MU FE Sem 1 Engineering Mechanics Most Important Questions with Solutions (Winter 2026)

Q.1 A force of magnitude 60 kN acts at point A(1, 2, 3) m and is directed towards point B(4, −2, 15) m. Find the components of the force along the X, Y and Z axes and write it in vector form.

Q.2 A force of 40 N acts at point P(2, 1, 3) m. The force makes angles of 60° and 45° with the X and Y axes respectively, and has a positive Z component. Express the force in vector form.

Q.3 A boat is pulled along a straight canal by two ropes, so that the resultant pull is 1500 N along the canal (the x-axis). Rope 1 makes 30° above the x-axis and rope 2 makes 20° below the x-axis, both pulling forward. Find the tension in each rope.

Q.4 A rectangular plate OBCD is 4 m long (OB, along the x-axis) and 3 m high (OD, along the y-axis). Determine the magnitude and direction of the resultant of the force system acting on it, and locate the point where the resultant cuts the x-axis (line OB) with respect to O.

Figure (described in words): O(0, 0) is the bottom-left corner, B(4, 0) the bottom-right, C(4, 3) the top-right and D(0, 3) the top-left corner; dimensions in metres. The forces are: 200 N vertically upward at B; 150 N horizontally to the left at C; 250 N at D directed along the diagonal DB (towards B); 100 N at O directed along the diagonal OC (towards C); and a clockwise couple of 150 N·m acting on the plate.

Q.5 Find the coordinates of the centroid of the shaded lamina with respect to the given reference axes.

Figure (described in words): Origin O at the bottom-left corner, x-axis to the right, y-axis upward; dimensions in cm. The lamina is made of: a rectangle with corners (0, 0), (8, 0), (8, 6) and (0, 6); a semicircle of radius 4 cm standing on the top edge of the rectangle (diameter from (0, 6) to (8, 6), curved side upward); and a right-angled triangle on the right side with vertices (8, 0), (12, 0) and (8, 6). A circular hole of diameter 3 cm centred at (4, 3) is cut out. The rest is shaded.

Q.6 Find the coordinates of the centroid of the shaded area with respect to the given reference axes.

Figure (described in words): Origin O, x-axis to the right, y-axis upward; dimensions in cm. In the first quadrant there is a quarter circle of radius 6 cm centred at O (bounded by the x-axis from 0 to 6, the y-axis from 0 to 6 and the arc). In the second quadrant there is a right-angled triangle with vertices O(0, 0), (−6, 0) and (0, 6). A semicircle of radius 2 cm, with its diameter on the x-axis from x = 1 to x = 5 (centre (3, 0)) and curved side upward, is cut out of the quarter circle. The remaining area is shaded.

Q.7 Prove that angle of friction is equal to angle of repose. (asked in MU Dec 2025)

Q.8 A 40 kg block hangs from joint B. Cord BA goes up to the left to a fixed support A, making 35° with the horizontal at B, and cord BC goes up to the right to a fixed support C, making 50° with the horizontal at B. Determine the tensions in cords AB and BC. (Take \(g=9.81\ \text{m/s}^2\).)

Q.9 Determine the force P applied at 30° to the horizontal just necessary to start a roller having 50 cm radius over a 12 cm high obstruction, if the roller is of 100 kg mass. Also find the magnitude and direction of P when it is minimum. (asked in MU Dec 2025)

Figure (described in words): A roller of radius 50 cm and centre O rests on a horizontal floor, touching on its right a rectangular step (obstruction) 12 cm high. The force P acts at the centre O, directed up and to the right at 30° to the horizontal.

Q.10 For the overhanging beam loaded as shown, calculate the support reactions at A and B.

Figure (described in words): Beam ABC is 10 m long. A (left end) is a hinged support and B is a roller support 8 m from A; BC is a 2 m overhang to the free end C. Loads, measured from A: at D (2 m) a 30 kN force acting downward and to the right, inclined at 60° to the beam; a uniformly distributed load of 10 kN/m from 3 m to 6 m; an anticlockwise couple of 20 kN·m at 7 m; and on the overhang BC a uniformly varying (triangular) load increasing from zero at B to 15 kN/m at C.

Q.11 Block A (mass 25 kg) rests on block B (mass 40 kg), which rests on a horizontal floor. A string attached to A runs up to the left to a fixed vertical wall, making 20° with the horizontal. A horizontal force P applied to B pulls it to the right. The coefficient of friction between A and B is 0.25 and between B and the floor is 0.3. Find the force P required to just move block B.

Q.12 The acceleration of a particle in rectilinear motion is given by \(a=64-4x^2\ \text{m/s}^2\), where x is in metres. At \(x=0\) the particle is at rest. Find the velocity at \(x=3\) m, and the position and value of the maximum velocity.

Q.13 The position vector of a particle moving in the X–Y plane is \(\vec r=(2t^3-3t)\,\hat i+(t^2+4)\,\hat j\) m, where t is in seconds. At \(t=2\) s, find (i) the velocity and acceleration, (ii) the tangential and normal components of acceleration, and (iii) the radius of curvature of the path.

Q.14 In the mechanism shown, the angular velocity of link AB is 5 rad/s anticlockwise. At the instant shown, determine the angular velocity of link BC and the velocity of piston C. (asked in MU Dec 2025)

Figure (described in words): Link AB hangs vertically from a fixed pivot A, with AB = 200 mm and B below A; AB rotates at 5 rad/s anticlockwise. The connecting rod BC runs from B down to the right to piston C, which slides in a vertical guide. At this instant C is 450 mm to the right of B and 150 mm below B.

Q.15 Collar B moves up a fixed vertical rod with a constant velocity of 3 m/s. Rod AB, 2 m long, is pinned to the collar at B and makes 45° with the horizontal at B, running down to the left to end A. End A slides on a fixed surface inclined at 30° to the horizontal (rising to the right). Using the instantaneous centre of rotation, find the angular velocity of rod AB, the velocity of end A and the velocity of the mid-point G of the rod.

Q.16 A cricket ball of mass 0.16 kg moving horizontally at 25 m/s is hit straight back by a bat and leaves at 35 m/s. If the bat is in contact with the ball for 0.02 s, find the impulse and the average force exerted by the bat.

Q.17 Explain with neat sketches (1) direct central impact and (2) oblique central impact. (asked in MU June 2025 KT)

Q.18 The system shown is initially at rest. Neglecting friction, determine the force P required if the velocity of collar B is 5 m/s after 2 s, and the corresponding tension in the cable. (asked in MU Dec 2025; also June 2025 KT)

Figure (described in words): A smooth horizontal rod is fixed to a vertical wall on the left. Collar B (mass 3 kg) slides on the rod and is pulled to the right by a horizontal force P. A small pulley is fixed to the left side of collar B. A cable is anchored to the wall, runs horizontally to the right, passes round the pulley on B and comes back horizontally to the left to a second pulley fixed to the wall by a bracket. From this fixed pulley the cable hangs vertically down to block A of mass 1.5 kg.

Q.19 A 4 kg block rests on a plane inclined at 20° to the horizontal. The coefficient of kinetic friction is 0.25. A variable force \(P=(2t+5)\) N, where t is in seconds, acts on the block parallel to the plane, pointing down the slope. If the block starts from rest at \(t=0\), find its velocity after 4 s using the impulse–momentum principle.

Q.20 A 60 kg crate rests on a horizontal floor for which the coefficient of kinetic friction is 0.25. It is pulled by a force of 400 N acting at 25° above the horizontal. Assuming the crate does not tip over, find its velocity 5 s after starting from rest, using the impulse–momentum principle.

Figure (described in words): A crate on a horizontal floor; a 400 N force acts on the crate pulling to the right at 25° above the horizontal.

Q.21 A planar elbow manipulator is shown. Write the D-H parameters and hence locate the position and orientation of the end E of the manipulator. Given \(L_1=L_2=250\) mm and \(\theta_1=\theta_2=30^\circ\). (asked in MU June 2025 KT)

Figure (described in words): Base frame \(X_0, Y_0\) at joint O. Link 1 (length \(L_1\)) goes from O to joint A at angle \(\theta_1\) measured anticlockwise from \(X_0\). Link 2 (length \(L_2\)) goes from A to the end point E; \(\theta_2\) is measured anticlockwise from the extension of link 1 to link 2.

Q.22 A three-link planar (3R) robot arm has link lengths \(L_1=0.5\) m, \(L_2=0.4\) m and \(L_3=0.2\) m. At an instant the joint angles are \(\theta_1=30^\circ\) (link 1 measured anticlockwise from the base \(X_0\) axis), \(\theta_2=45^\circ\) and \(\theta_3=-30^\circ\) (each measured from the extension of the previous link; negative means clockwise). Prepare the D-H parameter table and perform forward kinematics to find the position and orientation of the end effector.

Figure (described in words): Base joint O at the origin of \(X_0, Y_0\). Link 1 (0.5 m) rises from O at 30° to \(X_0\) to joint A. Link 2 (0.4 m) goes from A to joint B, turned 45° anticlockwise from the extension of link 1. Link 3 (0.2 m) goes from B to the end effector E, turned 30° clockwise from the extension of link 2.