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This exam covers essential concepts in mechanics, including motion, forces, energy, and friction. Key topics include the equations of motion, conservation of momentum, work-energy theorem, and properties of friction. Und…
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Find the value of 60 joule/ minute in a system, which has 100gm, 100cm and 1 min as a fundamental units.
Convert to fundamental units and calculate using dimensional analysis.
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Find the dimensions of momentum, in terms of frequency, velocity and density as the fundamental units.
Momentum dimensions are [M][L][T⁻¹].
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(i) Derive equation of motion: S = ut + ½ at² by calculus method. (ii) What is the retardation of a moving particle if the relation between time and position is: t = py² +Qy, (where p and Q are constants)
Use calculus to derive and differentiate to find retardation.
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(i) State and prove impulse momentum theorem. (ii) Fig. shows the position time graph of a particle of mass 4kg. What is the impulse at t = 0 and t = 4 sec. Assume motion is one dimensional.
Impulse equals change in momentum; calculate from graph.
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(i) State and prove work-energy theorem, for variable force. (ii) A car of mass 2000 kg is lifted up a distance of 30m by a crane in 1 min. A second crane does the same job in 2 min. Do the cranes consume the same or different amounts of fuel. What is the power supplied by each crane.
The work-energy theorem relates work done to energy; calculate power for each crane.
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(i) State the laws of kinetic friction. (ii) A block of mass m rest on a horizontal surface of coefficient of friction μ. A force F is applied at an angle θ as shown. Find the minimum value of F required to just move the block.
Kinetic friction laws state that fk = μkR; derive minimum F using equilibrium.
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Two masses M and m are connected at the two ends of an inextensible string. The string passes over a smooth frictionless pulley. Calculate the acceleration of the masses and tension in the string. Given M > m.
Use Newton's second law to derive acceleration and tension.
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Whenever a body moves or tends to move over the surface of another body, a force comes into play which acts parallel to the surface of contact and opposes the relative motion. This opposing force is called friction. Static friction fs opposes the impending relative motion while the kinetic friction fk opposes the actual relative motion. Static friction is a self-adjusting force. The maximum value of static friction (f_s^max) which comes into play when a body just starts moving over the surface of another body is called limiting friction. Two important constants which are characteristics of the pair of surfaces in contact are defined as follows: Coefficient of limiting friction, μs = f_s^max / R Coefficient of kinetic friction, μk = fk / R As fk < f_s^max or μkR < μsR implies μk < μs. where R is the normal reaction between the two surfaces in contact. Both static and kinetic friction are independent of the area of surfaces in contact.
Define and explain static and kinetic friction with coefficients.
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(a) Discuss the elastic collision of two bodies in one dimensions. Calculate the velocities of the bodies after the collision.
Use conservation of momentum and kinetic energy to derive final velocities.
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(b) Four identical balls A, B, C and D are placed in a line on a frictionless surface. A and D are moved with the same speed 'u' towards the middle as shown. Assuming elastic collision, final the find velocities.
Apply conservation laws to find final velocities after collisions.
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(a) A body of mass m is placed on the floor of a lift. Find the apparent weight when the lift is (i) moving upwards with uniform acceleration 'a'. (ii) moving downward with uniform acceleration 'a'. (iii) moving downward with acceleration a > g. (b) A pump on the ground floor of a building can pump up water to fill a tank of volume 30m³ in 15 min. If the tank is 40m above the ground and efficiency of the pump is 30% how much electric power is consumed by the pump.
Calculate apparent weight using F = ma; find power using work done over time.
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(a) State and prove triangle law of vector addition. (b) The resultant of two vectors A and B is perpendicular to the vector A and its magnitude is equal to half of the magnitude of the vector B. Find out the angle between A and B.
Use vector addition principles to prove and find angle.
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(a) A circular race trace of radius 300m is bunked at an angle of 15°. If the coefficient of friction is 0.2, what is the (i) optimum speed of the car to avoid wear and tear of tyres. (ii) maximum permissible speed to avoid slipping. (b) A body is projected making angle θ with horizontal with velocity u, show that path of projectile is parabola, derive maximum height and horizontal range.
Use dynamics to find speeds; derive projectile motion equations.
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(c) A fighter plane flying horizontally at a height of 1.5km with a speed 720km/h passes directly from overhead an antiaircraft gun. At what angle from the vertical should the gun be fired for the shell muzzle speed 600m/s to hit the plane. At what maximum height should the pilot fly the plane to avoid being hit.
Calculate firing angle using projectile motion; determine height.
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(a) Define centripetal acceleration, derive expression for it. (b) Define angle of friction and angle of repose establish relation between them.
Centripetal acceleration is a = v²/r; relate angles using friction definitions.
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A block of mass 40kg is raised by a 60kg man as shown in fig. What is the action on the floor by the man in each case. (g= 10 m/s²)
The action on the floor is the sum of the weight of the man and the block.
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Answer carefully, with reason: is the total linear momentum conserved during the short time of an elastic collision of two balls.
Yes, total linear momentum is conserved in elastic collisions.
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(i) Find the components of a = 2î + 3ĵ along the direction of vectors, î + ĵ. (ii) Do a + b and a − b lie in the same plane. Explain.
Components can be found using projection; they lie in the same plane.
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Find the dimensions of a/b in the equation: F = a√x + bt², where F → force, x → distance, t→ time.
Dimensions of a/b are [MLT⁻²].
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Draw position – time graph for an object: (i) moving with positive velocity (ii) moving with negative velocity.
Graph for positive velocity is a straight line with positive slope; negative velocity has a negative slope.
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The ratio of surface tension to viscosity can be represented by the dimensions of:
acceleration
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A body starts from rest. What is the ratio of distance travelled by it during 4th and the 3rd second.
7/5
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A particle experiences a constant acceleration for 20sec after starting from rest. If it travels a distance S1 in the first 10 sec and a distance S2 in the next 10 sec, then
S2 = 4S1
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A projectile can have same range R for two angles of projection. If t1 and t2 be the time of flights in the two cases, then the product of the two times of flight is proportional to:
R
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Two bodies have their masses m1/m2 = 3 and their kinetic energies E1/E2 = 1/3. The ratio of their velocities is:
1:3
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A particle moves along x-axis from x = x1 to x = x2 under the influence of a force, F = 2x. The work done in the process is
(x2²-x1²)
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p. e. of a long spring when stretched by 2cm is U. If spring is stretched by 8cm, the p. e. stored in it is:
16U
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A block B pushed momentarily along a horizontal surface with an initial velocity V, if μ is the coefficient of sliding friction, the block B will come to rest after a time.
V/gμ
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For what value of λ vectors Ā = 2î + λĵ + k and B = 4î – 2ĵ - 2k are perpendicular to each other.
3
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0.00792 contains significant figures:
3
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A particle is projected from ground at an angle of θ with horizontal with an initial speed u. Time after which velocity vector of the projectile is perpendicular to the initial velocity is:
u/(g sinθ)
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The force required to stretch a spring varies with distance as shown in the figure. If the experiment is performed with the above spring of half the length, the line OA will:
Shift towards F-axis
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A body dropped from the top of a tower clears (7/16)th of the total height in its last second of flight. The time taken by the body to reach the ground is
4 sec
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A spring of force constant K is cut into two pieces such that one piece is double the length of the other. Then the short piece will have a force constant of.
6K
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A mass m rests on a horizontal surface. The coefficient of friction between the mass and the surface is μs. If mass is pulled by a force F as shown in fig; the limiting friction between the mass and surface is.
μs mg
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A cricketer can throw a ball to a maximum horizontal distance of 100m. How much high above the ground can the cricketer throw the same ball in both cases initial speed is the same.
25m
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(i) Which of the following is the dimension of coefficient of friction?
[M⁰L⁰T⁰]
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(ii) A block of mass 50 kg just slides over a horizontal distance of 1 m. If coefficient of friction between their surfaces is 0.2, the work done against friction is
98 J
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(iii) If reaction is R and coefficient of friction is μ, what is work done against friction in moving a body by distance d?
μRd
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(iv) A cylinder of mass 10 kg is rolling on a place with an initial velocity of 10 m/s. If coefficient of friction between surface and cylinder is 0.5, then before stopping, the cylinder will cover a distance of
7.5 m
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(v) Two iron blocks of equal masses but with different surface areas slide down an inclined plane with friction coefficient μ. If the first block with surface area A experiences a frictional force f, then the second block with surface area 2 A will experience a frictional force
f
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(i) A particle moves along a curve of unknown shape but magnitude of force F is constant and always acts along tangent to the curve. Then,
F must be non-conservative
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(ii) Which one of the following is not a conservative force?
Frictional force
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(iii) The potential energy of a body is given by U = A - Bx² (where x is displacement). The magnitude of the force acting on the particle is
proportional to x
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(iv) The potential energy of a system increases work is done
upon the system by a conservative force
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(v) A particle is moving in a circular path of radius a the action of an attractive potential energy U = -k/(2a²). Its total energy is
-k/(2a²)
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