QUESTION 4 (Start on a new page.) A construction worker has to pump water from reservoir A to reservoir B over a hill using an electric water pump, as illustrated in the diagram below - NSC Technical Sciences - Question 4 - 2024 - Paper 1

The force applied (pumping force) on the water is equal to the weight of the water since the system is working at a constant velocity. Thus:

$F_p = F_w = 8330 ext{ N, upwards}$

To find the velocity, use the power equation:

$P = F imes v$

Rearranging gives:

$v = \frac{P}{F}$

Substituting the values:

$v = \frac{7200 ext{ W}}{8330 ext{ N}} = 0.865 ext{ m/s, upwards}$

The principle of conservation of mechanical energy states that in an isolated system, the total mechanical energy (the sum of potential and kinetic energy) remains constant unless acted upon by an external force.

The work done by the gravitational force can be calculated using the formula:

$W = F_g imes d imes ext{cos}(\theta)$

Here:

• $F_g = 8330$ N (weight of the water)
• $d = 6$ m (distance of pipeline)
• $\theta = 0º$ (since the direction of force and displacement are in the same direction)

Thus:

$W = 8330 ext{ N} \times 6 ext{ m} \times ext{cos}(0º) = 49980 ext{ J}$

The work done against gravity when moving the mass of water is eventually converted into potential energy as it gains height. The relevant formula showing the relation of work done and gravitational potential energy is:

$E_p = m imes g imes h$

Where:

• $m = 850$ kg (mass of water)
• $g = 9.8$ m/s² (acceleration due to gravity)
• $h = 12.58$ m (height)

The work done directly translates to the gravitational potential energy gained, thus confirming the principle of conservation of mechanical energy.

The gravitational potential energy (E_p) can be calculated using the formula:

$E_p = m imes g imes h$

Here:

• $m = 274$ kg
• $g = 9.8$ m/s²
• $h = 12.58$ m

Thus:

$E_p = 274 ext{ kg} \times 9.8 ext{ m/s}^2 \times 12.58 ext{ m} = 34276.7 ext{ J}$

At the height of 6 m, the mechanical energy consists of both potential and kinetic energy. The potential energy (E_p) at this height is:

$E_p = m imes g imes h = 274 ext{ kg} \times 9.8 ext{ m/s}^2 \times 6 ext{ m} = 16065.6 ext{ J}$

Assuming the water has a velocity $v$ at this point, and considering kinetic energy (E_k):

$E_k = \frac{1}{2} m v^2$

The total mechanical energy will then be:

$E_{total} = E_p + E_k$