University of Mumbai first-year engineering Semester I Engineering Mechanics paper, December 2025, NEP 2020 scheme, with step-by-step solutions written with AI help and checked.
Mumbai University FE Sem 1 Engineering Mechanics (ESC101) Question Paper December 2025
Written semester papers (NEP 2020 scheme): 60 marks and 2 hours for most first-year courses. Question 1 is compulsory, then any three of questions 2 to 6.
Sections
| Q.1 Solve any five (compulsory) | 6 questions | |
| Q.2 (attempt any three of Q.2 to Q.6) | 2 questions | |
| Q.3 | 2 questions | |
| Q.4 | 2 questions | |
| Q.5 | 2 questions | |
| Q.6 | 2 questions |
Chapters covered
Instructions
Solve on paper with a timer, then check against the solutions.
Q.1 Solve any five (compulsory)
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Q.1 (a) A car is made to move by applying resultant force R = 2000 N along the x-axis. This resultant is developed due to two pulling forces \(F_1\) and \(F_2\) on two ropes as shown in figure. Determine the tension in individual ropes.
[Figure: car with front point O; resultant R = 2000 N along the x-axis (to the right). Rope force \(F_1\) acts at O at 35° above the x-axis and rope force \(F_2\) at 25° below the x-axis.]
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Q.1 (b) For the member shown in figure find the magnitude and direction of the resultant reaction at the hinged support A.
[Figure: L-shaped member BAC. A is a hinge at the corner. AB is vertical, 3 m long, with its upper end B attached by a horizontal link to a vertical wall on the left. AC is horizontal, 4 m long, to the right of A; a vertical downward load of 30 N acts at C.]
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Q.1 (c) Prove that angle of friction is equal to angle of repose.
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Q.1 (d) A force 30 N is acting at point P(3, 2, 1) m. The force makes 35° & 85° with x & y axes respectively. Express the force in vector form.
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Q.1 (e) The acceleration of the particle is defined by the relation \(a = 25 - 3x^2\ \text{mm/s}^2\). The particle starts with no initial velocity at the position x = 0. Determine the position where the velocity is maximum and the corresponding maximum velocity.
[The exponent of x is partly obscured in the source scan; it is read as \(x^2\).]
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Q.1 (f) A cricket ball of mass 0.15 kg is moving with a velocity of 20 m/s. A player catches it and brings it to rest in 0.1 s. Find the impulse and average force applied.
Q.2 (attempt any three of Q.2 to Q.6)
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Q.2 (A) Find the coordinate of the centroid for the shaded part of the lamina with respect to given reference axis.
[Figure: origin O with x-axis to the right and y-axis upward. A semicircle of radius R = 8 cm (the digit is partly obscured in the scan; read as 8 cm, consistent with the drawing) centred at O lies below the x-axis, spanning x = −8 cm to x = 8 cm. A right-angled triangle lies above the x-axis with one vertex at O, base along the x-axis from O to x = 8 cm, vertical side at x = 8 cm, and hypotenuse from O at 60° to the x-axis. A semicircle of radius r = 3 cm, below the x-axis with its diameter on the x-axis from x = −8 cm to x = −2 cm (centre at (−5, 0)), is cut out. The remaining region (big semicircle + triangle − small semicircle) is shaded.]
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Q.2 (B) The motion of a particle is defined by the position vector, \(\bar r = 5t\,i + 3t^2\,j\) where ‘r’ is in meters and t is in seconds. At the instant when t = 2 sec, find i) Tangential and normal components of accelerations ii) Radius of curvature.
Q.3
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Q.3 (A) A system of forces acting on a bell crank is as shown in fig. Determine the magnitude, direction and the point of application of the resultant w.r.t. O.
[Figure: bell crank pivoted at O. A horizontal arm extends 300 mm to the left of O (two segments of 150 mm). A 1000 N vertical downward load acts at 150 mm left of O; a 500 N force acts at the left end (300 mm from O), directed downward to the right at 60° to the horizontal. A second arm rises from O to the upper right at 60° to the horizontal, 300 mm long (two segments of 150 mm). A 1200 N vertical downward load acts at its mid-point (150 mm from O) and a 700 N horizontal force acts to the left at its upper end (300 mm from O).]
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Q.3 (B) In the mechanism shown in fig, the angular velocity of link AB is 5 rad/sec anticlockwise. At the instant shown, determine the angular velocity of link BC and velocity of piston C.
[Figure: link AB hangs vertically from a fixed pivot A; AB = 200 mm, rotating at 5 rad/s anticlockwise. Connecting rod BC (mid-point M) runs from B down to the right to piston C, which slides in a vertical guide. C is 450 mm to the right of B and 150 mm below B.]
Q.4
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Q.4 (A) 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.
[Figure: roller of radius 50 cm, centre C, resting on a horizontal floor against a step (obstruction) 12 cm high on its right. Force P acts at the centre C, directed up and to the right at 30° to the horizontal.]
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Q.4 (B) The system shown in figure is initially at rest. Neglecting friction determine the force ‘P’ required if the velocity of the collar B is 5 m/s after 2 sec and corresponding tension in the cable.
[Figure: a horizontal smooth rod is fixed to a vertical wall on the left. Collar B (mass 3 kg) slides on this rod and is pulled to the right by a horizontal force P. A small pulley is attached to the left face of collar B by a short horizontal bar. A cable is fixed to the wall at its upper end, runs horizontally to the right, wraps around the pulley on B and returns to the left to a second pulley that is mounted on the wall by a horizontal bracket (lower left). From this fixed pulley the cable hangs vertically down to block A of mass 1.5 kg.]
Q.5
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Q.5 (A) Find the force P required to pull block B shown in figure. Coefficient of friction between A and B is 0.3 and between B and floor is 0.25. Mass of A = 20 kg and B = 30 kg.
[Figure: block B rests on a horizontal floor and block A rests on top of B. A string attached to A runs up to the left to a vertical wall, making 30° with the horizontal. A horizontal force P pulls B to the right. (The “B” before “= 30 kg” is obscured in the scan.)]
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Q.5 (B) A 3 kg mass kept on a 30° slope is subjected to a variable force \(P = (t^2 + 3)\), where P is in ‘Newton’ and t is in ‘second’ find velocity of the block after 5 second using Impulse momentum principle. At t = 0, V = 0.
[Figure: block on an incline rising to the right at 30°, \(\mu = 0.2\); force P acts on the block parallel to the incline, pointing down the slope.]
Q.6
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Q.6 (A) For the beam loaded as shown in figure, Calculate the support reactions.
[Figure: cantilever AB, 3 m long, fixed at A (left wall). A UDL of 10 kN/m acts over the first 2 m from A. At 2 m from A, a 20 kN force acts downward to the left, inclined at 60° to the beam (angle measured on the right side, between the force and the beam). A vertical load of 15 kN acts downward at the free end B (3 m from A).]
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Q.6 (B) A simple 2 DOF planar robot is shown. Prepare a D H parameter table and perform forward kinematics to find the location and orientation of end effector B.
[Figure: base joint O on a fixed support. Link OA, 3 m long, makes 110° measured anticlockwise from the horizontal x-axis at O. Link AB, 2 m long, is turned clockwise by 65° from the extension of OA (dashed line) at joint A. B is the end effector.]