Describe the basic tenets of mechanics and two dimensional motion (projectile motion, circular motion, rotation, gravitation and fluid mechanics)

Uniform circular motion occurs when an object moves in a circle at constant speed. Although speed is constant, the direction of velocity continuously changes, meaning there is acceleration directed toward the center of the circle.

Key Concepts

• Centripetal acceleration: $a_c = \frac{v^2}{r} = \omega^2 r$
• Centripetal force: $F_c = \frac{mv^2}{r} = m\omega^2 r$
• Angular velocity: $\omega = \frac{2\pi}{T} = 2\pi f$

where $v$ is linear velocity, $r$ is radius, $T$ is period, and $f$ is frequency.

Worked Example

A 500 g stone attached to a string is whirled in a horizontal circle at constant speed of 10 m/s. The string length is 1.0 m. Find: (a) Centripetal acceleration (b) Centripetal force

Solution

(a) $a_c = \frac{v^2}{r} = \frac{10^2}{1} = 100\,$m/s²

(b) $F_c = ma_c = 0.5 \times 100 = 50\,$N

Banking of Roads

For a vehicle on a banked curve without friction: $v = \sqrt{rg\tan\theta}$ where $\theta$ is the banking angle.

This allows vehicles to turn safely at higher speeds without relying solely on friction.

Newton's Law of Universal Gravitation states that every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them:

$F = G\frac{m_1 m_2}{r^2}$

where $G = 6.67 \times 10^{-11}\,$Nm²/kg² is the gravitational constant.

Acceleration Due to Gravity

On Earth's surface: g = \frac{GM_e}{r_e^2} = 9.8\,$m/s²

The value of $g$ decreases with altitude and depth:

• At height $h$: $g' = g\left(1 - \frac{2h}{r_e}\right)$ (approximation for $h \ll r_e$)
• At depth $d$: $g'' = g\left(1 - \frac{d}{r_e}\right)$

Satellites

For a satellite in circular orbit:

• Orbital velocity: $v = \sqrt{\frac{GM_e}{r}}$
• Period: $T = 2\pi\sqrt{\frac{r^3}{GM_e}}$
• Escape velocity: $v_E = \sqrt{\frac{2GM_e}{r_e}} = 11.2\,$km/s (from Earth's surface)

A geostationary satellite orbits at approximately 36,000 km above Earth's surface, matching Earth's rotation period.

Bernoulli's Principle

For steady, incompressible, non-viscous flow: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$

This states that the sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline.

Important applications:

• Torricelli's theorem (efflux from a tank): $v = \sqrt{2gh}$
• Venturi meter: Measures flow speed using pressure difference
• Lift on aircraft wings: Faster air flow over top creates lower pressure above

Viscosity

Newton's law of viscosity describes viscous flow: $F = \eta A\frac{dv}{dy}$

where $\eta$ is the coefficient of viscosity (unit: Pa·s or kg/m·s).

Stokes' law for a sphere falling through viscous fluid: $F = 6\pi\eta rv$

At terminal velocity: $v_t = \frac{2r^2(\rho - \sigma)g}{9\eta}$

Surface Tension

Surface tension $\gamma$ (unit: N/m) causes liquid surfaces to behave like elastic membranes. It arises from cohesive forces between molecules.

Capillary rise: $h = \frac{2\gamma\cos\theta}{\rho g r}$

Excess pressure inside a bubble:

• Air bubble in liquid: $P = \frac{2\gamma}{r}$
• Soap bubble (two surfaces): $P = \frac{4\gamma}{r}$

In Tanzania, understanding projectile motion helps farmers and gardeners use sprinkler irrigation systems effectively. When water is sprayed from a hosepipe at an angle, the parabolic trajectory determines how far the water reaches. If a gardener knows the initial speed of water from their pump (about 10 m/s), they can calculate the optimal spray angle to water crops at different distances from the water source. Similarly, understanding pressure in fluid systems helps in designing water storage tanks for homes and schools—ensuring taps on the top floor still have adequate water pressure when the tank is filled. This knowledge is applied by local plumbers and water engineers throughout Tanzania.