June 2011

Problem 325
Determine the amount and direction of the smallest force P required to start the wheel in Fig. P-325 over the block. What is the reaction at the block?
 

Problem 327
Forces P and F acting along the bars shown in Fig. P-327 maintain equilibrium of pin A. Determine the values of P and F.
 

Problem 329
Two cylinders A and B, weighing 100 lb and 200 lb respectively, are connected by a rigid rod curved parallel to the smooth cylindrical surface shown in Fig. P-329. Determine the angles α and β that define the position of equilibrium.
 

Problem 333
Determine the reactions R₁ and R₂ of the beam in Fig. P-333 loaded with a concentrated load of 1600 lb and a load varying from zero to an intensity of 400 lb per ft.
 

Problem 335
The roof truss in Fig. P-335 is supported by a roller at A and a hinge at B. Find the values of the reactions.
 

Problem 337
The upper beam in Fig. P-337 is supported at D and a roller at C which separates the upper and lower beams. Determine the values of the reactions at A, B, C, and D. Neglect the weight of the beams.
 

Properties of Logarithm

1. If   $y = a^x$,   then   $\log_a y = x$.   ← Definition of logarithm
2. $\log_a xy = \log_a x + \log_a y$
3. $\log_a \dfrac{x}{y} = \log_a x - \log_a y$
4. $\log_a x^n = n \log_a x$
5. $\log_a a = 1$
6. $\log_a 1 = 0$
7. $\log_{10} x = \log x$   ←   Common logarithm
8. $\log_e x = \ln x$   ←   Naperian or natural logarithm
9. $\log_y x = \dfrac{\log x}{\log y} = \dfrac{\ln x}{\ln y}$   ←   Change base rule
10. If   $\log_a x = \log_a y$,   then   $x = y$.
11. If   $\log_a x = y$,   then   $x = {\rm antilog}_a \, y$.

The Expansion of (a + b)ⁿ
If   $n$   is any positive integer, then

$(a + b)^n = a^n + {_nC_1}a^{n - 1}b + {_nC_2}a^{n - 2}b^2 + \, \cdots \, + {_nC_m}a^{n - m}b^m + \, \cdots \, + b^n$
 

Where
${_nC_m}$ = combination of n objects taken m at a time.