Advertisements
Advertisements
प्रश्न
Derive an expression for maximum speed moving along a horizontal circular track.
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 as per the given requirements. 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?
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 ]
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 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 ______.
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 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)
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 ______.
The two blocks, m = 10 kg and M = 50kg are free to move as shown. The coefficient of static friction between the blocks is 0.5 and there is no friction between M and the ground. A minimum horizontal force F is applied to hold m against M that is equal to ______.
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, ______.
Write about the kinetic friction between the road and the tyres.
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.
