Rotation problems

*Angular and tangential velocity.*

1 A child is riding on a merry-go-round a distance of 5.0 m from the axis of rotation. If it takes 10.0 s to complete one revolution, calculate the angular velocity and the tangential velocity of the child. Also find the centripetal acceleration.

*Centripetal force and acceleration. Balance of forces.*

2 A satellite is said to be in a *geostationary* orbit if it rotates above the earth with the same angular velocity as the earth. Calculate the correct height for geostationary orbit assuming that the centripetal force is supplied by gravity

where *M* is the mass of the earth, *m* the mass of the satellite, *G* the gravitational constant and *r* the distance from the centre of the earth.

3 In the Bohr model of the hydrogen atom the electron rotates in a circular orbit round the nucleus. The centripetal force is supplied by electrostatic attraction which is given by Coulomb’s law:

where *e* is the charge on the electron and ε0 is a fundamental physical constant called the permittivity of free space, or the electric constant. Find a relationship between the tangential velocity of the electron and the radius of the orbit, and hence between the potential energy and the rotational kinetic energy of the electron.

*Rotational kinetic energy.*

4 A conker of mass of 8.0 g is whirled around a string of length 30 cm. It executes 1.00 cycle per second. Calculate the centripetal acceleration, the centripetal force, and the rotational kinetic energy.

5 Just as a proper force is the derivative of a potential energy, show that (for a given angular velocity) the centripetal force is the derivative of the rotational kinetic energy with respect to *r*. In a rotating frame of reference the rotational energy therefore appears to be a potential energy. This principle is used in centrifuge equilibrium experiments.

*Moment of inertia (point masses only)*

6 The equilibrium separation between two hydrogen atoms in a hydrogen molecule is 80 pm. Calculate the moment of inertia of the molecule *I*.

If the rotational kinetic energy of the molecule is

,

calculate the time it takes for the molecule to complete one revolution.

7 The CC and CH bond lengths in benzene are 139 pm and 109 pm respectively. Assuming benzene to be perfectly hexagonal calculate the moment of inertia of the benzene molecule about the following three axes, each of which passes through the centre of mass of the molecule:

(*a*) an axis perpendicular to the plane of the molecule

(*b*) an axis in the plane of the molecule, passing through opposite C atoms.

(*c*) an axis in the plane of the molecule, bisecting opposite CC bonds.

8 The planar molecule 35Cl19F3 is T-shaped with the F atom at the mid-point of the cross-piece of the 'T'.

(a) If the F-Cl distance in the stem of the T is 160 pm and the F-Cl distances in the cross-bar of the T are 170 pm, find the position of the centre of mass. (You may assume a bond angle of 90°, which is almost correct.)

(b) Calculate the moment of inertia (in kg m2) about an axis normal to the molecular plane which passes through the centre of mass.

[Relative isotopic masses: 19F 19.00; 35Cl 34.97]

*Angular momentum.*

9 Find the angular momentum of the conker in problem 4.

10 Find an expression for the kinetic energy of a rotating object in terms of its angular momentum (as opposed to its angular velocity).

*Conservation of angular momentum*

11 (*a*) The angular velocity of an ice skater performing a spin increases as the skater’s arms are pulled in. Give a qualitative explanation of this phenomenon.

(*b*) A crude model of this phenomenon can be constructed as follows. Let the trunk of the skater be represented by a cylinder whose moment of inertia is *I*0, and let the arms be represented by two point masses *m* joined to the axis of rotation by a massless arm of length *r*1. Calculate the total moment of inertia of the skater.

(*c*) The skater rotates with an angular velocity ω. Find expressions for the angular velocity, angular momentum and rotational kinetic energy of the skater.

(*d*) The pulling in of the skater’s arms is modelled by sliding the two masses inwards along the arms. Explain how the three quantities in (*c*) will vary as a function of the position of the masses.

(*e*) If you decide that the energy changes, where does the extra kinetic energy come from / where does the lost kinetic energy go?

12 Why do helicopters need tail rotors?

13 A molecule of HCl rotates with an angular momentum 3*ħ*. It absorbs a photon which carries an angular momentum of ±*ħ* in the same direction. What is the angular momentum of the molecule after absorbing the photon?