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Sma2113:Maths For Science Question Paper
Sma2113:Maths For Science
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2012
QUESTION ONE (30 MARKS)
a) Find the impulse and its magnitude developed by the force = 4???? + 6??2 - 2 + 12 from ?? = 0 ?? ?? = 2 . (2 Marks) b) A particle moves in a force field given by = ??2 - 2????3 ?? + 3 + 2???? - ??23 + 6 - 3??2??2 i. Prove that is a conservative force field. (3 Marks) ii. Find the potential associated with the force field. (4 Marks) iii. Find the work done in moving the particle form (-2,-1,-2) to (-1, 3,-2) by this force field. (2 Marks) c) A particle of mass m slides without falling down a frictionless plane AB that forms an angle with the horizontal. If the particle starts from rest at the top end of the incline, find the acceleration, velocity and distance travelled by the particle at any time t. (6 marks) d) A particle of mass m moves along a space curve C of a force field. = 6?? - 8 ?? - 60??3 + 20??3 + 36??2 . Its initial position and velocity are 0 = 2?? - 3 and 0 = 5?? + 4 respectively. Find i. Acceleration, velocity and position of the particle at time t. (4 Marks) ii. The momentum of the particle at a time ?? = 2. (3 Marks) iii. Power applied to the particle at any time t. (3 Marks) iv. Work done by the force field in moving the particle from P1 at ?? = 0 to P2 at ?? = 2. (3 Marks)
2
QUESTION TWO (20 MARKS)
A projectile is launched from the top of an inclined plane at an angle with the horizontal. If the inclined makes an angle with the horizontal and also given that the initial velocity of the projectile is 0 and air resistance is negligible, show that;
a) The range down the plane is
=
20 2 sincos(-) cos 2
(13 Marks) b) The maximum range is 0 2 (1-sin ) (4 Marks) c) At what angle of launch is this achieved? (3 Marks)
QUESTION THREE (20 MARKS)
a) The figure below shows motion of a particle attached to one end of an elastic string.
' ' fixed
L L x
Given that the length of the string is L and its mass is negligible, FR is the restoring force due to tension and is the modulus of elasticity of the string at any time, show that the normal time and the time for simple harmonic motion is 1 = 4 ??0 ?? and = 2?? respectively. (14 Marks) b) One end of an elastic spring is fixed to a point O on a smooth horizontal table and a particle of mass m is attached to the other end A, which is stretched to a point B. if L is the natural length of the spring an its stretched within the natural limits, show that the time of a complete oscillation of the particle is; = 2?? ( + 2 ) where is the elasticity of the spring. (6 Marks)
QUESTION FOUR (20 MARKS)
a) At a time ?? = 0 a parachutist having weight = ?? is located at 0 = 0 and is travelling vertically downwards. If the initial velocity of the parachute is V0 and the air resistance acting on the parachute is proportional to its speed, find; i. The speed and distance travelled by the parachutists at any time t where ?? > 0. (12 Marks) ii. The acceleration of the parachutist at any time t where ?? > 0. (4 Marks) b) The position vectors of two particles are given as 1 = ???? - ??2 + 2?? + 3 ?????? 2 = 2?? - 3??2 ?? + 4?? - ??3 . Find the relative velocity and acceleration of the first particle with respect to the second one at ?? = 2. (4 Marks)
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