Question 1. The work energy theorem states that the work done by all the forces acting on an object is equal to the change in its kinetic energy. Use the concept of work and energy to answer the following question. (Note, this question can also be answered using Newton's laws and calculating accelerations. NO MARKS will be given for this type of solution).

A 25 kg object is sliding down on 30 m long ramp by a under the force of gravity. Note that the ramp has an angle with the horizontal of θ = 35?.

(a) What is the final velocity of the object if there is no friction?

(b) If the final velocity is 3 m/s calculate the frictional force and describe its direction.

(c) What is the coefficient of friction in part (b)?

Question 2. An electric motor rotating a grinding wheel at 85 rev/min is switched off. Assuming constant angular acceleration of magnitude 10 rad/s²,

(a) how long does it take for the wheel to stop?

(b) Through how many radians does it turn during the time found in (a).

Question 3. Three forces try to rotate a seesaw. For d₁ = 25 cm, d₂ = 10 cm, d₃ = 30 cm, θ = 50^o, Φ = 24^o, |F₁| = 18 N, |F₂| = 6 N, and |F₃| = 20 N.

(a) Find the moment arm for each of these forces.

(b) Determine the magnitude and direction of torque that each force exert on the wheel.

(c) Find (both the magnitude and the direction of) the net torque on the seesaw.

Question 4. A 75 kg man is sitting on the edge of a flat disks that weighs 340 kg with radius 4 m. The disc and the man are spinning at 0.25 revolutions per second about an axis through the centre of the disc. The moment of inertia for the disc is M R²/2 and the man can be treated as a point mass.

(a) Calculate the total angular momentum and kinetic energy of the system.

(b) A bear that weighs 60 kg touches down on the disc (straight from above) and lands on the edge. What is the final kinetic energy of the bear, man, and disc system? What was the change in total kinetic energy? You can treat the bear as a point mass.

Question 5. Two planets rotate around each other. Each planet has a mass of 5.40 × 10²⁴ kg. What is the tangential speed of the planet system if the distance between the planets is 6.00 × 10⁸ m? - Hint: the planets rotate around their centre of mass; because their mass is equal, the centre of mass is situated in between the two planets.

Question 6. Two astronauts, each having a mass of 120 kg, are connected by a 35 m rope of negligible mass. They are isolated in space, orbiting their centre of mass anticlockwise at speed of 0.5 m/s. Treat the astronauts as point particles and calculate

(a) the magnitude and direction of their angular momentum, and

(b) the rotational energy of the system.

(c) By loosening the rope, the astronauts increases the distance between them to 70 m. What is the new rotational energy of the system?

(d) What is the new angular momentum direction and magnitude of this system with the longer rope?

(e) How much work is done by the astronauts in increasing the length of the rope?

Question 7. A wheel is rotating anti-clockwise with radius r = 8 m on a frictionless axis. The wheel pulled by a rope with a mass of m = 15 kg attached to its end; the mass m falls under gravity. The rotational inertia of the disc is I = mr²/2.

(a) Compute the tension in the rope?

(b) Compute the acceleration of the mass.

(c) Hence or otherwise, determine the angular accel- eration of the wheel.

Question 8. A solid uniform ball of radius R and mass m (moment of inertia I = 2/3 mR²) rolls without slipping up a hill. It has an initial speed of vb = 32m/s at the bottom of the hill which is h = 40 m high as shown in the diagram. At the top of the hill, the ball is moving hor- izontally and then goes over a vertical cliff.

(a) Show that the initial kinetic energy KE of the ball at the bottom of the hill is given by K = 5/6mv_b².

(b) Determine the speed of the ball at the top of the hill.

(c) Show that the rotational kinetic energy of the ball at the top of the hill is given by KE_r = 1/3mv_t² where v_t is the speed of the ball at the top of the hill.

(d) Determine the speed of the ball just before it hits the ground. You can neglect any effect of air resistance.

(e) This speed is greater than the initial speed when it set off up the hill. Does this mean that the ball has somehow gained energy? Explain your answer.

Question 9. A three-particle system consists of masses m₁ = 4 kg, m₂ = 3 kg, and m₃ = 2 kg.

What are

(a) the x-coordinate,

(b) and the y-coordinate of the system's centre of mass?

(c) If m³ is gradually increased, does the centre of mass of the system shift toward or away from that particle, or does it remain stationary?

Question 10. Use the parallel-axis theorem to calculate the rotational inertia of a meter stick, with centre of mass inertia I_cm = ML²/12 and mass M = 25 kg, about an axis perpendicular to the stick and located at the 35 cm mark.

Question 11.  A ball of mass m is suspended from a ceil- ing. The ball is displaced from its equilib- rium position (Xeq ) by a man and released from rest at the tip of the handler's face (see picture). If the handler remains stationary,

(a) will the handler get struck by the ball on its return swing? Explain your reasoning.

(b) Calculate the maximum speed of the ball if m = 150kg, h = 12 m and θ = 10?.

Question 12. A 4 kg solid ball strike a wall with a speed 30 m/s at an angle 55? with the surface on a space station. It bounces off with the same speed and angle. If the ball is in contact with the wall for 0.75s, what is the impulse that the ball experience, and what is the average force exerted on the ball by the wall.

Question 13. A blue ball moving with 50 m/s strikes a red stationary ball of identical mass. After the collision the blue ball moves at 20 m/s, at an angle θ = 25? with respect to the original line of motion.

(a) Find the velocity of the red ball after the collision, and find the angle Φ.

(b) Is the collision elastic or inelastic? - you must show a calculation to get marks for this question.

(c) Does it make sense to have more total kinetic energy after the collision than before collision?

Question 14. A m = 80 g bullet is fired horizontally into a M = 5 kg block of wood suspended by a long cord. The bullet sticks in the block. Compute the ve- locity of the bullet if the impact causes the block to swing h = 50 cm above its initial level. Is the collision between the block and the bullet elastic or inelastic? - please explain in detail.

Static equilibrium, and gravitation

Question 15. A 100 kg projectile is shot from the surface of the Earth. If itl takes off at 55? to the ground, at what speed does the ball escape the Earth's gravitational pull and never return to the ground. Assume that it experiences negligible friction, and ignore all effects due the the Earth's rotation. The Earth's radius is r = 6371 kms; the Earth's mass is M = 5.97219 × 10²⁴ kg, and the universal gravitational constant is G = 6.67384 × 10^-11 m³kg^-1s^-2.

Question 16. (a) Explain the superposition principle and the shell theorem in the context of gravity. Hence solve the next problem.

(b) Four 40 kg spheres are located at the distance d1 = 0.2 m and d2 = 0.8 m as shown. What are the magnitude and direction of the net gravitational force on sphere B due to spheres A, C and D? Hint: Remember the gravitational force is a vector quantity, the answer must be a vector and its magnitude!

Question 17. Two hypotheticala planets of mass m₁ = 8.58 × 1024 kg and m₂ = 9.49 × 1022 kg and radii r₁ = 7100 km and r₂ = 4500 km respectively, are at rest when they are a large (assume infinite) distance apart. Because of their gravitational attraction, they head toward each other on a collision course.

(a) When their centre-to-centre separation is distance d, find the speed of each planet

(b) Hence or otherwise, at that separation d, find their relative velocity; that is the velocity of one planet as it appears from the perspective of another planet.

(c) Find the kinetic energy of each planet just before they collide.

Hint: Use conservation of momentum and energy!!

Question 18. A block A (mass M = 80 kg) is in equilibrium, but it would slip if block B (mass m = 20 kg) were any heavier. The mass m hangs on three ropes that have negligible mass and θ = 45^o.

(a) Draw a free body diagram of the forces and tensions involved.

(b) Find the tensions T₁, T₂, and T₃.

(c) What is the coefficient of static friction between block A and the surface below it?

Simple harmonic motion, and Rolling

Question 19. (a) Give the definition of simple harmonic motion, and give an example of simple harmonic motion. Then, solve the following questions:

(b) If (x - x0) = ?x is the displacement from some reference position x0, and the total spring force acting on the body is F = -k?x, then Newton's law implies that -k?x = ma. This motion creates simple harmonic motion (SHM). Show that Newton's law for SHM is satisfied by the function x = A sin(ωt + Φ) + x0, where A, ω, and Φ are constants and ω =√(k/m). Hint: remember F = ma = mx¨

Question 20. A 0.7 kg mass oscillates horizontally without friction at the end of a horizontal spring with spring- constant k = 10 N/m. The mass is displaced 1.2 m from equilibrium and released. Find

(a) its maximum speed, and

(b) its speed when it is 0.4 m from equilibrium.

(c) What is the magnitude of acceleration in each of these cases?

(d) What is the frequency and period of the motion?

(e) How can you tell if this system oscillates with a simple harmonic motion?

Question 21. A uniform ball, of mass m = 10 kg and radius R, rolls smoothly from rest down a ramp at an angle θ = 15^o. Assume that the ball has the centre of mass inertia Icom = (2mR²)/5.

(a) If the ball descends a vertical height h = 8m to reach the bottom of the ramp, what is its speed at the bottom of the ramp?

(b) What are the magnitude and direction of frictional force that acts on the ball as it rolls down the ramp?

Question 22. (a) Explain, what is the difference between stress and strain?

(b) Explain the meaning of the Bulk modulus.

Question 23. A brass sphere has a radius of 6.8 m. How much stress must be applied to the sphere to reduce the radius to 6.75 m? The bulk modulus of brass is B = 6.10 × 10¹⁰ N/m2.

Question 24. (a) What is Young's modulus? How is it determined?

(b) A circular copper rod 60 m long is placed vertically under the roof of a building to support it. The radius of the rod is r = 500 mm, and its Young's Modulus is E = 117 GPa. A sensitive strain gauge indicates that the rod's length decreases by 0.2 mm. What is the magnitude of the load that the copper rod supports? Hint: Make sure that you match the units in your calculation.