c) State four limitations of Bernoulli's equation. (4 marks) d) For a hydraulic device, a piston has a cross-sectional area of 30cm^2moving an incompressible liquid with a force of 60 N.The other end of the hydraulic pipe is attached to 2^nt piston with a 60cm^2 cross-sectional area. Determine the force on the second piston? (2 marks) QUESTION SIX (14 marks) a) An object of mass 3 kg is attached to a string of length 0.4 m which will cut if the tension exceeds 300 N. The object is whirled in a vertical circle whose centre is 4.5m above the ground. The angular velocity is gradually increased until the string cuts. (i) In what position is this likely to occur and what is the angular velocity? (4 marks) (ii) What time will the stone take to reach the ground. (3 marks) (iii) Where will the stone hit the ground? (3 marks) b) State four examples of applications of circular motion in real life. (4 marks)

### Question 5<br /><br />#### (c) **State four limitations of Bernoulli's equation.** <br />(4 marks)<br /><br />1. **Incompressible Flow Assumption**: Bernoulli's equation assumes that the fluid is incompressible, which is not valid for gases at high velocities or under significant pressure changes.<br /><br />2. **No Viscosity**: The equation neglects the effects of viscosity, making it unsuitable for flows where viscous forces are significant, such as in highly viscous fluids or near solid boundaries.<br /><br />3. **Steady Flow Assumption**: It applies only to steady flow conditions, meaning the velocity and other properties of the fluid do not change with time.<br /><br />4. **Irrotational Flow**: Bernoulli's equation assumes irrotational flow, so it cannot be applied to flows with significant rotational effects, such as turbulent or swirling flows.<br /><br />---<br /><br />#### (d) **Determine the force on the second piston.** <br />(2 marks)<br /><br />We use Pascal's principle, which states that pressure is transmitted equally throughout a confined fluid. The pressure exerted by the first piston is equal to the pressure exerted on the second piston:<br /><br />\[<br />P_1 = P_2<br />\]<br /><br />Pressure is defined as force per unit area:<br /><br />\[<br />\frac{F_1}{A_1} = \frac{F_2}{A_2}<br />\]<br /><br />Where: <br />- \( F_1 = 60 \, \text{N} \) <br />- \( A_1 = 30 \, \text{cm}^2 = 30 \times 10^{-4} \, \text{m}^2 = 0.003 \, \text{m}^2 \) <br />- \( A_2 = 60 \, \text{cm}^2 = 60 \times 10^{-4} \, \text{m}^2 = 0.006 \, \text{m}^2 \) <br /><br />Rearranging for \( F_2 \):<br /><br />\[<br />F_2 = F_1 \cdot \frac{A_2}{A_1}<br />\]<br /><br />Substitute the values:<br /><br />\[<br />F_2 = 60 \cdot \frac{0.006}{0.003}<br />\]<br /><br />\[<br />F_2 = 60 \cdot 2 = 120 \, \text{N}<br />\]<br /><br />**Answer**: The force on the second piston is **120 N**.<br /><br />---<br /><br />### Question 6<br /><br />#### (a) **An object of mass 3 kg is attached to a string of length 0.4 m...**<br /><br />##### (i) **In what position is this likely to occur and what is the angular velocity?** <br />(4 marks)<br /><br />The tension in the string is maximum when the object is at the **bottom of the vertical circle**, as both the centripetal force and the weight of the object act in the same direction.<br /><br />The tension \( T \) at the bottom of the circle is given by:<br /><br />\[<br />T = m \cdot g + m \cdot r \cdot \omega^2<br />\]<br /><br />Where: <br />- \( T = 300 \, \text{N} \) (maximum tension before the string cuts) <br />- \( m = 3 \, \text{kg} \) <br />- \( g = 9.8 \, \text{m/s}^2 \) <br />- \( r = 0.4 \, \text{m} \) <br />- \( \omega \) is the angular velocity (to be determined).<br /><br />Rearranging for \( \omega \):<br /><br />\[<br />300 = 3 \cdot 9.8 + 3 \cdot 0.4 \cdot \omega^2<br />\]<br /><br />\[<br />300 = 29.4 + 1.2 \cdot \omega^2<br />\]<br /><br />\[<br />300 - 29.4 = 1.2 \cdot \omega^2<br />\]<br /><br />\[<br />270.6 = 1.2 \cdot \omega^2<br />\]<br /><br />\[<br />\omega^2 = \frac{270.6}{1.2} = 225.5<br />\]<br /><br />\[<br />\omega = \sqrt{225.5} \approx 15.02 \, \text{rad/s}<br />\]<br /><br />**Answer**: The string is likely to cut at the **bottom of the circle**, and the angular velocity is approximately **15.02 rad/s**.<br /><br />---<br /><br />##### (ii) **What time will the stone take to reach the ground?** <br />(3 marks)<br /><br />When the string cuts, the stone becomes a projectile. At the bottom of the circle, the stone has an initial velocity equal to the tangential velocity:<br /><br />\[<br />v = r \cdot \omega<br />\]<br /><br />Substitute the values:<br /><br />\[<br />v = 0.4 \cdot 15.02 = 6.008 \, \text{m/s}<br />\]<br /><br />The height from which the stone falls is the sum of the radius of the circle and the height of the center above the ground:<br /><br />\[<br />h = r + 4.5 = 0.4 + 4.5 = 4.9 \, \text{m}<br />\]<br /><br />The time to fall is determined using the equation of motion:<br /><br />\[<br />h = \frac{1}{2} g t^2<br />\]<br /><br />Rearranging for \( t \):<br /><br />\[<br />t = \sqrt{\frac{2h}{g}}<br />\]<br /><br />Substitute the values:<br /><br />\[<br />t = \sqrt{\frac{2 \cdot 4.9}{9.8}} = \sqrt{1} = 1 \, \text{s}<br />\]<br /><br />**Answer**: The stone will take **1 second** to reach the ground.<br /><br />---<br /><br />##### (iii) **Where will the stone hit the ground?** <br />(3 marks)<br /><br />The horizontal distance traveled by the stone is given by:<br /><br />\[<br />x = v \cdot t<br />\]<br /><br />Substitute the values:<br /><br />\[<br />x = 6.008 \cdot 1 = 6.008 \, \text{m}<br />\]<br /><br />**Answer**: The stone will hit the ground **6.01 m horizontally** from the point where the string cuts.<br /><br />---<br /><br />#### (b) **State four examples of applications of circular motion in real life.** <br />(4 marks)<br /><br />1. **Centrifuges**: Used in laboratories and industries to separate substances based on density.<br />2. **Amusement Park Rides**: Such as Ferris wheels and roller coasters.<br />3. **Vehicle Turning**: Cars taking turns on curved roads involve circular motion.<br />4. **Satellites**: Orbiting Earth due to gravitational circular motion.<br /><br />