1 a a massless rope is stretched horizontally

Question

1. a) A massless rope is stretched horizontally between two supports that are 3.0 m apart. When an object of weight 3200 N is hung at the center of the rope, the rope is observed to sag by 50 cm. Calculate the tension in the rope. 

b) A ball of mass kg, 5.0 x10^-2 starting from rest, falls a height of 4.0 m and then collides with the ground. The ball bounces up to a height of 2.0 m. The collision with the ground takes place over a time s. 4.0 x10^-3 Determine (i) the momentum of the ball immediately
before the collision, (ii) the momentum of the ball immediately after the collision and

(iii) the average force of the ground on the ball. 

2. a) The speed of an aeroplane is 1200 ms^-1. The engines take in 80 kg of air per second and mix it with 40 kg of fuel. This mixture is expelled after it ignites and it moves at a velocity of 3000 ms^-1 relative to the aeroplane. Calculate the thrust of the engine. 

b) A 2400 W engine pulls a 200 kg block at constant speed up a 12.0 m long, 25.0° incline. Determine long does it takes to cover this distance. 

3. a) A ball has an angular velocity of 0.5 rads^-1 counterclockwise. Some time later, after rotating through a total angle of 4.5 radians, the ball has an angular velocity of 5.1 rads^-1 clockwise. Determine the angular acceleration, the average angular velocity and how much time it takes for the ball to attain this velocity. 

b) When a high diver wants to execute a flip in midair, she draws her legs up against her chest. Why does this make her rotate faster? What should she do when she wants to come out of her flip? 

4. a) What is the maximum torque exerted by a 60 kg person riding a bike, if the rider puts all his weight on each pedal when climbing a hill? The pedals rotate in a circle of radius 17 cm. 

b) A ball of mass 1.5 kg rolling to the right with a speed of , ms 3.6 ms^-1 collides head-on with a spring with a spring constant of 2.0 Nm^-2. Determine the maximum compression of the spring and the speed of the ball when the compression of the spring is 0.10 m. 

5. a) A force F = A(x/a - 1) is acting on a particle along the x-axis. Determine the work done by the force in moving the particle from x = 0 to x = 2a.

b) A block of mass 6.0 kg slides from rest at a height of 2.0 m down to a horizontal surface where it passes over a 1.5 m rough patch. After crossing this patch it climbs up another incline which is at an angle of 30° to the ground. The rough patch has a coefficient of kinetic friction µ_k = 0.25 What height does the block reach on the incline before it comes to rest? 

6. a) The distance between the oxygen molecule and each of the hydrogen atoms in a water (H₂O) molecule is 0.96 Å and the angle between the two oxygen-hydrogen bonds is 105°. Treating the atoms as particles, find the centre of mass of the system. 

b) A stationary ball, with a mass of 0.2 kg, is struck by an identical ball moving at 4.0 ms^-1 After the collision, the second ball moves 60° to the left of its original direction. The stationary ball moves 30° to the right of the moving ball's original direction. What is the velocity of each ball after the collision? 

7. a) An L-shaped object of uniform density is hung over a nail so that it is free to rotate. Determine the angle that the long side of the object makes with the vertical. The long side of the L-shaped object is given to be twice as long as the short side. 

b) A solid cylinder rolls up an inclined plane without slipping. If the incline makes an angle of 30º to the horizontal and the coefficient of static friction is µ_s = 0.40 find its acceleration. Also determine the angle of the inclined plane at which the object will start to slip. 

8. a) A particle of mass 4.0 kg, initially moving with a velocity of 1ms 0.5 -collides elastically with a particle of mass 6.0 kg, initially moving with a velocity of -0.8 ms^-1 What are the velocities of the two particles before and after the collision in the centre of mass frame of reference? What are the velocities of the two particles after the collision in the laboratory frame? 

b) A 30.0 kg girl stands at the rim of a merry-go-round that has a moment of inertia of 2m kg 500 and a radius of 3.00 m. The merry-go-round is initially at rest. The woman then starts walking around the rim clockwise at a constant speed of 0.2^-1ms.

i) In what direction and with what angular speed does the merry-go-round rotate?

ii) How much work does the girl do to set herself and the merry-go-round into motion? 

9. a) The comet Encke has an aphelion distance of 6.1 x 10¹¹ m and perihelion distance of 5.1 x 10¹¹ m. The mass of the sun is 2.0 x 10³⁰ kg. Find the speed of the comet at the perihelion and the aphelion. 

b) The planet earth is m 1.5 x 10¹¹ from the sun and orbits the sun in one year. The planet Pluto takes 248 years to orbit the sun. How far is Pluto from the sun? 

10. a) Consider a simple pendulum of mass m mounted inside a railroad car that is accelerating to the right with constant acceleration a. Analyse this problem in the non inertial frame of reference to find the angle φ with the vertical direction at which the pendulum will remain at rest relative to the moving car. 

b) On Jupiter a day lasts for 9.9 earth hours and the circumference at the equator is 448600 km. If the measured value of gravitational acceleration at the equator is 24.6 ms^-2 what is the true gravitational acceleration and the centrifugal acceleration.