Simple Chemical Conversion

From the results of chemical experimentation of substance converted into another substance, it was found that the rate of change of unconverted substance is proportional to the amount of unconverted substance.

If x is the amount of unconverted substance, then

$\dfrac{dx}{dt} = -kx$

with a condition that x = xo when t = 0.

$\dfrac{dx}{dt} = -kx$

$\dfrac{dx}{x} = -k \, dt$

$\ln x = -kt + \ln C$

$\ln x = \ln e^{-kt} + \ln C$

$\ln x = \ln Ce^{-kt}$

Newton's Law of Cooling

Newton's Law of Cooling states that the temperature of a body changes at a rate proportional to the difference in temperature between its own temperature and the temperature of its surroundings.

We can therefore write

$\dfrac{dT}{dt} = -k(T - T_s)$

T = temperature of the body at any time, t
Ts = temperature of the surroundings (also called ambient temperature)
To = initial temperature of the body
k = constant of proportionality

Problem 1010 and Problem 1011 | Investigation of timber reinforced by two steel channels

Problem 1010
A pair of C250 × 30 steel channels are securely bolted to wood beam 200 mm by 254 mm, as shown in Fig. P-1010. From Table B-2 in Appendix B, the depth of the channel is also 254 mm.) If bending occurs about the axis 1-1, determine the safe resisting moment if the allowable stresses σs = 120 MPa and σw = 8 MPa. Assume n = 20.



Problem 1011
In Problem 1010, determine the safe resisting moment if bending occurs about axis 2-2.

Problem 1009 | Width of aluminum plate reinforcement for the wood section to resist 14 kN-m moment

Problem 1009
A timber beam 150 mm wide by 200 mm deep is to be reinforced at the top and bottom by aluminum plates 6 mm thick. Determine the width of the aluminum plates if the beam is to resist a moment of 14 kN·m. Assume n = 5 and take the allowable stresses as 10 MPa and 80 MPa in the wood and aluminum, respectively.