Advertisements
Advertisements
प्रश्न
Derive an expression for maximum speed moving along a horizontal circular track.
Advertisements
उत्तर
When an object moves along a horizontal circular track, the force providing the necessary centripetal force is the frictional force between the object and the surface. The object will move with maximum speed when the frictional force is at its maximum value, which is given by limiting friction.
Consider an object of mass m moving on a horizontal circular track of radius r. The forces acting on the object are:
- Normal Reaction Force: N exerted by the surface (acts vertically upward).
- Gravitational Force: mg (acts vertically downward).
- Frictional Force: f (acts towards the center of the circular path to provide the required centripetal force).
Since the object is moving on a horizontal surface, the normal reaction is equal to the weight of the object:
N = mg
For circular motion, the centripetal force required is:
`Fc = (mv_"max"^2)/r`
Here, vmax is the maximum speed of the object.
The maximum frictional force available to provide this centripetal force is given by:
fmax = μN = μmg
where μ is the coefficient of friction between the object and the surface.
At maximum speed, the entire frictional force provides the necessary centripetal force:
`(mv_"max"^2)/r = μmg`
Canceling m from both sides:
`v_"max"^2/r = μg`
`v_max = sqrt(μgr)`
This formula indicates that the maximum speed is determined by the track's radius, the gravitational force, and the friction between the track and the object.
APPEARS IN
संबंधित प्रश्न
A road is constructed part of a racing tracks to be designed with radius of curvature 72 m. We are not recommending the vehieles to drive faster than 216 kmph.. The coefficient of static friction between the tyres of a vehicle on this road is 0.8, will there be any lower speed limit? By how much can the upper speed limit exceed in this case?
(Given: r = 72 m, vo = 216 km/h, w = 10 m, θ = 78°4', h = 9.805 m)
Using the energy conservation, derive the expressions for the minimum speeds at different locations along a vertical circular motion controlled by gravity. Is zero speed possible at the uppermost point? Under what condition/s?
A vehicle of mass m is moving with momentum p on a rough horizontal road. The coefficient of friction between the tyres and the horizontal road is µ. The stopping distance is ____________.
(g = acceleration due to gravity)
For a body moving with constant speed in a horizontal circle, which of the following remains constant?
A cyclist with combined mass 80 kg goes around a curved road with a uniform speed 20 m/s. He has to bend inward by an angle `theta` = tan-1 (0.50) with the vertical. The force of friction acting at the point of contact of tyres and road surface is______.
[g = 10 m/s2 ]
The maximum safe speed, for which a banked road is intended, is to be increased by 20 %. If the angle of banking is not changed, then the radius of curvature of the road should be changed from 30 m to ____________.
A pendulum has length of 0.4 m and maximum speed 4 m/s. When the length makes an angle 30° with the horizontal, its speed will be ______.
`[sin pi/6 = cos pi/3 = 0.5 and "g" = 10 "m"//"s"^2]`
A car moves at a speed of 36 km hr-1 on a level road. The coefficient of friction between the tyres and the road is 0.8. The car negotiates a curve of radius R. If g = 10 ms-2 , then the car will skid (or slip) while negotiating the curve, if the value of R is ____________.
A body of mass 10 kg is attached to a wire 0.3 m long. Its breaking stress is 4.8 × 107 N/m2. The area of cross-section of the wire is 10-6m2. The maximum angular velocity with which it can be rotated 111 a horizontal circle is ______.
A motorcyclist rides in a horizontal circle about central vertical axis inside a cylindrical chamber of radius 'r'. If the coefficient of friction between the tyres and the inner surface of chamber is 'µ', the minimum speed of motorcyclist to prevent him from skidding is ______.
('g' =acceleration due to gravity)
In the case of conical pendulum, if 'T' is the tension in the string and 'θ' is the semi-vertical angle of cone, then the component which provides necessary centripetal force is ______.
A flat curved road on highway has radius of curvature 400 m. A car rounds the curve at a speed of 24 m/s. The minimum value of coefficient of friction to prevent car from sliding is ______.
(take g = 10 m/s2)
A particle rotates in horizontal circle of radius ‘R’ in a conical funnel, with speed ‘V’. The inner surface of the funnel is smooth. The height of the plane of the circle from the vertex of the funnel is ______.
(g = acceleration due to gravity)
If friction is made zero for a road, can a vehicle move safely on this road?
What is banking of a road?
The centripetal acceleration of the bob of a conical pendulum is, in the usual notation, ______.
A string of length 0.5 m carries a bob of mass 0.1 kg at its end. If this is to be used as a conical pendulum of period 0.4 π sec, the angle of inclination of the string with the vertical is ______. (g = 10m/s2)
Write about the kinetic friction between the road and the tyres.
The radius of curvature of road is 60 m. If angle of banking is 27°, find maximum speed with which vehicle can tum along this curve. . (g = 9.8 m/s2)
A body performing uniform circular motion has ______.
A horizontal force of 0.5 N is required to move a metal plate of area 10−2 m2 with a velocity of 3 × 10−2m/s, when it rests on 0.5 × 10−3 m thick layer of glycerin. Find the coefficient of viscosity of glycerin.
The radius of a circular track is 200 m. Find the angle of banking of the track, if the maximum speed at which a car can be driven safely along it is 25 m/sec.
In case of well of death which is a vertical cylindrical wall of radius ‘r’ inside which vehicle is driven in horizontal circles. If ‘m’ is mass of vehicle, ‘V’ is the velocity and u, is the coefficient of static friction between the wheels of vehicle and walls then correct relation is ______.
[g = acceleration due to gravity]
