Newton's third law links interaction forces
Definition
Newton's third law
Newton's third law states that when two objects interact, they exert forces on each other that are equal in magnitude and opposite in direction.
Definition
Momentum
Momentum is the product of an object's mass and velocity, measured in kilogram metres per second (kg m/s).
- A Newton's third-law pair acts on two different objects during the same interaction.
- The forces are the same type, equal in magnitude, opposite in direction and present for the same time.
- The pair does not cancel because the two forces act on different objects.
- For a book resting on a table, the book pushes down on the table and the table pushes up on the book with an equal force.
Common Mistake
- Do not pair the weight of a book with the table's normal contact force because both forces act on the book.
- The third-law partner of the book's weight is the gravitational force that the book exerts on Earth.
Momentum includes direction
- Momentum is calculated using p=mvp=mvp=mv, where ppp is momentum in kg m/s\text{kg m/s}kg m/s, mmm is mass in kg\text{kg}kg and vvv is velocity in m/s\text{m/s}m/s.
- Momentum is a vector, so choose one direction as positive and give motion in the opposite direction a negative velocity and momentum.
- An object at rest has zero momentum because v=0v=0v=0.
Example
- Moving car: a 1200 kg1200\ \text{kg}1200 kg car travels east at 15 m/s15\ \text{m/s}15 m/s.
- Taking east as positive, p=mv=1200×15=1.8×104 kg m/sp=mv=1200\times15=1.8\times10^4\ \text{kg m/s}p=mv=1200×15=1.8×104 kg m/s.
- Its momentum is 1.8×104 kg m/s1.8\times10^4\ \text{kg m/s}1.8×104 kg m/s east.
Momentum is conserved in a closed system
Definition
Conservation of momentum
Conservation of momentum states that in a closed system the total momentum before an interaction equals the total momentum after the interaction.
- During a collision, each object experiences an equal and opposite force for the same time.
- Each object therefore has an equal and opposite change in momentum, so the total momentum of the system does not change.
- Write conservation as ∑pbefore=∑pafter\sum p_{\text{before}}=\sum p_{\text{after}}∑pbefore=∑pafter and include signs for direction.
- Kinetic energy does not have to be conserved in a collision because some energy may be transferred to thermal stores, sound or deformation.
Example
- Before: a 2.0 kg2.0\ \text{kg}2.0 kg trolley moving right at 4.0 m/s4.0\ \text{m/s}4.0 m/s hits a stationary 3.0 kg3.0\ \text{kg}3.0 kg trolley and they stick together.
- Initial momentum: p=(2.0×4.0)+(3.0×0)=8.0 kg m/sp=(2.0\times4.0)+(3.0\times0)=8.0\ \text{kg m/s}p=(2.0×4.0)+(3.0×0)=8.0 kg m/s.
- Combined mass: m=2.0+3.0=5.0 kgm=2.0+3.0=5.0\ \text{kg}m=2.0+3.0=5.0 kg.
- Conservation gives 8.0=5.0v8.0=5.0v8.0=5.0v, so v=1.6 m/sv=1.6\ \text{m/s}v=1.6 m/s to the right.
Explosions also conserve momentum
- If two objects start at rest, their total initial momentum is zero.
- After they push apart, their momenta are equal in magnitude and opposite in direction, so their vector sum remains zero.
Exam technique
- State a positive direction before calculating and keep negative signs until the final answer.
- For a third-law explanation, name both objects, both forces, and state that the forces are equal in magnitude and opposite in direction.
- For conservation, calculate the total momentum of the whole system before and after the interaction.
Interaction forces explain momentum changes
- A larger force or a longer interaction time produces a larger change in momentum.
- Equal and opposite changes in momentum preserve the total momentum of a closed system.
Self review
- State Newton's third law.
- Define momentum and give its unit.
- Why do third-law forces not cancel each other?
- What condition is needed for total momentum to be conserved?
- How should direction be represented in a momentum calculation?
