July 2011

Change of Scale Property
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   then,
 

Problem 01
Find the Laplace transform of   $g(t) = \begin{cases} f(t - 2)^3 & t \gt 2 \\ 0 & t \lt 2 \end{cases}$
 

Problem 01
Find the Laplace transform of   $g(t) = \begin{cases} f(t - 1)^2 & t \gt 1 \\ 0 & t \lt 1 \end{cases}$
 

Second Shifting Property
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   and   $g(t)
= \begin{cases} f(t - a) & t \gt a \\ 0 & t \lt a \end{cases}$
 

then,

Problem 361
Referring to Problem 359, if T = 30 kN and x = 1 m, determine the angle θ at which the bar will be inclined to the horizontal when it is in a position of equilibrium.
 

Problem 360
Referring to Problem 359, what value of T acting at x = 1 m from B will keep the bar horizontal.
 

Problem 359
A 4-m bar of negligible weight rests in a horizontal position on the smooth planes shown in Fig. P-359. Compute the distance x at which load T = 10 kN should be placed from point B to keep the bar horizontal.
 

Problem 358
A bar AE is in equilibrium under the action of the five forces shown in Fig. P-358. Determine P, R, and T.
 

Problem 357
The uniform rod in Fig. P-357 weighs 420 lb and has its center of gravity at G. Determine the tension in the cable and the reactions at the smooth surfaces at A and B.