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COLLEGE OF SCIENCES DEPARTMENT OF MATHEMATICAL AND PHYSICAL SCIENCES PHY 214 ANALYTICAL MECHANICS 2016/17 SESSION TUTORIAL QUESTIONS (TO BE SUBMITTED ON/BEFORE THURSDAY 15TH JUNE 2017) 1. (a) (b) (c) What is rectilinear motion? Hence, for a particle traveling along the π₯-axis, write the expressions which describe its (i) average velocity, (ii) instantaneous velocity, (iii) average acceleration, and (iv) instantaneous acceleration. A particleβs position at any time π‘ is given by: π₯ = (2π‘ 4 + 5π‘ 2 + 3π‘ )π. Obtain (i) the velocity and acceleration at any time, (ii) the position, the velocity and acceleration at π‘ = 2π πππ 3π , and (iii) the average velocity and average acceleration between π‘ = 2π πππ 3π . If the particle in (b) above moves under the influence of a constant force πΉ and its initial speed is ππ , show that the speed of the particle at any position π₯ is given by: 2πΉ π = βππ2 + ( ) π₯ π where π is the mass of the particle. 2. (a) (b) (c) What is referred to as a freely falling body? Write the differential equation from Newtonβs second law for such body. An object of mass π is thrown vertically upward from the earthβs surface with speed ππ . Find (i) the position at any time, (ii) the time taken to reach the highest point, (iii) the maximum height reached Given that a uniform force with linear frictional damping acts on mass m in ππ£ (b) above, such that its equation of motion is π = βππ β π½π, show that ππ‘ the position of m at any time is: π½ π π π π§ = (ππ + π) (1 β π βππ‘ ) β ππ‘ π½ π½ π½ Page 1 of 3 3. (a) (b) When is a particle said to be in equilibrium? A particle of mass π is suspended in equilibrium by two inelastic strings of lengths π and π which are separated by a distance π apart as shown below. Find the tension in each string. 4. (a) Show that the magnitude of centripetal acceleration of a particle undergoing circular motion is: π2 π= π A particle moves so that its position vector is given by πβ = cos ππ‘ πΜ + ββ of the particle sin ππ‘ πΜ where π is a constant. Show that (π) the velocity π is perpendicular to πβ, (π) the acceleration πβ is directed toward the origin ββ = and has magnitude proportional to the distance from the origin, (π ) πβ × π π ππππ π‘πππ‘ π£πππ‘ππ. (b) 5. 6. (a) (i) (ii) (iii) (b) (i) (ii) (a) (b) Describe a constrained motion? A particle π of mass π slides without rolling down a frictionless plane inclined at angle πΌ. If it starts from rest at the top of the incline, find (π) the acceleration, (ππ) the velocity, and (πππ) the distance traveled after time π‘. Suppose the incline in (b) has a coefficient of friction π. Find (i) the acceleration, (ii) the speed, and (iii) the distance travelled by the particle after time π‘. (c) 7. (a) State and prove the work-energy theorem. Under what conditions is a force field said to be conservative. Apply the conditions in (ii) to show that πΉβ = (π¦ 2 π§ 3 β 6π₯π§ 2 )πΜ + 2π₯π¦π§ 3 πΜ + (3π₯π¦ 2 π§ 2 β 6π₯ 2 π§)πΜ is a conservative force field State and prove the principle of conservation of angular momentum βββ. Obtain the relationship between Torque πβ and angular momentum Ξ© The equation of motion of a simple harmonic oscillator is given by π2 π₯ π 2 πΜ = βππ₯πΜ ππ‘ Page 2 of 3 (i) (ii) (b) (c) 8. (a) (b) Obtain the general solution to the equation and the boundary conditions. Obtain the period (P) and frequency (f) of the motion. Assuming the harmonic oscillator in (a) above has a damping force ββ acting on it. Obtain the solutions for the following cases πΉβπ· = βπ½π (π) Over-damped motion (ππ) Critically damped motion, and (πππ) Underdamped motion Show that the system π₯Μ + 4π₯Μ + 3π₯ = 0 is over-damped. Distinguish between Constrained motion and Free fall motion At time π‘ = 0 a parachutist having weight of magnitude ππ is located at π§ = 0 and is travelling vertically downward with speed ππ . If the force or air resistance acting on the parachute is proportional to the instantaneous speed, find (π) the speed at any time π‘ > 0, (ππ) the distance travelled at any time π‘ > 0, (πππ) the acceleration at any time π‘ > 0, and ππ (ππ£) show that the parachutist approaches a limiting speed given by βπ½. Page 3 of 3