Determine the force p and the tension of the rope and

Question

Q1.

i.   A particle is moving in a straight line has an acceleration a = 2/(0.1V+1) (m/s²). It starts from rest at time t= 0 and at time t the velocity is V.m/s and the displacement is D m. Also D=0 at t=0 sec.

ii. Derive a relationship between time t and the velocity V.

iii. Derive a relationship between displacement D and the velocity V.

iv. Draw V Vs t for 0

Q2.

A rocket's trajectory shown in Figure Q2, details the rocket's state of motion after 10 seconds from its launching point A (@ time t= 0). Following are the measured parameters of the state of motion after 10 secs:

r = 2100 m             r^. = 500 m/s,                  r^.. = 4.66m/s²   

θ = 21.5⁰               θ^. = 0.078 rad/s            θ^.. = - 0.0341 rad/s² 

i. For this instant determine the angle β between the horizontal and the direction of the rocket's trajectory and calculate the velocity and the acceleration of the rocket in i and j directions.

ii. At  this  instant  a  rocket  detaches  it's  1st   stage  rocket  engine  module. Calculate the horizontal distance from point A to the 1st stage rocket engine module when it landed on the earth's surface (Assume launching pad and the landing point of 1st stage of rocket are  at the same horizontal plane and neglect the air resistance)

Q3.

Figure Q3, shows a small cart of mass M has velocity v m/s enters a smooth circular path (vertical) at A.  The cart accelerates along the path and leaves it as a projectile at point B.

I.  Determine an expression for angle α.

II. Determine an expression for minimum velocity v that cause cart to leave point A as a projectile.

III. Calculate angle α if the carts initial velocity at v=0 m/s.

Q4.

Figure Q4 shows an arrangement of an elevator which is moving upward with a constant acceleration of m/s² starting from rest at time t=0. An electric motor (M) attached to the carriage drives the elevator. The electric motor has an efficiency of 80%. Total mass of the elevator including the pay load is 1850 kg. (Neglect masses of pulleys).

I. Calculate the tension in the rope when the elevator is moving upward at a constant acceleration of 0.25 m/s².

II.  the power drawn by the motor M after  time t=8 seconds.

Q5.

Figure Q5 shows an arrangement of a carriage system. A force P is applied on the carriage B and moving rightward at 3 m/s² constant acceleration.  At the instant shown the velocity of B is 2 m/s.

I. Determine the velocity of B relative to A.

II. Determine the absolute velocity of C

III. Determine the absolute acceleration of carriage A

IV. Determine the force P and the tension of the rope if mass A= 50 kg and mass B=40 kg (neglect the masses of pulleys, rope and the frictional forces at the wheels)