Problems on circular motion
A circular motion is motion of a body along a circular path. A circular motion is governed by following characteristics:-
1. The ‘linear speed’ of the remains constant.
2. The ‘angular speed’ of the body also remains constant.
3. The body faces/suffers from a fictitious force arising due to the motion. This force is called as the centripetal force and the acceleration thus generated is centripetal acceleration.
4. A body oscillating about a mean position can be imagined to be performing a circular motion. For example look the following figures:-
The first figure explains the particle performing a oscillatory motion. The graph is of position vs. time when the time is zero; the position of particle is zero. When one period is complete the position is again zero. At half of the period the value of position is maximum. The same can be visualized as a circle. (Imagine the figure on left folded.) So all the equations of circular motion can be applied to oscillatory motion.
5. The centripetal force acting on the body is equal to the tension on the string.
6. The angular velocity of the body is defined as
The value of the centripetal acceleration is given as: - ; R is the radius of the circle.
To obtain the condition for circular motion of any particle one has to equate the forces acting on the particle with the centripetal acceleration. By equating the forces in such a manner the condition obtained will involve the parameter for the circular motion. These parameters will generally be the radius, frequency, velocity, acceleration.
Q1) Which of the following statements are correct?
A1) The velocity of the body performing circular motion is constant.
A2) The speed of the body performing circular motion is constant.
A3) The body is acted upon by a force.
A4) The angular velocity of the body is constant.
Q2) There are satellites that complete one revolution around earth in one day. Such kinds of satellites are called as geosynchronous satellites. The typical distance of such satellites from earth is 36,000 km. Calculate the velocity of these satellites.
Q3) The radius of electron in a hydrogen atom is given by the formula: -
From the values of the constants calculate the radius of the electron in first orbit. The typical speed of the electron in this orbit is 0.9c. What is the acceleration of the electron?
Q4) An astronaut is rotated in a horizontal centrifuge at a radius of 5m.
A1) What is the astronaut’s speed if the centripetal acceleration is 7g?
A2) How many revolutions per minute are required to produce such acceleration?
A3) What is the period of motion?
Q5) Calculate the following:-
A1) What is the centripetal acceleration of an object on the earth’s equator owing to the rotation of the earth?
A2) What would the period of rotation of earth have to be for objects on the equator to have a centripetal acceleration equal to 9.8m/s2?
Q6) An electron is moving in a magnetic field. The force acting on such an electron is given as , which can be visualized as follows:-
The diagram explains the meaning of cross product. If the direction of v and B is as shown in the figure the force will be perpendicular to the plane formed by v, B. This force is balanced by the centripetal force and thus a particle performs circular motion. Using this information calculate the following;-
A1) Draw a representation of this electron performing circular motion.
A2) Write down the formula for the frequency for the electron in such a magnetic field of intensity B.
Q7) To do experiments and to verify to the properties of the particle scientists need high energies of particles. To do so accelerators are used. An accelerator is a instrument used to increase the energy of the particles. One such oscillator is a cyclotron. In a cyclotron a particle is rotated around in a circular motion and the radius of the motion is continuously increased. This gives a modest “kick” to the particle and the velocity of the particle increases, increasing the energy of the particle. The key to the cyclotron mechanism lies in the fact that the frequency of the particle should be equal to the frequency of the oscillator. (Resonance condition) The region through which the particle is circulated is called as Dee. Using this information write down the equation for the resonance condition.
A1) Also given that cyclotron is operated at 12 MHz with a Dee of radius of 53 cm. Calculate the magnetic field applied.