Advertisements
Advertisements
Question
A racing car travels on a track (without banking) ABCDEFA (Figure). ABC is a circular arc of radius 2 R. CD and FA are straight paths of length R and DEF is a circular arc of radius R = 100 m. The co-efficient of friction on the road is µ = 0.1. The maximum speed of the car is 50 ms–1. Find the minimum time for completing one round.
Solution
Balancing frictional force for centripetal force = `(mv^2)/r` = f = μN = μmg
Where N is the normal reaction
∴ `v = sqrt(μrg)` ....(Where, r is radius of the circular track)
For path ABC, Path length = `3/4 (2π 2R) = 3π R = 3π xx 100`
= 300 πm
`v_1 = sqrt(μ2Rg) = sqrt(0.1 xx 2 xx 100 xx 10)`
= 14.14 m/s
∴ `t_1 = (300π)/(14.14)` = 66.6 s
For path DEF, Path length = `1/4 (2πR) = (π xx 100)/2` = 50π
`v_2 = sqrt(μRg) = sqrt(0.1 xx 100 xx 10)` = 10 m/s
`t_2 = (50π)/10` = 5π s = 15.7 s
For path, CD and FA
Path length = R + R = 2R = 200 m
`t_3 = 200/50` = 4.0 s
∴ Total time for completing one round t = t1 + t2 + t3 = 66.6 + 15.7 + 4.0 = 86.3 s
APPEARS IN
RELATED QUESTIONS
A stone of mass m tied to a string of length l is rotated in a circle with the other end of the string as the centre. The speed of the stone is v. If the string breaks, the stone will move
A motorcycle is going on an overbridge of radius R. The driver maintains a constant speed. As the motorcycle is ascending on the overbridge, the normal force on it
A car of mass M is moving on a horizontal circular path of radius r. At an instant its speed is v and is increasing at a rate a.
(a) The acceleration of the car is towards the centre of the path.
(b) The magnitude of the frictional force on the car is greater than \[\frac{\text{mv}^2}{\text{r}}\]
(c) The friction coefficient between the ground and the car is not less than a/g.
(d) The friction coefficient between the ground and the car is \[\mu = \tan^{- 1} \frac{\text{v}^2}{\text{rg}.}\]
A hemispherical bowl of radius R is rotated about its axis of symmetry which is kept vertical. A small block is kept in the bowl at a position where the radius makes an angle θ with the vertical. The block rotates with the bowl without any slipping. The friction coefficient between the block and the bowl surface is μ. Find the range of the angular speed for which the block will not slip.
A table with smooth horizontal surface is placed in a circle of a large radius R (In the following figure). A smooth pulley of small radius is fastened to the table. Two masses m and 2m placed on the table are connected through a string going over the pulley. Initially the masses are held by a person with the string along the outward radius and then the system is released from rest (with respect to the cabin). Find the magnitude of the initial acceleration of the masses as seen from the cabin and the tension in the string.
A particle of mass m is performing UCM along a circle of radius r. The relation between centripetal acceleration a and kinetic energy E is given by
Two identical masses are connected to a horizontal thin (massless) rod as shown in the figure. When their distance from the pivot is D, a torque τ produces an angular acceleration of α1. The masses are now repositioned so that they are 2D from the pivot. The same torque produces an angular acceleration α2 which is given by ______
A particle performs uniform circular motion in a horizontal plane. The radius of the circle is 10 cm. If the centripetal force F is kept constant but the angular velocity is halved, the new radius of the path will be ______.
A body is moving along a circular track of radius 100 m with velocity 20 m/s. Its tangential acceleration is 3 m/s2 then its resultant accelaration will be ______.
When a body slides down from rest along a smooth inclined plane making an angle of 45° with the horizontal, it takes time T. When the same body slides down from rest along a rough inclined plane making the same angle and through the same distance, it is seen to take time pT, where p is some number greater than 1. Calculate the co-efficient of friction between the body and the rough plane.
