mechanics4

Q 1. A smooth platform with length L is fixed on the ground as shown in the diagram. Two small objects, A and B, are placed at the centre of the platform. The two objects are in contact. The upper surface of object A is a semi-circle rail with radius R. (R height h relative to the platform. The masses of A, B and C are all the same ( = m). Object C slides down on the rail starting from zero velocity. AC are always in contact during the motion.

Find

1) The speed of object B when A and B separate.

2) The maximum height which can be attained by object C (relative to the platform) after A and B separates.

3) Determine whether object A will fall from the left side or the right side of the platform. Meanwhile, estimate the time it takes for A to leave the platform after the separation of A and B.

Q 2. A cylinder of mass M is placed on a smooth horizontal surface. The cross section of M is a quarter-circle A with Radius R. The surface of the cylinder is smooth and a small object B with mass m is placed on its top. Initially both A and B are stationary in the fixed x-O-y coordinate system shown in the diagram. If the small object slides down from the top of the cylinder, find the equation of

motion of the small object before the object detaches from the arc of circle.

Q 3. Two identical small beads with mass m are attached on a smooth circular ring that stands vertically on the ground. Initially the beads are resting on the top of the ring. They then start to slide down. Derive the relationship between the mass of the ring and the mass of the beads such that the ring can jump up from the ground. Also, find the location of the beads when the ring jumps up.

Q 4. The perigee and apogee of the first artificial satellite of our mother country is 87km and 6809km respectively. Find the speed of the satellite when it passes through the perigee and apogee. What is the period of the satellite?

Q 5. This problem models the lounging of a detector from the surface of the earth to Mars. We assume that the earth and Mars are moving around the sun in circles resting on the same plane. The radius of the orbit of Mars is Rm, which is 1.5 times larger than that of the earth. An economical and simple way to lounge the detector consists of 2 steps. First, a rocket is used to accelerate the detector on the earth’s surface such that it acquires enough kinetic energy to overcome the gravitational force of the earth and becomes a satellite moving around the earth. Second, at a suitable time, an engine that connects to the detector ignites for a short instant and accelerates the detector along its direction of motion. After a short (negligible) time, the speed of the detector increases to a suitable value such that the detector moves along an elliptical orbit that connects the earth and Mars, with the two planets located at the end points of the ellipse (see Diagram A)

(1) In Step I, what is the minimum speed needed for the detector to become an artificial satellite that moves along the earth’s orbit?

(2) After the detector becomes a satellite moving around the earth, on 1 March 00.00am of a certain year, the angular distance between the detector and Mars is measured to be 60o (diagram B). What is the date that the engine of the detector should be fired so that the detector can fall on the surface of Mars (Correct to day)? Given: radius of the earth: 6.4x106 m, acceleration due to gravity :9.8m/s2 .

Q.6 A solid ball with uniform density and radius R (originally resting on a desk) is hit by a horizontal force F at height h ( <2R ). The force lasts for a small time ∆t. Afterward, the ball is found to roll without sliding. What is h?

Q.7 A small ring with radius r =10cm falls on a desk from a height h = 20cm. Initially, the ring is rotating upon its axis with angular frequency ωo= 21 rads-1 (see figure). Assuming that the collision between the ring and desk is inelastic such that the ring stays on the desk after collision. How long will it takes for the ring to stop rotating if the coefficient of friction between the ring and desk is µ=0.3.