Advertisements
Advertisements
Question
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.
Advertisements
Solution
Consider a small body (or particle) of mass m tied to a string and revolved in a vertical circle of radius r at a place where the acceleration due to gravity is g. At every instant of its motion, the body is acted upon by two forces, namely, its weight `vec"mg"` and the tension `vec"T"` in the string.
Let `"v"_2` be the speed of the body and `"T"_2` be the tension in the string at the lowest point B. We take the reference level for zero potential energy to be the bottom of the circle. Then, the body has only kinetic energy `1/2"mv"_2^2` at the lowest point.
∴ `"T"_2 = "mv"_2^2/"r" + "mg"` ...(1)
and the total energy at the bottom = KE + PE
`= 1/2"mv"_2^2 + 0`
`= 1/2 "mv"_2^2` ...(2)
Let v1 be the speed and T1 the tension in the string at the highest point A. As the body goes from B to A, it rises through a height h = 2r.
∴ `"T"_1 = "mv"_1^2/"r" - "mg"` .....(3)
and the total energy at A = KE + PE
`= 1/2 "mv"_1^2 + "mg"(2r)` ...(4)
Then, from Eqs. (1) and (3),
`"T"_2 - "T"_1 = "mv"_2^2/"r" + "mg" - ("mv"_1^2/"r" - "mg")`
`= "m"/"r" ("v"_2^2 - "v"_1^2) + 2 "mg"` ...(5)
Assuming that the total energy of the body is conserved, the total energy at the bottom = total energy at the top
Then, from Eqs. (2) and (4),
`1/2 "mv"_2^2 = 1/2"mv"_1^2 + "mg"(2r)`
∴ `"v"_2^2 - "v"_1^2 = 4 "gr"` ...(6)
Substituting this in Eq. (5),
`"T"_2 - "T"_1 = "m"/"r" (4 "gr") + 2 "mg"`
`= 4 "mg" + 2 "mg"`
= 6 mg
Therefore, the difference in the tensions in the string at the highest and the lowest points is 6 times the weight of the body.
APPEARS IN
RELATED QUESTIONS
Answer in brief:
A uniform disc and a hollow right circular cone have the same formula for their M.I. when rotating about their central axes. Why is it so?
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 metallic ring of mass 1 kg has a moment of inertia 1 kg m2 when rotating about one of its diameters. It is molten and remolded into a thin uniform disc of the same radius. How much will its moment of inertia be, when rotated about its own axis.
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 block of mass m is moving on rough horizontal surface with momentum p. The coefficient of friction between the block and surface is µ. The distance covered by the block before it stops is [g =acceleration due to gravity)
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?
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 ____________.
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 ______.
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 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.
