Advertisements
Advertisements
Questions
Obtain an expression for maximum safety speed with which a vehicle can be safely driven along a curved banked road.
Show that the angle of banking is independent of the mass of the vehicle.
Advertisements
Solution
- The vertical section of a vehicle on a curved road (considering friction) of radius ‘r’ banked at an angle θ with the horizontal is shown in the figure.
If the vehicle is running exactly at the optimum speed, then the forces acting on the vehicle are
- Weight mg acting vertically downwards
- Normal reaction N acting perpendicular to the road.
- But in practice, vehicles never travel exactly with this speed.
- Hence, for speeds other than this, the component of the force of static friction between the road and the tires helps us, up to a certain limit.
- For maximum possible speed,
The component N sinθ is less than the centrifugal force `(mv^2)/r`.
∴ `(mv^2)/r` > N sinθ
Banked road: upper-speed limit - In this case, the direction of the force of static friction (fs) between the road and the tires is directed along with the inclination of the road downwards.
- The horizontal component (fs cos θ) is parallel to Nsinθ.
These two forces take care of the necessary centripetal force (or balance the centrifugal force).
∴ `(mv^2)/r = N sinθ + f_s cos θ` …(1) - The vertical component, N cosθ balances the component fs sin θ and weight mg.
∴ N cos θ = fs sinθ + mg
∴ mg = N cos θ − fs sin θ ...(2) - For maximum possible speed, fs is maximum and equal to μsN. From equations (1) and (2),
`v_max = sqrt(rg((tanθ + mu_s)/(1 - mu_stanθ)))` ...(3)
This is an expression for maximum safety speed with which a vehicle can be safely driven along a curved banked road (considering friction). - If µs = 0, then equation (3) becomes,
`v_max = sqrt(rg[(0 + tanθ)/(1 - 0tanθ)]`
∴ `v_max = sqrt(rg tanθ)` ...(4)
This is an expression of maximum safety speed with which a vehicle can be safely driven along a curved banked road (neglecting friction). - From equation (3) and equation (4), we can write,
`((tanθ + mu_s)/(1 - mu_s tanθ)) = V_max^2/(rg)` ...(5)
and `tan theta = V_max^2/(rg)`
∴ `theta = tan^-1(V_max^2/(rg))` ...(6)
From equation (5) and equation (6), angle of banking is independent of the mass of the vehicle.
RELATED QUESTIONS
A thin walled hollow cylinder is rolling down an incline, without slipping. At any instant, without slipping. At any instant, the ratio "Rotational K.E.: Translational K.E.: Total K.E." is ______.
Answer in brief:
Why are curved roads banked?
On what factors does the frequency of a conical pendulum depend? Is it independent of some factors?
While driving along an unbanked circular road, a two-wheeler rider has to lean with the vertical. Why is it so? With what angle the rider has to lean? Derive the relevant expression. Why such a leaning is not necessary for a four wheeler?
Somehow, an ant is stuck to the rim of a bicycle wheel of diameter 1 m. While the bicycle is on a central stand, the wheel is set into rotation and it attains the frequency of 2 rev/s in 10 seconds, with uniform angular acceleration. Calculate:
- The number of revolutions completed by the ant in these 10 seconds.
- Time is taken by it for first complete revolution and the last complete revolution.
The coefficient of static friction between a coin and a gramophone disc is 0.5. Radius of the disc is 8 cm. Initially the center of the coin is 2 cm away from the center of the disc. At what minimum frequency will it start slipping from there? By what factor will the answer change if the coin is almost at the rim? (use g = π2m/s2)
Starting from rest, an object rolls down along an incline that rises by 3 in every 5 (along with it). The object gains a speed of `sqrt 10` m/s as it travels a distance of `5/3` m along the incline. What can be the possible shape/s of the object?
During ice ballet, while in the outer rounds, why do the dancers outstretch their arms and legs.
A hollow sphere has a radius of 6.4 m. what is the minimum velocity required by a motorcyclist at the bottom to complete the circle.
Derive an expression for maximum safety speed with which a vehicle should move along a curved horizontal road. State the significance of it.
Derive an expression for the kinetic energy of a rotating body with uniform angular velocity.
A rigid body rotates with an angular momentum L. If its kinetic energy is halved, the angular momentum becomes, ______
When a mass is rotating in a plane about a fixed point, its angular momentum is directed along, ______
Give any two examples of torque in day-to-day life.
What is the relation between torque and angular momentum?
What are the rotational equivalents for the physical quantities, (i) mass and (ii) force?
Discuss conservation of angular momentum with example.
A flywheel rotates with uniform angular acceleration. If its angular velocity increases from `20pi` rad/s to `40pi` rad/s in 10 seconds. Find the number of rotations in that period.
A uniform metallic rod rotates about its perpendicular bisector with constant angular speed. If it is heated uniformly to raise its temperature to a certain value, its speed of rotation ______.
A wheel of radius 2 cm is at rest on the horizontal surface. A point P on the circumference of the wheel is in contact with the horizontal surface. When the wheel rolls without slipping on the surface, the displacement of point P after half rotation of wheel is ______.
A ring and a disc of different masses are rotating with the same kinetic energy. If we apply a retarding torque τ on the ring, it stops after completing n revolution in all. If the same torque is applied to the disc, how many revolutions would it complete in all before stopping?
What is the difference between rotation and revolution?
