Chapter 5, Problem 5.33P

(a)

Interpretation Introduction

Interpretation:

The power transmitted to the cart is shown to be maximum when $\frac{v_c}{v}=\frac{1}{3}$ .

Concept Introduction:

The general equation written for the overall linear momentum balance for a controlled volume is:

  ${\displaystyle \sum F}={\displaystyle {\iint }_{C.S}v\rho \left(v\cdot n\right)dA}+\frac{\partial }{\partial t}{\displaystyle {\iiint }_{C.V}\rho vdV}$......... (1)

Here, C.S is the control surface over which the integral dA is taken, C.V is the control volume over which the integral dV is taken, $\rho$ is the density of the fluid, $v$ is the velocity vector, $n$ is the direction of the vector $v$ , The product $v\cdot n$ is scalar defined as:

  $v\cdot n=\left|v\right|\left|n\right|\cos \theta$

Power required per unit time is defined in terms of force ( F ) and linear speed $\left(v\right)$ is:

  $P=F\cdot v$......... (2)

(b)

Interpretation Introduction

Interpretation:

The power transmitted to the cart is shown to be maximum when $\frac{v_c}{v}=\frac{1}{2}$ .

Concept Introduction:

Torque is defined as the turning effect due to the rotational force being applied on an object. It is also known as moment or moment of forces.

The general equation written for the total moment of all forces acting on a controlled volume is:

  ${\displaystyle \sum M}={\displaystyle {\iint }_{C.S}\left(r\times v\right)\rho \left(v\cdot n\right)dA}+\frac{\partial }{\partial t}{\displaystyle {\iiint }_{C.V}\left(r\times v\right)\rho dV}$......... (3)

Here, C.S is the control surface over which the integral dA is taken, C.V is the control volume over which the integral dV is taken, $\rho$ is the density of the fluid, $v$ is the velocity vector, $n$ is the direction of the vector $v$ , $r$ is the position vector, $v\cdot n$ is a dot product and $\left(r\times v\right)$ is a cross product.

Power required per unit time is defined in terms of moment of force ( M ) and angular speed $\left(\omega \right)$ is:

  $P=M\cdot \omega$......... (4)