Advertisements
Advertisements
प्रश्न
A turn of radius 20 m is banked for the vehicles going at a speed of 36 km/h. If the coefficient of static friction between the road and the tyre is 0.4, what are the possible speeds of a vehicle so that it neither slips down nor skids up?
Advertisements
उत्तर
Given:
Speed of vehicles = v = 36 km/hr = 10 m/s
Radius = r = 20 m
Coefficient of static friction = μ = 0.4
Let the road be banked with an angle \[\theta\]
We have :
\[\theta = \tan^{- 1} \frac{\text{v}^2}{\text{rg}}\]
\[ = \tan^{- 1} \frac{100}{20 \times 10}\]
\[ = \tan^{- 1} \left( \frac{1}{2} \right)\]
\[ \Rightarrow \text{tan } \theta = 0 . 5\]
When the car travels at the maximum speed, it slips upward and μN1 acts downward.
Therefore we have :
\[\text{ N}_1 - \text{mg}\cos\theta - \frac{\text{mv}_1^2}{r}\sin\theta = 0 . . . \left( \text{i} \right)\]
\[\mu \text{N}_1 + \text{mg }\sin\theta - \frac{\text{mv}_1^2}{r}\cos\theta = 0 . . . \left(\text{ ii} \right)\]
On solving the above equations, we get :
\[\text{v}_1 = \sqrt{\text{rg}\frac{\mu + \tan\theta}{1 - \mu\tan\theta}}\]
\[ = \sqrt{20 \times 10 \times \frac{0 . 9}{0 . 8}}\]
\[ = 15 \text{ m/s = 54 km/hr }\]
Similarly, for the other case, it can be proved that :
\[\text{v}_2 = \sqrt{\text{rg}\frac{\text{tan}\theta - \mu}{\sqrt{1 - \mu \text{ tan }\theta}}}\]
\[ = \sqrt{20 \times 10 \times \frac{0 . 1}{1 . 2}}\]
\[ = 4 . 08 \text{ m/s = 14 . 7 km/hr}\]
Thus, the possible speeds are between 14.7 km/hr and 54 km/hr so that the car neither slips down nor skids up.
APPEARS IN
संबंधित प्रश्न
A thin circular loop of radius R rotates about its vertical diameter with an angular frequency ω. Show that a small bead on the wire loop remains at its lowermost point for `omega <= sqrt(g/R)` .What is the angle made by the radius vector joining the centre to the bead with the vertical downward direction for `omega = sqrt("2g"/R)` ?Neglect friction.
When a particle moves in a circle with a uniform speed
Three identical cars A, B and C are moving at the same speed on three bridges. The car A goes on a place bridge, B on a bridge convex upward and C goes on a bridge concave upward. Let FA, FB and FC be the normal forces exerted by the car on the bridges when they are at the middle of bridges.
A rod of length L is pivoted at one end and is rotated with a uniform angular velocity in a horizontal plane. Let T1 and T2 be the tensions at the points L/4 and 3L/4 away from the pivoted ends.
A simple pendulum having a bob of mass m is suspended from the ceiling of a car used in a stunt film shooting. the car moves up along an inclined cliff at a speed v and makes a jump to leave the cliff and lands at some distance. Let R be the maximum height of the car from the top of the cliff. The tension in the string when the car is in air is
The position vector of a particle in a circular motion about the origin sweeps out equal area in equal time. Its
(a) velocity remains constant
(b) speed remains constant
(c) acceleration remains constant
(d) tangential acceleration remains constant.
A particle is going in a spiral path as shown in figure with constant speed.
A particle moves in a circle of radius 1.0 cm at a speed given by v = 2.0 t where v is cm/s and t in seconds.
(a) Find the radial acceleration of the particle at t = 1 s.
(b) Find the tangential acceleration at t = 1 s.
(c) Find the magnitude of the acceleration at t = 1 s.
Suppose the bob of the previous problem has a speed of 1.4 m/s when the string makes an angle of 0.20 radian with the vertical. Find the tension at this instant. You can use cos θ ≈ 1 − θ2/2 and SINθ ≈ θ for small θ.
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?
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?
A child starts running from rest along a circular track of radius r with constant tangential acceleration a. After time the feels that slipping of shoes on the ground has started. The coefficient of friction between shoes and the ground is [g = acceleration due to gravity].
The centripetal force of a body moving in a circular path, if speed is made half and radius is made four times the original value, will ____________.
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?
Statement I: A cyclist is moving on an unbanked road with a speed of 7 kmh-1 and takes a sharp circular turn along a path of radius of 2 m without reducing the speed. The static friction coefficient is 0.2. The cyclist will not slip and pass the curve. (g = 9.8 m/s2)
Statement II: If the road is banked at an angle of 45°, cyclist can cross the curve of 2 m radius with the speed of 18.5 kmh-1 without slipping.
In the light of the above statements, choose the correct answer from the options given below.
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 ______.
A particle at rest starts moving with constant angular acceleration ‘α’ in circular path. At what time the magnitude of centripetal acceleration is half the tangential acceleration?
