Relate the torque $T$ to the other variables shown.

A vacuum cleaner creates a pressure drop $\Delta p$ across its fan. Relate this pressure drop to the impeller diameter $D$ and width $h$, its rotational speed $\omega$, the air density $\rho$, and the inlet and outlet diameters $d_i$ and $d_o$. Also, find an expression for the power requirement of the fan.

The cables holding a suspension bridge are observed to undergo large vibrations under certain wind conditions. Select a set of variables that influence the periodic force acting on a cable and write a simplified relationship of dimensionless parameters.

Derive an expression for the torque $T$ needed to rotate a disk of diameter $d$ at the angular velocity $\omega$ in a fluid with density $\rho$ and viscosity $\mu$ if the disk is a distance $t$ from a wall. Also, find an expression for the power requirement.

The lift $F_L$ on an airfoil depends on the aircraft speed $V$, the speed of sound $c$, the fluid density, the airfoil chord length $l_c$, the airfoil thickness $t$, and the angle of attack $\alpha$. Find an expression for $F_L$.

The frequency $f$ with which vortices are shed from a cylinder depends on the viscosity, density, velocity, and diameter. Derive an expression for $f$.

The drag force $F_D$ on a golf ball depends on the velocity, the viscosity, the density, the diameter, the dimple depth, the dimple radius, and the concentration $C$ of dimples measured by the number of dimples per unit area. What expression relates $F_D$ to the other variables?

The drag force $F_D$ on the smooth sphere of fig. falling in a liquid depends on the constant sphere speed $V$, the solid density $\rho_s$, the liquid density $\rho$ and viscosity $\mu$, the sphere diameter $D$, and gravity $g$. Find an expression for $F_D$ using (a) the $M-L-T$ system, and (b) the $F-L-T$ system.

The thrust force $T$, in newtons, from an airplane propeller is a function of propeller diameter $D$, air density $\rho$, air viscosity $\mu$, and airplane speed $V$. Determine a suitable set of dimensionless numbers to describe the relationship $T=f(\rho, D, \mu, V)$.

The drag force acting on a ship is considered to be a function of the fluid density $\rho$, viscosity $\mu$, gravity $g$, ship velocity $V$, and a characteristic length $l$. Determine a set of suitable dimensionless numbers to describe the relationship

$$F_D=f(\rho, \mu, g, V, l)$$

The drag force $F_D$ on a sphere depends on the velocity $V$, the viscosity $\mu$, the density $\rho$, the surface roughness height $e$, the freestream fluctuation intensity $I$ (a dimensionless quantity), and the diameter $D$. Find an expression for $F_D$.

The velocity of propagation of ripples in a shallow liquid depends on the liquid depth $h$, gravity $g$, surface tension $\sigma$, and the liquid density $\rho$. Find an expression for the propagation velocity $V$. See Fig.

The flow rate $Q$ in an open channel depends on the hydraulic radius $R$, the cross-sectional area $A$, the wall roughness height $e$, gravity $g$, and the slope $S$. Relate $Q$ to the other variables using (a) the $M-L-T$ system, and (b) the $F-L-T$ system.

Select an appropriate set of variables that influence the drag force $F_D$ on an airfoil and write the final form in terms of dimensionless parameters.