11  Problems 3

Week 6

11.1 Vector product

Find the vector product of \(\vec{a} \times \vec{b}\) where \[ \vec{a} = 2\hat{\imath} + 3\hat{\jmath} \] \[ \vec{b} = 3\hat{\imath} + 4\hat{\jmath} \]

Draw a sketch to check your answer – does the magnitude of the vector correspond to the area you expect, and is it in the direction you expect?

11.2 Lorentz force

The force on a particle of charge \(q\) moving with velocity \(\vec{v}\) due to an electric field \(\vec{E}\) and a magnetic field \(\vec{B}\) is given by the Lorentz force, defined as follows. \[ \vec{F} = q (\vec{E} + \vec{v} \times \vec{B}). \]

A particle of charge \(+1\) coulomb is moving with velocity \(\vec{v}\) in an electric field \(\vec{E}\) and magnetic field \(\vec{B}\).

  1. Calculate the force on the particle if \[ \vec{v} = 8.3 \hat{\imath} + 5.6 \hat{\jmath} \text{ m s}^{-1} \] \[ \vec{E} = \vec{0} \text{ V m}^{-1} \] \[ \vec{B} = 1.7\hat{\imath} + 5.0\hat{\jmath} \text{ T}\]

    Is any work being done on the particle by the Lorentz force in this case?

  2. Calculate the force on the particle if \[ \vec{v} = −9\hat{\imath} + 6\hat{\jmath} − 3\hat{k} \text{ m s}^{-1} \] \[ \vec{E} = 6\hat{\imath} − 4\hat{\jmath} + 2\hat{k} \text{ V m}^{-1} \] \[ \vec{B} = 3\hat{\imath} − 2\hat{\jmath} + \hat{k} \text{ T}\]

11.3 Circular motion in a synchrotron – angular SUVAT

Synchrotrons are used to generate high intensity X-ray beams. The Diamond light source at Harwell in Oxfordshire is one of the most powerful X-ray sources available for physics research. At Diamond, electrons are first accelerated in a linear accelerator and then injected into a synchrotron ring where they are further accelerated (by a constant angular acceleration α) up to their final speed, which is a substantial fraction of the speed of light.

Suppose the electrons are travelling with a speed \(v\) when they first enter the synchrotron ring which has a radius \(R\). This means they have an initial angular speed \(\omega_0 = v/R\) (measured in rad s\(^{-1}\)) and are subjected to a constant angular acceleration \(\alpha\) measured in rad s\(^{-2}\), until they reach their final speed. You may also assume that the initial angle that the electrons enter the ring \(\phi_0 = 0\) rad. You should ignore relativistic effects.

  1. Write down an equation describing the angular velocity \(\omega\) of an electron as a function of time since its injection.
  2. By integration, find an equation describing the angular position \(\phi\) (in radians) of the electron as a function of time.
  3. Using the result of the previous part, find the total distance (in m) that the electron will move around the synchrotron ring as a function of time.

11.4 Moment of inertia of a cylinder

Note: I have changed the wording of this question a little bit after feedback from the problems class in an attempt to make it clearer. Please get in touch if anything isn’t clear; I am happy to discuss it with you.

Consider a thin cylindrical shell of density \(\rho\), length \(L\), radius \(a\), and thickness \(\delta a\), where \(\delta a\) is small.

  1. Show that the mass of the shell is given by \[ \delta m = \rho L 2\pi a \delta a + \underbrace{\rho L \pi (\delta a)^2}_{\text{\color{blue} we can ignore this term (see below)}} \] and its moment of inertia about the axis of the cylinder is given by \[ \delta I = \rho L 2\pi a^3 \delta a + \underbrace{\rho L \pi a^2 (\delta a)^2}_{\text{\color{blue} we can ignore this term (see below)}} \]

    Hence, in the limit as \(\delta a \to 0\), \[ \frac{dI}{da} = \rho L 2\pi a^3 \]

  2. The moment of inertia about the axis of a uniform, thick cylindrical tube of density \(\rho\), length \(L\), inner radius \(r_i\) and outer radius \(r_o\) can be calculated by considering the tube as the sum of many thin cylindrical shells. Explain, with the aid of a diagram, why the moment of inertia of the tube is given by \[ I = \int_{I_i}^{I_o} dI = \int_{a = r_i}^{a = r_o} \frac{dI}{da} da \] and hence show that \[ I = \frac{\pi\rho L}{2} (r_o^4 - r_i^4) \]

  3. Find an expression for \(\rho\) in terms of \(M\), \(r_i\), \(r_o\), and \(L\), and show that \[ I = \frac{1}{2} M (r_i^2 + r_o^2)\]

  4. Write down expressions for the linear and rotational kinetic energy of such a cylinder when the centre of mass has a speed of \(v_\text{com}\). Assume the cylinder is rolling and not slipping.

    Hint: you can write \(v_\text{com} = \omega r_o\).

  5. Such a hollow cylinder is released at the top of a slope and rolls down the slope, under gravity, without slipping. Show that the speed of the centre of mass after the cylinder has dropped by a vertical distance \(h\) is given by \[ v_\text{com} = \sqrt{\frac{4gh}{3+\left(\frac{r_i}{r_o}\right)^2}} \]

  6. A hollow cylinder of mass 1 kg with inner radius 5 cm and outer radius 8 cm is rolling along the ground; the centre of mass has a speed of \(1.6 \text{ m s}^{-1}\).

    1. What is the angular velocity of the cylinder?
    2. What is the linear kinetic energy of the cylinder?
    3. What is the rotational kinetic energy of the cylinder?