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
Answer in Brief:
Part of a racing track is to be designed for curvature of 72m. We are not recommending the vehicles to drive faster than 216 kmph. With what angle should the road be tilted? By what height will its outer edge be, with respect to the inner edge if the track is 10 m wide?
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)
During a stunt, a cyclist (considered to be a particle) is undertaking horizontal circles inside a cylindrical well of radius 6.05 m. If the necessary friction coefficient is 0.5, how much minimum speed should the stunt artist maintain? The mass of the artist is 50 kg. If she/he increases the speed by 20%, how much will the force of friction be?
Using energy conservation, along a vertical circular motion controlled by gravity, prove that the difference between the extreme tensions (or normal forces) depends only upon the weight of the objects.
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)
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 horizontal circular platform of mass 100 kg is rotating at 5 r.p.m. about vertical axis passing through its centre. A child of mass 20 kg is standing on the edge of platform. If the child comes to the centre of platform then the frequency of rotation will become ______.
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 ______.
In the case of conical pendulum, if T is the tension in the string and θ is the semivertical angle of cone, then the component of tension which balances the centrifugal force in equilibrium position is ______.
A particle moves along a circular path of radius 'r' with uniform speed 'V'. The angle described by the particle in one second is ______.
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 executes uniform circular motion with angular momentum 'L'. Its rotational kinetic energy becomes half when the angular frequency is doubled. Its new angular momentum is ______.
If friction is made zero for a road, can a vehicle move safely on this road?
Why it is necessary banking of a road?
A curved road 5 m wide is to be designed with a radius of curvature 900 m. What should be the elevation of the outer edge of the road above the inner edge optimum speed of the vehicles rounding the curve is 30 m/s.
The centripetal acceleration of the bob of a conical pendulum is, in the usual notation, ______.
A cyclist is undertaking horizontal circles inside a cylindrical well of radius 5 m. If the friction coefficient is 0.5, what should be the minimum speed of the cyclist?
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)
Why does a motorcyclist moving along a level curve at high speed have to lean more than a cyclist moving along the same curve at low speed?
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.
