1. Given mv = Ft, where m is mass, v is speed, F is force, and t is time, what are the dimensions of each side of the equation? Is the equation dimensionally correct? [left side] = M·L/T
[right side] = M·L/T
Therefore the equation is dimensionally correct.
2. Given H = mCDT, where H is in joules, m in kilograms, and DT in kelvin, what are the SI units and dimensions of C? Since C = H/(mDT), the SI units are J·kg-1·K-1.
[C] = (M·L2·T-2)·M-1·q-1 = L2·T-2·q-1
3. Given P = kADT/l, where A is the area, DT is difference in temperature, l is length, and k is a constant with SI units of watts pr (metre·kelvin), what are the SI units for P (rate of thermal energy flow)?  Recall that watt (W) is joules per second, so [k] = M·L·T-3·q-1.
[A] = L2, [DT] = q , and [l] = L
[right side] = M·L2/T3
Therefore, [P] = M·L2/T3, and SI units are  kg·m2/s3, or J/s.
4. Given E = a  sin (bt), where E is energy,  is length and t is time: (a) What are the dimensions and SI units of b?  [b] = T-1
Remember that the argument of the sine function must be dimensionless. Since the argument in this case is an unknown (b) multiplied by time (t), then b must have dimensions of inverse time.  The SI units of "b" are s-1.
(b) What are the dimensions and SI units of a?  [a] = [E/] = M·L/T2, since sine is dimensionless.  The SI units of "are" are kg·m/s2, or newton.
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