06 Aug 5Q-applied maths
1. A mass on the end of a string of length 30 cm is spun around in a circle in a horizontal plane. The centre of the circle about which the mass rotates is directly below the other end of the string. If the angle of the string below the horizontal is α, where tanα = 4/3, how long does it take for the mass to go around the circle once?
2. The radius of the orbit of Mars is 228 million kilometres and it takes 687 days orbit the sun. Calculate the angular velocity of Mars in radians/s, the circumference of its orbit in km and its speed in m/s. What is the magnitude and direction of its acceleration?
3. A fairground ride consists of a large drum (radius 4 m). People stand around the edge of the drum, which begins to rotate about a vertical axis. When the drum is rotating fast enough, the floor drops away, but participants do not fall: they feel that they are pressed against the wall of the drum, and are held in place by friction. If the coefficient of friction on the wall is µ = 0.4, how long should the drum take to complete one revolution so that the participants are just prevented from slipping down? Draw a diagram showing all the forces acting on each person, and find the speed at which they are moving.
4. The coefficient of friction between the road and the tyres of a car is 0.7. At what speed can the car drive round a corner of radius 140 m without skidding, if the road is horizontal?
5. A road is banked at an angle α so that a car can round a corner of radius 100 m at a speed of 25 m/s without skidding, no matter how smooth the road is (see below). Calculate the angle of the bank.
Car
α
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