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PHYS1250 Fundamental Physics

Assignment Two

Due Date: October 12, 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 semi-circular track A of radius R is at rest on the horizontal plane, where A is adjacent to block B. A small block C is released from the top position of the curved surface of A, as shown in the figure. The initial height of block C is h above the horizontal plane. Assume that objects A, B, and C have the same mass m and all surfaces are smooth, solve the following problems.

(a) Find the speed of B when A and B just separate. (10 marks)

(b) Find the maximum height that C can reach. Your measurement should be made from the horizontal plane. (10 marks)

2. A cubical box is constructed from identical metal plate of uniform density and negligible thickness. The box is open at the top and has edge length 40 cm. Find the coordinates of the center of mass of the box. If the base plate is replaced by an alloy plate of the same size, uniform density and negligible thickness, find the ratio of the density of metal to that of alloy such that the center of mass of the box is located 10 cm above the base plate. (15 marks)

3. The uniform solid sphere of radius R has a spherical hole in it, locate the center of mass of the object using the coordinate system provided in the figure. The center of the solid sphere is located at (0, 0, 0). The spherical hole has radius a and its center is located at (b, 0, 0). (15 marks)

4. A 100 g mass is attached to one end of a 1.0-m-long massless string which is suspended vertically from a fixed point. The mass is pulled 8.0 ◦ to one side and released. How long does it take for the pendulum to reach 4.0 ◦ on the opposite side? (15 marks)

5. A uniform turntable is a circular disk of mass M and radius R1. It is free to rotate about a smooth and vertical axis at its center. Initially, the turntable is at rest and a man of mass m walks on the turntable and traces a circular path of radius R2 < R1 about the axis. If he walks with a speed v relative to the turntable, find the angular speed of the turntable? The moment of inertia of the turntable about the axis is MR1 2/2. (15 marks)

6. A uniform rod of length l and mass m is pivoted at its end and is free to rotate about a horizontal axle. Initially, the rod is at rest and vertical. A bullet of mass m moving with speed v collides normally and perfectly inelastically to the rod. The point of collision is at a distance 3 l/4 from the pivot. The moment of inertia of the rod about the pivot is ml2/3.

(a) Find the angular speed of the rod just after the collision. (5 marks)

(b) Find the impulse exerted on the rod by the bullet due to the collision. (5 marks)

(c) Find the impulse exerted on the pivot by the axle due to the collision. (5 marks)

(d) Find the height climbed by the center of mass of the rod. (5 marks)