Question 13 of 13: Magnus Effect — Why Topspin Makes a Table Tennis Ball Dip
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
04-BS-7 Mechanics of Fluids — National Examination, 2018-May. Three (3) hours duration, closed book. Section A (Calculative, 9 questions, do 7) and Section B (Graphical & Analytical, 4 questions, do 3); every question is answered below regardless of the exam's "do N of M" instruction, so the set is a complete study resource.
Reference texts: White, F.M., Fluid Mechanics (8th ed.) — fluid statics and manometry (Ch. 2), Bernoulli and the energy equation (Ch. 3), viscous flow in ducts and the Moody chart (Ch. 6), flow past immersed bodies and drag (Ch. 7), potential flow and the Magnus effect (Ch. 8), open-channel flow and the hydraulic jump (Ch. 10), turbomachinery and jet propulsion (Ch. 11).
Check — assumptions used across this paper:
Q1's manometer chain is read off the extraction as a two-stage water–mercury–glycerine–(air)–glycerine–mercury system. The enclosed air pocket between the two glycerine columns is treated as weightless (uniform pressure), so only the one described open end is needed to close the hydrostatic chain back to pipe P; the second "opening" is not load-bearing for this calculation.
Q5's wave/hydraulic-jump analysis takes the depth "in front of the wave" (0.15 m, undisturbed, at rest) as the upstream state and "behind the wave" (0.75 m) as the downstream state, per the question's own prose (the raw figure-label ordering in the extraction is a reconstruction and is not used to override the stated text). The classic hydraulic-jump head-loss formula is applied to the given depths, and the swept flow rate uses the measured wave celerity directly — a standard engineering estimate, not a fully momentum-self-consistent bore solution.
Q6's air properties are taken at 20°C (domestic ambient, ρ=1.19 kg/m³) since no duct-air temperature is stated.
Q7(c)'s "discharged at right angles to the initial direction (20° becomes 0°)" is read as: the reverser's exhaust jet is normally angled 20° forward of the fully-radial (right-angle) direction; part (c) removes that forward lean entirely, leaving a purely radial (90° to the engine axis) discharge with zero axial velocity component.
Q8's terminal velocity and Q9's cable drag coefficient are obtained from the Reynolds-number relations the attached charts themselves plot (Morrison's sphere-drag correlation for Q8; the flat subcritical Cd≈1.2 plateau of the smooth-cylinder curve for Q9, since Re≈3×104 falls solidly within it).
Question 13: Magnus Effect — Why Topspin Makes a Table Tennis Ball Dip (5 marks)
Given. A table tennis ball travelling forward with topspin (the top surface of the ball rotates in the same sense as the ball's forward travel, as in a wheel rolling forward).
In the ball-fixed frame (hint), air streams past from front to back. Topspin carries the top surface in the same direction as that local relative flow, entraining and accelerating it (streamlines crowd, pressure drops); the bottom surface moves against the local flow, retarding it (streamlines spread, pressure rises). The pressure imbalance pushes the ball downward.
Find. Physical explanation, with streamlines and pressure variation, for the extra downward curvature caused by topspin.
Approach. Move into the ball's own reference frame (hint) so the ball is a stationary spinning sphere with air streaming past it from front to back; compare how the spinning surface locally speeds up or slows down the boundary-layer flow on the top versus the bottom, then apply Bernoulli's equation to convert that velocity asymmetry into a pressure asymmetry.
With topspin, the top of the ball's surface moves in the same direction as the ball's forward travel — which, viewed in the ball's own frame (hint), is the SAME direction the local air is already streaming past the top of the ball (from front to back). Viscous entrainment means the moving surface drags the adjacent boundary-layer air along with it, effectively ADDING the surface speed to the local relative flow: the air over the top of the ball is accelerated. By Bernoulli's equation, a faster local flow at essentially the same elevation means a LOWER local pressure over the top of the ball.
The bottom surface, by contrast, moves OPPOSITE to the local relative flow direction there (the ball's rotation carries the bottom surface forward relative to the ball's centre, i.e. against the backward-streaming relative air), so it retards the boundary layer instead of accelerating it: the air near the bottom is slowed, and by Bernoulli's equation the local pressure there RISES.
The result is a higher pressure on the bottom of the ball and a lower pressure on the top — a net downward force (the Magnus force) superimposed on gravity. This is why a heavily top-spun table tennis (or tennis, or golf) shot dips into the table noticeably faster and more sharply than gravity alone would produce; the opposite spin sense (backspin) produces the opposite pressure asymmetry and an upward Magnus force that makes a ball "float" or hang in the air longer than expected.
