Q3. A platform of volume \( 1.00 \times 10^{5} \mathrm{~m}^{3} \) and density \( 927 \mathrm{~kg} / \mathrm{m}^{3} \) floats in seawater (density \( =1024 \mathrm{~kg} / \mathrm{m}^{3} \) ). Draw and label a free body diagram representing the platform in the seawater. Calculate the volume of the platform that is above the water-line (i.e. not submerged). Q4. A racing car is designed to produce down-force as it moves. Air is directed beneath the chassis and between the wheels as shown in the simplified diagram below. The cross-sectional area at the very front of the car is \( A_{0}=0.0330 \mathrm{~m}^{2} \) and at the centre, it is \( A_{1}=0.0310 \mathrm{~m}^{2} \). The car moves forward at 30.0 \( \mathrm{m} / \mathrm{s} \). Draw and label a free body diagram that includes the weight of the car, down-force and normal force. Calculate the difference in pressure between the air under the middle of the car (with cross-sectional area \( A_{1} \) ) and atmosphere. State whether the pressure under the car is higher or lower than atmospheric pressure. You can treat the air flow under the car as laminar and as though it occurs through a closed pipe that decreases in cross-sectional area. Density of air \( =1.21 \mathrm{~kg} / \mathrm{m}^{3} \) Area \( A_{0} \) Area \( A_{1} \)
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Step 1: Free body diagram for the platform in seawater The free body diagram for the platform in seawater will have two forces acting on it: Show more…
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Floating Car A car has a total mass of $1800 \mathrm{~kg}$. The volume of air space in the passenger compartment is $5.00 \mathrm{~m}^{3} .$ The volume of the motor and front wheels is $0.750 \mathrm{~m}^{3}$, and the volume of the rear wheels, gas tank, and trunk is $0.800 \mathrm{~m}^{3}$; water cannot enter these areas. The car is parked on a hill; the handbrake cable snaps and the car rolls down the hill into a lake (Fig. $15-43$ ). (a) At first, no water enters the passenger compartment. How much of the car, in cubic meters, is below the water surface with the car floating as shown? (b) As water slowly enters, the car sinks. How many cubic meters of water are in the car as it disappears below the water surface? (The car, with a heavy load in the trunk, remains horizontal.)
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Supratim P.
Untethered helium balloons, floating in a car that has all the windows rolled up and outside air vents closed, move in the direction of the car's acceleration, but loose balloons filled with air move in the opposite direction. To show why, consider only the horizontal forces acting on the balloons. Let $a$ be the magnitude of the car's forward acceleration. Consider a horizontal tube of air with a cross-sectional area $A$ that extends from the windshield, where $x=0$ and $p=p_{0},$ back along the $x$ -axis. Now consider a volume element of thickness $d x$ in this tube. The pressure on its front surface is $p$ and the pressure on its rear surface is $p+d p$ . Assume the air has a constant density $\rho$ . (a) Apply Newton's second law to the volume element to show that $d p=\rho a d x$ . (b) Integrate the result of part (a) to find the pressure at the front surface in terms of $a$ and $x$ . (c) To show that considering $\rho$ constant is reasonable, calculate the pressure difference in atm for a distance as long as 2.5 $\mathrm{m}$ and a large acceleration of 5.0 $\mathrm{m} / \mathrm{s}^{2}$ . (d) Show that the net horizontal force on a balloon of volume $\dot{V}$ is $\rho V a$ . (e) For negligible friction forces, show that the acceleration of the balloon (average density $\rho_{\text { bal }} )$ is $\left(\rho / \rho_{\text { bal }}\right) a,$ so that the acceleration relative to the car is $a_{n e 1}=\left[\left(\rho / \rho_{\text { bel }}\right)-1\right] a$ (f) Use the expression for $a_{\mathrm{rel}}$ in part (e) to explain the movement of the balloons.
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