Mechanics Questions. (Throughout this document take the acceleration under gravity to be g = 9.81ms 2 ).

Transcription

1 Mechanics Questions (Throughout this document take the acceleration under gravity to be g = 9.81ms 2 ). 1. A block of mass m = 10kg is resting on a rough plane inclined at θ = 30 to the horizontal. A light, frictionless rope attached to the block passes over a pulley at the corner of the block as shown. If the coefficient of friction, µ = 0.1, what must the force F be so that the block stays at rest? 2. Four ropes are tied to a very small sphere of m = 0.1kg, suspending it in the air. The sphere is in equilibrium. For angles α = 30, β = 20, δ = 10 and γ = 60, and F 1 = 10N, with forces F 1 andf 2 in a vertical plane and F 3 and F 4 in the horizontal plane. Find the other three forces. 3. A painter is standing on a plank, supported by two ropes. Each rope is tied to a corner of the plank, goes up and over a pulley, coming down to the near corner where it is tied over a hook. It can then be easily adjusted to change the height of the plank. 1

2 The painter and his equipment have mass 175kg, while the plank has mass 50kg. (a) Given that the painter has to move side to side on the plank, what is the maximum tension the ropes will have to carry? (b) The maximum load of the ropes is now 150N. If the painter were to wrap the rope around a nail on the side of the building he was painting, the rope would snap. Why? 4. A lift, with mass m = 1000kg is travelling down to the bottom floor of a building at v = 10ms 1. It is brought to rest by a constant acceleration, over a distance of d = 10m. What is the tension, T, in the lift cable when the lift is stopping? 5. There is an infinitely long chain, with mass per unit length ρ = 1kgm 1, held vertically above a table. The chain is allowed to fall, but held so that it falls at v = 5ms 1. After 5s what is the force exerted downwards on the table? Again there is a chain held above a table, but this time it is 5m long and is allowed to fall freely under gravity. The moment the last piece of the chain touches the table, what is the total force being exerted downwards on the table? 2

3 6. A metal hoop is rotating about its centre at an angular velocity ω = 2πn, where n is the revolutions per second. We want to know how quickly our hoop can rotate without shattering. To do this, look at a small element of the hoop: Given that the hoop is rotating at a constant angular velocity, show that the tension holding the elements together, T, is given by: T = 4π 2 n 2 r 2 ρa. (1) The hoop will break if the stress, (stress = F orce Area ), is greater than a number, the ultimate tensile strength, which is different for each material. 3

4 For steel, this is about 10 9 Nm 2. If the hoop has radius r = 1m and density ρ = 7600kgm 3, what is the maximum revolutions per second the hoop can go at before breaking? 7. A light elf is sitting on a wheeled water butt in the rain. The elf is filling water balloons with mass m w = 0.5kg from the butt and throwing them away at v w = 5ms 1, to get the water butt moving. The rain is filling the butt so quickly that we take it s mass to be constant at M b = 100kg. Neglecting friction, how many balloons must the elf throw to get the water butt moving at 1ms 1? (For simplicity, assume that the rain falling into the water butt is moving in the same speed and direction as the butt, but does not exert any force on the butt). 8. The coefficient of restitution, C r, is the ratio of the speeds before and after an impact along the line of the collision. So if a ball was thrown at a wall at 2ms 1, and rebounded at 2ms 1, C r = 0.5. A ball is dropped from one metre, and rebounds to 45cm. What is the coefficient of restitution for the ball and the floor? How high will it rebound after its next collision? 9. A rocket with initial mass M 0 = kg and exhaust speed v = 2000ms 1 is sitting on its launch pad. Its engines eject mass at a constant rate dm dt = µ. (a) What, in symbols, is the initial acceleration? (b) If we want a = 0.5ms 2, what must µ be? 4

5 10. Find the centre of momentum 2-vector velocity of the three body system shown below. Sources Cavendish Residential Course Mechanics questions, Dr Lisa Jardine-Wright. Introduction to Classical Mechanics, French and Ebison, Chapman & Hall Problems in Physics, F Tyler, Edward Arnold Ltd