15  Workshop 2

Week 5

15.1 Part 1 - Preliminary questions

Formative (i.e., not assessed), for practice, and does not need to be submitted

15.1.1 Work done in circular motion

What is the rate at which work is done on a particle executing motion in a circle at a constant speed?

15.1.2 Neutron colliding with carbon nucleus

A neutron of mass \(m_n\) and speed \(v_{ni}\) undergoes a head-on, perfectly elastic collision with a carbon nucleus of mass \(m_C\) initially at rest. You may assume Newtonian mechanics applies.

  1. What are the final velocities of the neutron and carbon nucleus?
  2. What fraction of its initial energy does the neutron lose?

15.1.3 Vector products

Find the vector product \(\vec{a} \times \vec{b}\) of the following vectors.

  1. \(\vec{a} = 4 \hat{\imath}\) and \(\vec{b} = 6 \hat{\imath} + 6\hat{\jmath}\).
  2. \(\vec{a} = 2 \hat{\imath} + 3 \hat{\jmath} − \hat{k}\) and \(\vec{b} = 3\hat{\imath} + \hat{\jmath} + 4\hat{k}\).

If you haven’t seen the vector product before, we will go through this in lectures. You can find out more by looking at Section 8.1 of the notes. Here is the information you need to answer this question.

The vector product of two vectors \(\vec{a}\) and \(\vec{b}\) produces a third vector \(\vec{c}\) that is perpendicular to both \(\vec{a}\) and \(\vec{b}\). The magnitude of \(\vec{c}\) is equal to the area of the parallelogram formed by \(\vec{a}\) and \(\vec{b}\).

There are various ways to calculate \(\vec{c}\) including the determinant method described below.

\[ \vec{c} = \vec{a} \times \vec{b} = \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix} \]

Illustration of the pattern you can remember for calculating the determinant. Each circled unit basis vector, with sign above it, is multiplied by the cofactor indicated by the dotted lines to get the vector product.

\[ \vec{a} \times \vec{b} = {\color{green} +(a_y b_z - b_y a_z) \hat{\imath}} {\color{blue} -(a_x b_z - b_x a_z) \hat{\jmath}} {\color{red} +(a_x b_y - b_x a_y) \hat{k}} \]

Note the “\(-\)” sign associated with the \(\hat{\jmath}\) term.

15.1.4 Person pulling sled

A girl pulls a sled in which her brother sits, over the snow. She pulls a rope over her shoulder which is attached to the sled. The boy has mass 19 kg, the sled has mass 1.4 kg, the girl is 1.2 m high at the shoulder and the rope is 2.4 m long. You may assume that the coefficient of kinetic friction \(\mu k\) between the sled and the snow is 0.1.

  1. Draw a diagram of the forces acting on the sled, assuming it travels at a constant speed. What are the forces acting on the girl?
  2. What is the tension in the rope?
  3. The girl changes the position she is carrying the rope, dragging the rope now at a height of 0.6 m. Does she have to increase or decrease the force with which she pulls on the rope to maintain a constant speed?
  4. The girl is now pulling the sled over grass for which \(\mu_k = 0.5\). Plot a graph of the tension versus the angle between the rope and the ground, and explain what you see.

15.2 Part 2 - Problem solving question

Summative (i.e., counts 3.75% towards final grade), please submit PDF individually

The problem solving question will be provided during the workshop session.