PHYS1250 Fundamental Physics

Assignment One

Due Date: October 28, 2024 at 11:00 pm

The magnitude of gravitational acceleration is denoted by g throughout this paper.

The problems in this paper have a wide range of difficulties. They are specially designed to strengthen students’ concepts and to develop the problem solving skills. Students are expected to drill the prob-lems with perseverance.

1. A massless L-shaped framework has two arms. The length of longer arm is 2L and that of the shorter arm is L. Particle A, of mass 2 m, and particle B, of mass 3 m are attached to the longer arm and shorter arm respectively, as shown in the figure. The system is installed on a vertical plane such that it can rotate about point O which is the junction of arms.

Initially, OA is held horizontally, then the system is released. Please solve the following problems.

(a) Find the speed of A when it is at the lowest position. (6 marks)

(b) Find the greatest height that B can reach. (6 marks)

(c) Find the maximum speed of B. (6 marks)

Useful formulae:

sin(A + B) = sin A cos B + cos A sin B and sin(A ? B) = sin A cos B ? cos A sin B

2. A light inextensible string has one end connected to particle A and another end connected to particle B on the smooth table. Particle A is hanging over a smooth and fixed ring O when particle B moves in a horizontal circle of radius h/2 on the table and particle A is at equilibrium, as shown in the figure. Given that the ring is above B by h and particles A and B have masses m and 2 m respectively. Show that the time particle B takes to describe the circle once is (10 marks)

3. A particle is projected with speed u at an angle α above the horizontal from the top of a pole of height h and strikes the horizontal ground at a point P, where P has a horizontal distance R from the pole and 0? < α < 90?.

(a) If and R = 2 h, find the two possible values of α. (15 marks)

(b) If find the maximum value of R and the corresponding value of α. (15 marks)

4. Three identical particles A, B, and C, each of mass m, are placed on a smooth horizontal table. A is joined to B and C by light threads AB, AC, and the angle BAC is 60?. An impulse I is applied to A in the direction BA. Find the initial velocities of the particles and show that A begins to move in a direction making an angle of with BA. (14 marks)

5. A wedge of mass m and vertex angle ? has a smooth surface in contact with table, where 0? < α < 90?. A particle of mass m is placed on the inclined surface of wedge. This inclined surface is rough and the coefficient of kinetic friction is tan λ. The system is released from rest and both the particle and the wedge move afterwards. Write down the equations of motion of the particle along and perpendicular to the rough inclined surface with respect to an inertial observer. Show that the acceleration of the wedge is given by (14 marks)

[Useful formula: sin(A ? B) = sin A cos B ? cos A sin B]

6. Masses m1 and m2 are connected to the two ends of a light string which passes over a smooth and massless pulley. As shown in the below figure, the pulley is fixed on the top of a smooth incline which has an elevation angle 30?. Initially, m2 is at rest at the bottom of the incline and m1 is hanging freely. The system is released and the time taken for m2 to reach the top of the incline is t. The measurement is repeated for the case when the positions of m1 and m2 are exchanged. The required time for m1 to climb up the incline is t/3. Find m1 : m2.? ? (14 marks)