Sheet 03 Friction and Dynamics of Uniform Circular Motion.
A particle of mass \(42g\) attached to a string of \(60cm\) length is whirled in a horizontal circle. If, period of revolution is \(2s\) then find tension in string.
Ans: \(T(tension)=24846dyne\).
A string breaks under a load of \(50kg\). A mass of \(1kg\) is attached to one end of the string \(10m\) long and rotated in a horizontal circle, calculate: the greatest number of revolutions that the mass can make without breaking the string?
Ans: \(f=1.114\frac{rev}{Sec}\).
A bob of mass \(200g\) is suspended by a string \(20cm\) long. Keeping the string always taut, the ball describes a horizontal circle of radius \(10cm\). What will be angular speed of bob?
Ans: \(w=5.31\frac{rad}{s}\).
A car of mass \(2000kg\) is moving with a speed of \(50\frac{m}{s}\) on a circular path of radius \(20m\) on a level road. What will be the frictional force between the car and the road so that car may not slip? Also, find the value of co-efficient of friction to attain this force?
Ans: \(\mu=1.147\).
A train has to negotiate a curve of radius \(400m\). By how much the outer edge of the rail be raised with respect to the inner rail for a speed of \(54\frac{km}{hr}\)? The distance between the rails is \(1m\).
Ans: \(h=0.0573m\).
A circular track of radius \(400m\) is banked at an angle of \(10^{o}\). The co-efficient of friction between the wheels of a racing car and the track is \(0.2\). Then; what will be (i) Optimal speed of the racing car to avoid wear and tear on its tyres? and (ii) Maximum permissible speed to avoid slipping?
Ans:(i) \(v_{optimal}=26.3\frac{m}{s}\) and \(v_{max}=39.1\frac{m}{s}\).
A car is speeding on a horizontal road curving round with a radius of \(80m\). The co-efficient of friction between the wheels and road is \(0.5\). The height of the center of gravity of the car from the road level is \(0.3m\) and the distance between the wheels is \(1m\). calculate: the maximum safe velocity for the vehicle to negotiate the curve with out skidding and with out toppling?
Ans: \(v_{s}=19.79\frac{m}{s}\) and \(v_{t}=36.14\frac{m}{s}\).
A sphere of mass \(400g\) is attached to an inextensible string of length \(130cm\) whose upper end is fixed to a celling. The sphere is made to describe a horizontal circle of radius \(50cm\). calculate: the periodic time of this conical pendulum and tension in the string?
Ans: \(T=2.19s\) and \(T=4.24N\).
A particle describe a horizontal circle on a smooth surface of an inverted cone. The height of the plane of the circle above the vertex is \(10cm\). Find: the speed of the particle. Take: \(g=10\frac{m}{s}\).
Ans: \(v=1\frac{m}{s}\).
What should be the angle through which a cyclist bends from the vertical when he crosses a circular path of radius \(34.3m\) in circumference in \(\sqrt{22}s\). Take: \(g=10\frac{m}{s^{2}}\).
Ans: \(\theta=45^{o}\).
A cyclist speeding \(6\frac{m}{s}\) in a circular turn of radius \(18m\) makes an angle \(\theta\) with the vertical. Calculate: \(\theta\)? Also, determine the minimum possible value of the co-efficient of friction between the tyre and the road?
Ans: \(\theta=11^{o}\) and \(\mu=0.2041\).
A railway carriage has its center of gravity at a height of \(1m\) above the rails which are \(1m\) apart. Calculate: the maximum safe speed at which it can travel round an unbanked curve of radius \(80m\).
