Advertisements
Advertisements
प्रश्न
Answer the following question.
Show that the centripetal force on a particle undergoing uniform circular motion is -mrω2.
Advertisements
उत्तर
- Suppose a particle is performing U.C.M in anticlockwise direction. The co-ordinate axes are chosen as shown in the figure.
Let, A = initial position of the particle which lies on positive X-axis
P = instantaneous position after time t
θ = angle made by radius vector
ω = constant angular speed
`vec"r"` = instantaneous position vector at time t - From the figure,
`vec"r" = hat"i""x" + hat"j""y"`
where, `hat"i" and hat"j"` are unit vectors along X-axis and Y-axis respectively.
- Also, x = r cos θ and y = r sin θ
∴ `vec"r" = ["r"hat"i"cos theta + "r"hat"j"sin theta]`
But θ = ωt
∴ `vec"r" = ["r"hat"i" cos omega"t" + "r"hat"j" sin omega"t"]` ....(1) - Velocity of the particle is given as rate of change of position vector.
∴ `vec"v" = (vec"dr")/"dt" = "d"/"dt" ["r"hat"i" cos omega"t" + "r" hat"j" sin omega"t"]`
`= "r"["d"/"dt" cos omega"t"]hat"i" + "r"["d"/"dt" sin omega"t"]hat"j"`
∴ `vec"v" = - "r"omega hat"i" sin omega"t" + "r"omega hat"j" cos omega"t"`
∴ `vec"v" = "r"omega (-hat"i" sin omega"t" + hat"j" cos omega"t")` - Further, instantaneous linear acceleration of the particle at instant t is given by,
`vec"a" = (vec"dv")/"dt" = "d"/"dt"["r"omega(- hat"i" sin omega "t" + hat"j" cos omega"t")]`
`= "r"omega ["d"/"dt" (- hat"i" sin omega"t" + hat"j" cos omega "t")]`
`= "r"omega ["d"/"dt" (- sin omega"t")hat"i" + "d"/"dt"(cos omega "t")hat"j"]`
`= "r"omega(-omega hat"i" cos omega"t" - omega hat"j" sin omega "t")`
`= - "r"omega^2(hat"i" cos omega "t" + hat"j" sin omega"t")`
∴ `vec"a" = - omega^2("r"hat"i" cos omega"t" + "r" hat"j" sin omega "t")` ....(2) - From equation (1) and (2),
`vec"a" = - omega^2vec"r"` ....(3)
The negative sign shows that the direction of acceleration is opposite to the direction of the position vector. Equation (3) is the centripetal acceleration. - The magnitude of centripetal acceleration is given by,
a = `omega^2"r"` - The force providing this acceleration should also be in the same direction, hence centripetal.
∴ `vec"F" = "m"vec"a" = - "m"omega^2vec"r"`
This is the expression for the centripetal force on a particle undergoing uniform circular motion. - Magnitude of F = `"m"omega^2"r" = "mv"^2/"r"` = mωv
APPEARS IN
संबंधित प्रश्न
A particle starts from the origin at t = 0 s with a velocity of 10.0 `hatj "m/s"` and moves in the x-y plane with a constant acceleration of `(8.0 hati + 2.0 hatj) ms^(-2)`.
- At what time is the x-coordinate of the particle 16 m? What is the y-coordinate of the particle at that time?
- What is the speed of the particle at the time?
Differentiate between uniform linear motion and uniform circular motion.
A piece of stone tied at the end of a thread is whirled in a horizontal circle with uniform speed by hand. Answer the following questions:
- Is the velocity of stone uniform or variable?
- Is the acceleration of stone uniform or variable?
- What is the direction of acceleration of stone at any instant?
- Which force provides the centripetal force required for circular motion?
- Name the force and its direction which acts on the hand.
In a uniform circular motion, the speed continuously changes because of the direction of motion changes.
Solve the following problem.
A car moves in a circle at a constant speed of 50 m/s and completes one revolution in 40 s. Determine the magnitude of the acceleration of the car.
Solve the following problem.
A particle moves in a circle with a constant speed of 15 m/s. The radius of the circle is 2 m. Determine the centripetal acceleration of the particle.
Which of the following graph represents uniform motion of a moving particle?
The angle subtended by the vector A = `5hat"i" + 3hat"j" + 12hat"k"` with the X-axis is ______.
A particle goes round a circular path with uniform speed v. After describing half the circle, what is the change in its centripetal acceleration?
The ratio of angular speed of a hour-hand to the second-hand of a watch is ____________.
The ratio of the angular speed of minute hand and hour hand of a watch is ____________.
At any instant, the magnitude of the centripetal force on a particle of mass 'm' performing circular motion is given by (ω = angular velocity and v = linear velocity of the particle) ______.
A string of length 'l' fixed at one end carries a mass 'm' at the other end. The string makes `3/pi` revolutions/second around the vertical axis through the fixed end as shown in figure. The tension 'T' in the string is ______.
It is possible to have objects moving with uniform velocity but non-uniform acceleration.
A flywheel at rest is to reach an angular velocity of 24 rad/s in 8 second with constant angular acceleration. The total angle turned through during this interval is ______.
If a body is moving in a circle of radius r with a constant speed v, its angular velocity is ______.
For a particle performing uniform circular motion, choose the correct statement(s) from the following:
- Magnitude of particle velocity (speed) remains constant.
- Particle velocity remains directed perpendicular to radius vector.
- Direction of acceleration keeps changing as particle moves.
- Angular momentum is constant in magnitude but direction keeps changing.
A particle is given an initial speed u inside a smooth spherical shell of radius R = 1 m that it is just able to complete the circle. Acceleration of the particle when its velocity is vertical is ______.
A disc of radius 5 cm rolls on a horizontal surface with linear velocity v = 1`hat"i"` m/s and angular velocity 50 rad/s. Height of particle from ground on rim of disc which has velocity in vertical direction is ______ cm.
A wet open umbrella is held upright and is rotated about the handle at a uniform rate of 21 revolutions in 44 s. If the rim of the umbrella is a circle of 1 metre in diameter and the height of the rim above the floor is 1.5 m, the drops of water spun off the rim and hit the floor at a horizontal ______ m from the umbrella.
A particle is performing a uniform circular motion along a circle of radius R. In half the period of revolution, its displacement and distance covered are respectively.
Two bodies of masses 10 kg and 5 kg moving in concentric orbits of radii R and r such that their periods are the same. Then the ratio between their centripetal accelerations is ______.
The acceleration of a point on the rim of a flywheel 1 m in diameter, if it makes 1200 rpm is ______.
The kinetic energy K of a particle moving along a circle of radius R depends on the distance covered s as K = as2, where a is a constant. The force acting on the particle is ______.
Explain the meaning of uniform circular motion.
What is the direction of the velocity of an object at any point during uniform circular motion?
In uniform circular motion, although the speed is constant, why does acceleration occur?
What is the direction of the velocity of an object at any point during uniform circular motion?
By applying Kepler's third law to the formula for centripetal force, Newton concluded that the force acting on a planet is:
