Advertisements
Advertisements
प्रश्न
A motorcycle has to move with a constant speed on an over bridge which is in the form of a circular arc of radius R and has a total length L. Suppose the motorcycle starts from the highest point.(a) What can its maximum velocity be for which the contact with the road is not broken at the highest point? (b) If the motorcycle goes at speed 1/√2 times the maximum found in part (a), where will it lose the contact with the road? (c) What maximum uniform speed can it maintain on the bridge if it does not lose contact anywhere on the bridge?
Advertisements
उत्तर
R = Radius of the bridge
L = Total length of the over bridge
(a) At the highest point:
Let m be the mass of the motorcycle and v be the required velocity.
\[\text{mg }= \frac{\text{mv}^2}{\text{R}}\]
\[ \Rightarrow \text{v}^2 = \text{Rg}\]
\[ \Rightarrow \text{v} = \sqrt{\text{Rrg}}\]
\[\left( b \right) \text {Given :} \]
\[ \text{v} = \left( \frac{1}{\sqrt{2}} \right)\sqrt{\text{Rg}}\]
Suppose it loses contact at B.
\[\text {At point B, we get : }\]
\[\text{mg}\cos\theta = \frac{\text{mv}^2}{R}\]
\[ \Rightarrow \text{v}^2 = \text{Rg}\cos\theta\]
\[\text {Putting the value of v}, \text {we get : } \]
\[\sqrt{\left( \frac{Rg}{2} \right)^2} = Rg\cos\theta\]
\[ \Rightarrow \frac{Rg}{2} = Rg\cos\theta\]
\[ \Rightarrow \cos\theta = \frac{1}{2}\]
\[ \Rightarrow \theta = 60^\circ = \frac{\pi}{3}\]
\[ \because \theta = \frac{L}{R}\]
\[ \therefore L = R\theta = \frac{\pi R}{3}\]
So, it will lose contact at a distance \[\frac{\pi R}{3}\] from the highest point.
(c) Let the uniform speed on the bridge be v. The chances of losing contact is maximum at the end bridge. We have :
\[\alpha = \frac{L}{2R}\]
\[\text{So}, \frac{\text{mv}^2}{R} = \text{mg}\cos\alpha\]
\[ \Rightarrow v = \sqrt{\text{gRcos}\left( \frac{L}{2R} \right)}\]
APPEARS IN
संबंधित प्रश्न
You may have seen in a circus a motorcyclist driving in vertical loops inside a ‘death-well’ (a hollow spherical chamber with holes, so the spectators can watch from outside). Explain clearly why the motorcyclist does not drop down when he is at the uppermost point, with no support from below. What is the minimum speed required at the uppermost position to perform a vertical loop if the radius of the chamber is 25 m?
When a particle moves in a circle with a uniform speed
A particle is kept fixed on a turntable rotating uniformly. As seen from the ground the particle goes in a circle, its speed is 20 cm/s and acceleration is 20 cm/s2. The particle is now shifted to a new position to make the radius half of the original value. The new value of the speed and acceleration will be
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 simple pendulum is suspended from the ceiling of a car taking a turn of radius 10 m at a speed of 36 km/h. Find the angle made by he string of the pendulum with the vertical if this angle does not change during the turn. Take g = 10 m/s2.
A person stands on a spring balance at the equator. By what fraction is the balance reading less than his true weight?
A car goes on a horizontal circular road of radius R, the speed increasing at a constant rate \[\frac{\text{dv}}{\text{dt}} = a\] . The friction coefficient between the road and the tyre is μ. Find the speed at which the car will skid.
In a children's park a heavy rod is pivoted at the centre and is made to rotate about the pivot so that the rod always remains horizontal. Two kids hold the rod near the ends and thus rotate with the rod (In the following figure). Let the mass of each kid be 15 kg, the distance between the points of the rod where the two kids hold it be 3.0 m and suppose that the rod rotates at the rate of 20 revolutions per minute. Find the force of friction exerted by the rod on one of the kids.
A person stands on a spring balance at the equator. If the speed of earth's rotation is increased by such an amount that the balance reading is half the true weight, what will be the length of the day in this case?
When seen from below, the blades of a ceiling fan are seen to be revolving anticlockwise and their speed is decreasing. Select the correct statement about the directions of its angular velocity and angular acceleration.
A particle of mass 1 kg, tied to a 1.2 m long string is whirled to perform the vertical circular motion, under gravity. The minimum speed of a particle is 5 m/s. Consider the following statements.
P) Maximum speed must be `5sqrt5` m/s.
Q) Difference between maximum and minimum tensions along the string is 60 N.
Select the correct option.
In a certain unit, the radius of gyration of a uniform disc about its central and transverse axis is `sqrt2.5`. Its radius of gyration about a tangent in its plane (in the same unit) must be ______.
Two particles A and B are located at distances rA and rB respectively from the centre of a rotating disc such that rA > rB. In this case, if angular velocity ω of rotation is constant, then ______
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 acceleration will be ______.
The real force 'F' acting on a particle of mass ' m' performing circular motion acts along the radius of circle 'r' and is directed towards the centre of circle. The square root of the magnitude of such force is (T = periodic time).
Angular displacement (θ) of a flywheel varies with time as θ = at + bt2 + ct3 then angular acceleration is given by ____________.
An engine is moving on a c1rcular path of radius 200 m with speed of 15 m/s. What will be the frequency heard by an observer who is at rest at the centre of the circular path, when engine blows the whistle with frequency 250 Hz?
A block of 200 g mass moves with a uniform speed in a horizontal circular groove, with vertical side walls of radius 20 cm. If the block takes 40 s to complete one round, the normal force by the side walls of the groove is ______.
A racing car is travelling along a track at a constant speed of 40 m/s. A T.V. cameraman is recording the event from a distance of 30 m directly away from the track as shown in the figure. In order to keep the car under view in the position shown, the angular speed with which the camera should be rotated is ______.
A bob is whirled in a horizontal plane by means of a string with an initial speed of ω rpm. The tension in the string is T. If speed becomes 2ω while keeping the same radius, the tension in the string becomes ______.
