Common Forces, Free-Body Diagrams and Connected Bodies: Quick Revision

A brief note on Common Forces, Free-Body Diagrams and Connected Bodies. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Weight $W = mg$, always down. Mass stays the same everywhere; weight does not.
• The normal reaction pushes at right angles to a surface, only as hard as it needs to.
• Tension pulls along a string, the same all along a light string, even over a smooth pulley.
• Hooke's law: a spring pushes or pulls with $F = kx$.
• Free-body diagram: one body, every force on it, then $F_{\text{net}} = ma$ along each axis.
• Connected bodies: the whole system for $a$, then one body alone for the force between them.

The facts at a glance

• Scale in a lift (up positive): $R = m(g + a)$; zero in free fall
• Load on a rope (up positive): $T = m(g + a)$
• Two equal strings at $\theta$ to the vertical: $2T\cos\theta = mg$
• Atwood machine: $a = \dfrac{(m_1 - m_2)g}{m_1 + m_2}$, $T = \dfrac{2m_1m_2g}{m_1 + m_2}$
• Table block $M$, hanging $m$: $a = \dfrac{mg}{M + m}$, $T = Ma$
• Three forces in equilibrium: they form a closed triangle

A worked example

Three trolleys, of $2.5\ \text{kg}$, $1.5\ \text{kg}$ and $1.0\ \text{kg}$, are joined in a row and pulled along a smooth floor with $15\ \text{N}$ on the first one.

1. System. $a = \dfrac{15\ \text{N}}{5.0\ \text{kg}} = 3.0\ \text{m/s}^2$.
2. First string. It pulls the last two trolleys: $T_1 = (1.5\ \text{kg} + 1.0\ \text{kg}) \times 3.0\ \text{m/s}^2 = 7.5\ \text{N}$.
3. Second string. It pulls only the last trolley: $T_2 = 1.0\ \text{kg} \times 3.0\ \text{m/s}^2 = 3.0\ \text{N}$.

Each string pulls just enough to speed up what lies behind it.

Practice with new numbers

A program to try

The program adds up all the forces on one body, part by part, and finds its acceleration.

Mistakes to avoid

• $N = mg$ every time. A slanting pull, a push, a slope or a lift all change $N$.
• A "force of motion" drawn along the velocity. Only real pushes and pulls go on the diagram.
• Tension equal to a weight when the bodies accelerate. It lies between the two weights in an Atwood machine.
• Dividing by one mass when a force speeds up several bodies.
• $\sin$ and $\cos$ swapped. Check whether the angle is from the vertical or the horizontal.

Check yourself

1. A scale under a $45\ \text{kg}$ girl in a lift reads $495\ \text{N}$. What is the lift's acceleration? ($g = 9.8\ \text{m/s}^2$.)

2. A spring $25\ \text{cm}$ long becomes $30\ \text{cm}$ long under an $8.0\ \text{N}$ load. What is its spring constant?

3. A box rests on a rough slope. Name every force on it.

4. A $20\ \text{N}$ sign hangs from a cord. You push it sideways with $15\ \text{N}$ and hold it still. What is the tension in the cord?

5. In an Atwood machine the tension is $30\ \text{N}$, and the lighter mass is $2.5\ \text{kg}$. What is the acceleration? ($g = 9.8\ \text{m/s}^2$.)