Answer the following questions: i) State Hooke's law. ii) State the principle of conservation of energy. iii) Explain how a simple pendulum can be used to illustrate the Principle of conservation of energy.

Given: Work done ($W$) = $1200\,\text{J}$, Time ($t$) = $30\,\text{s}$.
The formula for power ($P$) is $P = \frac{W}{t}$.

The power output is $\boxed{\text{40 W}}$.

10.

Given: Work done ($W$) = $4000\,\text{J}$, Time ($t$) = $50\,\text{s}$.
The formula for power ($P$) is $P = \frac{W}{t}$.

The power is $\boxed{\text{80 W}}$.

11.

Given: Power ($P$) = $250\,\text{W}$, Time ($t$) = $40\,\text{s}$.
The formula for work done ($W$) is $W = P \times t$.

The work done is $\boxed{\text{10000 J}}$.

12.

Given: Work done ($W$) = $900\,\text{J}$, Power ($P$) = $45\,\text{W}$.
The formula for time ($t$) is $t = \frac{W}{P}$.

The time taken is $\boxed{\text{20 s}}$.

13.

a) What is energy?
Energy is the capacity to do work. It exists in various forms such as kinetic, potential, thermal, electrical, chemical, and nuclear energy.

b) Name the energy changes that take place in each of the following:

i) When an electric bell rings.
Electrical energy is converted into sound energy and kinetic energy.

ii) Someone speaks into a microphone.
Sound energy is converted into electrical energy.

iii) A solid metallic ball is thrown upwards.
Kinetic energy is converted into gravitational potential energy.

iv) A torch is on to give light.
Chemical energy (from the battery) is converted into electrical energy, which then converts into light energy and heat energy.

v) In green plants.
Light energy is converted into chemical energy through photosynthesis.

14.

Given: Mass ($m$) = $4\,\text{kg}$, Height ($h$) = $6\,\text{m}$, Acceleration due to gravity ($g$) = $10\,\text{m/s}^2$.
The formula for gravitational potential energy ($GPE$) is $GPE = mgh$.

The gravitational potential energy is $\boxed{\text{240 J}}$.

15.

Given: Mass ($m$) = $2\,\text{kg}$, Speed ($v$) = $5\,\text{m/s}$.
The formula for kinetic energy ($KE$) is $KE = \frac{1}{2}mv^2$.

The kinetic energy is $\boxed{\text{25 J}}$.

16.

Given: Mass ($m$) = $10\,\text{kg}$, Speed ($v$) = $8\,\text{m/s}$.
The formula for kinetic energy ($KE$) is $KE = \frac{1}{2}mv^2$.

The kinetic energy is $\boxed{\text{320 J}}$.

17.

Given: Potential energy ($PE$) = $500\,\text{J}$, Height ($h$) = $5\,\text{m}$, Acceleration due to gravity ($g$) = $10\,\text{m/s}^2$.
The formula for potential energy is $PE = mgh$. Rearranging for mass ($m$): $m = \frac{PE}{gh}$.

The mass is $\boxed{\text{10 kg}}$.

18.

Given: Mass ($m$) = $3\,\text{kg}$, Height ($h$) = $10\,\text{m}$, Acceleration due to gravity ($g$) = $10\,\text{m/s}^2$.
The formula for potential energy ($PE$) is $PE = mgh$.

The potential energy before it falls is $\boxed{\text{300 J}}$.

19.

a) The work done

Given: Force ($F$) = $80\,\text{N}$, Distance ($d$) = $12\,\text{m}$.
The formula for work done ($W$) is $W = F \times d$.

The work done is $\boxed{\text{960 J}}$.

b) The power developed.

Given: Work done ($W$) = $960\,\text{J}$ (from part a), Time ($t$) = $6\,\text{s}$.
The formula for power ($P$) is $P = \frac{W}{t}$.

The power developed is $\boxed{\text{160 W}}$.

20.

Given: Work done ($W$) = $3000\,\text{J}$, Time ($t$) = $25\,\text{s}$.
The formula for power ($P$) is $P = \frac{W}{t}$.

The power is $\boxed{\text{120 W}}$.

21.

a) The work done.

Given: Mass ($m$) = $15\,\text{kg}$, Height ($h$) = $4\,\text{m}$, Acceleration due to gravity ($g$) = $10\,\text{m/s}^2$.
The work done ($W$) against gravity is equal to the change in gravitational potential energy, $W = mgh$.

The work done is $\boxed{\text{600 J}}$.

b) The power output.

Given: Work done ($W$) = $600\,\text{J}$ (from part a), Time ($t$) = $12\,\text{s}$.
The formula for power ($P$) is $P = \frac{W}{t}$.

The power output is $\boxed{\text{50 W}}$.

22.

Given: Mass ($m$) = $1000\,\text{kg}$, Speed ($v$) = $20\,\text{m/s}$.
The formula for kinetic energy ($KE$) is $KE = \frac{1}{2}mv^2$.

The kinetic energy is $\boxed{\text{200000 J}}$.

23.

a) The work done.

Given: Mass ($m$) = $200\,\text{kg}$, Height ($h$) = $10\,\text{m}$, Acceleration due to gravity ($g$) = $10\,\text{m/s}^2$.
The work done ($W$) against gravity is equal to the change in gravitational potential energy, $W = mgh$.

The work done is $\boxed{\text{20000 J}}$.

b) The power of the pump.

Given: Work done ($W$) = $20000\,\text{J}$ (from part a), Time ($t$) = $50\,\text{s}$.
The formula for power ($P$) is $P = \frac{W}{t}$.

The power of the pump is $\boxed{\text{400 W}}$.

24.

i) State Hooke's law.
Hooke's Law states that the extension of a spring is directly proportional to the force applied to it, provided the elastic limit is not exceeded.

ii) State the principle of conservation of energy.
The principle of conservation of energy states that energy cannot be created or destroyed, but it can be transformed from one form to another. The total energy in an isolated system remains constant.

iii) Explain how a simple pendulum can be used to illustrate the Principle of conservation of energy.
When a simple pendulum is pulled to one side and released, its energy continuously transforms. At its highest point, it has maximum gravitational potential energy and zero kinetic energy. As it swings downwards, potential energy converts to kinetic energy, reaching maximum kinetic energy and minimum potential energy at the lowest point. As it swings upwards again, kinetic energy converts back to potential energy. Ignoring air resistance and friction, the total mechanical energy (potential + kinetic) remains constant, illustrating the conservation of energy.