12PHYS - Mechanics

Finn LeSueur

2021

A ball is thrown upwards with an initial speed of \(161.3km/hr\) (\(44.8ms^{-1}\)).

- How long does it take for the ball to reach its highest point?
- How high does the ball rise?

**Remember:** Knowns, Unknowns, Formula, Substitute, Solve

**How long does it take for the ball to reach its highest point?**

Think in the \(y-direction\)!

**How high does the ball rise?**

Think in the \(y-direction\)!

- Be able to calculate gravitational potential energy (\(E_{p} = mgh\))
- Be able to calculate kinetic energy (\(E_{k} = \frac{1}{2}mv^{2}\))
- Be able to use the law of conservation of energy

Write the date and ngā whāinga ako in your book

Energy is a quantity that must be transferred/transformed to do

work.

**Pātai**: What is energy measured in?**Whakatika**: Joules (J)

- Light
**Heat****Sound**- Electrical
- Radiation
**Kinetic**- Nuclear potential
- Chemical potential
**Gravitational potential****Elastic potential**

Energy can neither be created nor destroyed, it can only be

transformedortransferred.

This tells us that: **the total energy in the system is always conserved**. The system might be a collision/explosion, a beaker, Earth or the whole Universe!

- For example, in this simulation, the skater will never go higher than they started.
- This is because they cannot
*get*extra energy from the surroundings. - In a frictionless world, they will also reach the same height because no energy is lost to the surroundings!

- Kinetic
- Gravitational potential
- Elastic potential

\[\begin{aligned} & E_{k} = \frac{1}{2}mv^{2} \newline & \text{m = mass of the moving object} \newline & \text{v = speed of the moving object} \end{aligned}\]The energy that a moving object has. Related to its velocity and mass.

\[\begin{aligned} & E_{p} = mg \Delta h \newline & \text{m = mass of the object} \newline & \text{g = acceleration due to gravity } 9.8ms^{-2} \downarrow \newline & \text{h = height of the object} \end{aligned}\]Energy an object has by being displaced from

groundin a gravitational field. Related to its mass and height.

When an object falls from a height, its **gravitational potential energy** is transformed into **kinetic energy**, but the total energy in the system is **constant**.

In the real world some energy is lost due to friction as heat, light or sound. In the ideal world 100% of the energy is transformed.

Therefore when comparing an object at the top of its fall, to the bottom of its fall we can say:

\[\begin{aligned} & E_{total} = E_{k} + E_{p} \newline & E_{k} = E_{p} && \text{they are equal} \newline & \frac{1}{2}mv^{2} = m g \Delta h && \text{substitute in the equations} \newline \end{aligned}\]A bullet of mass \(30g\) is fired with a speed of \(400ms^{-1}\) into a sandbag. The sandbag has a mass of \(10kg\) and is suspended by a rope so that it can swing.

Calculate the maximum height that the sandbag rises as it recoils with the bullet lodged inside.

**Step 1.** Find kinetic energy of the bullet

**Step 2.** Equate this with potential energy of sandbag & bullet

- Homework Booklet: Q70, Q66a, Q67a
- Textbook:

- What force do we know is
**not**acting due to the cars movement? - Therefore, what three forces
**are**acting? - Draw a force diagram illustrating these forces and their relative magnitude. Ensure you label them!

- What force do we know is
**not**acting due to the cars movement?*Thrust, because it is not accelerating.* - Therefore, what three forces
**are**acting?*Weight, friction and support.* - Draw a force diagram illustrating these forces and their relative magnitude. Ensure you label them!

- Be able to calculate the energy stored in a spring (\(E_{p} = \frac{1}{2}kx^{2}\))
- Be able to use Hooke’s Law (\(F=-kx\))

Write the date and ngā whāinga ako in your book

\[\begin{aligned} & E_{p} = \frac{1}{2}kx^{2} \newline & \text{k = spring constant} \newline & \text{x = spring compression/stretch (displacement)} \end{aligned}\]A spring displaced from equilibrium will store some potential energy (to return to equilibrium).

- The spring constant is a measure of the stiffness of the spring.
- Low constant \(\rightarrow\) easy to displace
- High constant \(\rightarrow\) difficult to displace

We can relate the displacement of a spring to its spring constant and the force required to create the displacement using **Hooke’s Law**.

- \(F\): the force displacing the spring (Newtons)
- \(x\): the displacement of the spring (meters)
- \(k\): the spring constant (\(Nm^{-1}\))

Paris has a mass of \(55kg\) and she is a spectator at a sports game. She steps onto a bench to get a good view. The bench is \(4m\) long and it is displaced by \(3mm\) in the middle when she stands on it.

- Calculate the spring constant of the bench.
**(M)** - Give correct SI units for the spring constant.
**(A)** - Calculate the elastic potential energy stored in the bench.
**(A)**

- Calculate the spring constant of the bench + unit

- Calculate the elastic potential energy stored in the bench.
**(A)**

A toy aeroplane (\(500g\)) is hanging at the end of a spring. The spring is \(48.0cm\) long when hanging vertically. When the aeroplane is hung from the end of the spring, the length of spring becomes \(80.0cm\).

- Calculate the spring constant.
**(M)** - Write a unit with your answer.
**(A)** - Calculate the energy stored in the spring when a second toy of mass \(400g\) is also hung along with the aeroplane.
**(M)** - The \(500g\) aeroplane is now hung on a stiffer spring, which has double the spring constant. Discuss how this affects the extension and the elastic potential energy in the spring.
**(E)**

- Calculate the spring constant.
**(M)**

- Calculate the energy stored in the spring when a second toy of mass \(400g\) is also hung along with the aeroplane.
**(M)**

- The \(500g\) aeroplane is now hung on a stiffer spring, which has double the spring constant. Discuss how this affects the extension and the elastic potential energy in the spring.
**(E)**

It halves the amount that the spring extends, and reduces the amount of energy stored by a lot.

\[\begin{aligned} & E_{p} = \frac{1}{2}kx^{2} \newline & E_{p} = \frac{1}{2} \times 30.62 \times 0.16^{2} \newline & E_{p} = 0.39J \end{aligned}\]- Homework Booklet: 47-48
- Textbook:

- Give the equations for kinetic, gravitational potential and elastic potential energy.
- Give the name and formula for the law that you can use to relate
**force, spring constant and displacement**. - Lachie is going to football in the weekend. The van he rides in with some of his teammates has suspension on each wheel. Lachie and his teammates weight \(357kg\) in total and their weight is spread evenly across all four springs. The springs have a spring constant of \(2.26 \times 10^{4}Nm^{-1}\). Calculate how much the car
**sinks down**when they get into the car.**(E)** - How much energy is stored in each spring if car sinks \(0.12m\)?
**(A)**

- Give the equations for kinetic, gravitational potential and elastic potential energy.

- Give the name and formula for the law that you can use to relate
**force, spring constant and displacement**.

- Calculate how much the car
**sinks down**when they get into the car.**(E)**

Step 1: Weight per Spring

\[\begin{aligned} & F = \frac{357 \times 9.8}{4} \newline & F = 874.65N \end{aligned}\]Step 2: Displacement

\[\begin{aligned} & F = kx && \text{Hooke's Law} \newline & x = \frac{F}{k} \newline & x = \frac{874.65}{2.26 \times 10^{4}} \newline & x = 0.0387m \newline \end{aligned}\]- How much energy is stored in each spring if they are compressed by \(0.12m\)?
**(A)**\[\begin{aligned} & E_{p} = \frac{1}{2}kx^{2} \newline & E_{p} = \frac{1}{2} 2.26 \times 10^{4} \times 0.12^{2} \newline & E_{p} = 160J \end{aligned}\]

- Be able to define & calculate work done (\(W=Fd=mgh\))
- Be able to define & calculate power (\(P=\frac{W}{t}\))

Write the date and ngā whāinga ako in your book

The amount of energy transferred/transformed (Joules, J).

One joule of work is done when a force of one newton moves an object one meter.

\[\begin{aligned} & W = Fd \newline & work = force \times distance \end{aligned}\]- Consider moving an object through a gravitational field
- You have to exert a force against a
**weight force**, to lift it some distance

- Work is done
**only**when energy is transferred or transformed. **Work Done**: Lifting an object and placing it on a shelf transfers energy to that object into the form of gravitational potential energy.**No Work Done**: Moving an object horizontally where it starts and finishes with \(v=0ms^{-1}\).- Work only depends on the
*start*and*finish*position, the path does not matter (path independence).

In 2016 weightlifter Eddie Hall set a new (at the time) world record for heaviest deadlift of \(500kg\). If he lifted the weights to a height of \(1.25m\), how much work did Eddie do?

\[\begin{aligned} & P = \frac{W}{t} \newline & power = \frac{work}{time} \newline & power = \frac{Joules}{seconds} \newline & power = Js^{-1} && \text{also known as a Watt (W)} \end{aligned}\]The rate at which energy is transferred/transformed (the rate at which work is done).

If it took Eddie \(7s\) to do \(6125J\) of work on the weights, what power was he exerting?

- Homework Booklet: 65, 66b, 69, 68
- Textbook: