 1 - Dec 18

### First Year Engineering (Semester 1)

Total marks: 80

Total time: 3 Hours
INSTRUCTIONS

(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Draw neat diagrams wherever necessary.

**Find the resultant of the parallel force system shown in figure 1 and locate the same with respect to point C**

**1.a.**** 1.b.**
Using Instantaneous Centre of rotation (ICR) method,Find the velocity of point A for the instant shown in figure 2.Collar B moves along vertical rod,Whereas link AB moves along the plane which is inclined at 25°

(5 marks) 00

** 1.c.**
If the support reaction A,for the beam shown in figure 3 is zero, then find force 'P' and support reaction at B.

** 1.d.**
From the top of a tower ,28 m high a stone is thrown vertically up with a velocity of 9m/s after how much time will the stone reach ground? with what velocity does it strike the ground.

** 1.e.**
For the truss shown in figure 4 find zero force members justify answer with FBD ,support reactions at C and D.

** 2.a.**
For the compsite lamina shown in figure 5,determine the co ordinates of its centroid.

**2.b.**

Replace the force system shown in figure 6 with a single force and couple system acting at point B.

** 2.c.**
The link CD of the mechanism shown in figure 7 is rotating in counterclockwise direction at an angular velocity of 5 rad/s For the given instance,determine the angular velocity of link AB.

(7 marks) 00

** 3.a.**
Cylinder A (diameter 1m,weight 20kN) and cylinder B(diameter 1.5m,weight 40kN) are arranged as shown in figure 8. Find the reactions at all the contact points.all contains are smooth.

** 3.b.**
Using Principle of virtual work determine force P which will keep the weightless bar AB in equilibrium Take length AB as 2m and length AC at 8m . The bar makes an angle of 30° with horizontal .All the surface in contact are smooth .Refer figure 9

** 3.c.**
Velocity time diagram for particle travelling along a straight line as shown in figure 10 Draw acceleration time and displacement time diagram for particle.Also find important values of acceleration and displacement.

(8 marks) 00

** 4.a.**
Find the force 'F' to have motion of block.A impeding up the plane.Take coefficient of friction for all the surfaces in contact as 0.2 .Consider the wedge B as weightless Refer Figure 11.

(7 marks) 00

** 4.b.**
3 forces f1,f2,f3 act at origin of Cartesian coordinate axes system.The force f1(=70N) acts along OA whereas f2(=80) acts along OB and f3(=100N) acts along OC the coordinates of points A,B,C are (2,1,3) ,(-1,2,0) and (4,-1,5) respectively .find resultant of this force system.

(5 marks) 00

** 4.c.**
A 75 kg person stands on a weighing scale in an elevator.3 seconds after the motion starts from rest the tension in the hoisting cable was found to be 8300N find reading of the scale in kg during this interval .Also find the velocity of elevator at the end of this interval.The total mass of elevator including mass of the person and the weighing scale, is 750kg.If the elevator is now moving in the opposite direction,with same magnitude of acceleration,what will be the new reading of the scale?

** 5.a.**
The cylinder B,diameter 400mm and weight 5kN,is held inposition as shown in figure 12 with the help of cable Ab .Find the tension in the cable and the reaction developed at contact C.

(4 marks) 00

** 5.b.**
Find the weight Ws so as to have its impending motion down the plane .Take weight of block A as 2kN. The pin connected rod AB is initially is in horizontal position.Refer figure 13

.

(5 marks) 00

** 5.c**
2 springs,each having stiffness of 0.6 N/cm and length 20 cm are connected to the ball B of weight 50 N .The initial tension developed in each springs is 1.6 N.The arrangement is initially horizonatal as shown in figure 14 .If the ball is allowed to fall from rest,What will be its velocity at D,after it has fallen through a height of 15cm?

(5 marks) 00

** 5.d**
2 balls A (mass 3kg) and B (mass 4kg), are moving with velocities 25m/s and 40 m/s respectively (refer figure 15). Before impact the direction of velocity of two balls are 30 and 50 with the line joining their centers as shown in figure 15 .If coefficient of restitution for impact is 0.78, find magnitude and the direction of velocities of the balls after the impact.

(6 marks) 00

** 6.a.**
For the truss shown in figure 16 ,find forces in members DE>BD and CB.

(5 marks) 00

** 6.b.**
A particle moves in x-y plane with acceleration components ax = -3m/s^2 and ay = -16 m/s^2 If its initial velocity is Vo= 50m/s^2 directed at 35 ° to x-axis compute the radius of curvature of the path at t = 2 sec

** 6.c.**
A force of magnitude of 20kN acts at point A(3,4,5)m and its line of action passing through B(5,-3,4)m,Calculate the moment of this force about line passing through points S(2,-5,3)m and T(-3,4,6)m.

** 6.d.**
Find an expression for maximum range of a particle which is projected with an initial velocity of 'u' inclined at an angle of 'β' with horizontal.