Problem 12: In this problem, the symbols M, L, and T represent the dimensions mass, length, and time, respectively. Consider the physical quantities V, ρ, and t with dimensions [V] = L3, [ρ] = ML-3, and [t] = T. (Here, the square bracket means “the dimensions of” so, for example, [V] represents the dimensions of the quantity V.) Part (a) What are the dimensions of ∫ρdV? dimensions = ______ Part (b) What are the dimensions of dV/dt? dimensions = ______ Part (c) What are the dimensions of ρ(dV/dt)? dimensions = ______

(a) How many significant figures are in the numbers 999.0 and 1000.0? 999.01000.0 (b) If the uncertainty in each number is 10, what is the percent uncertainty in each? (Give your answer to 4 significant figures.) 999.0  %1000.0  % (c) Which is a more meaningful way to express the accuracy of these two numbers, significant figures or percent uncertainties?

3.

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A rectangle has a length of (5.1 ± 0.2) m and a width of (2.3 ± 0.1) m. Calculate: (a) the area and (b) the perimeter of the rectangle, Give the uncertainty in each value.

A rectangle has a length of (2.8 ± 0.2) m and a width of (1.7 ± 0.2) m. Calculate the area and the perimeter of the rectangle, and give the uncertainty in each value. (a) Calculate the area and give its uncertainty. (Enter your answers in m2.)   m2 ±  m2 (b) Calculate the perimeter of the rectangle and give its uncertainty. (Enter your answers in m.)   m ±  m

a) Acceleration is described as space divided by time squared. Determine the dimensions of acceleration and its SI units.b) Kinetic Energy K has dimensions kg m2/s2. It can be written in terms of momentum p and mass m following the equation below. Determine the dimensions of the momentum and its SI units. c) The unit of force is the newton N, where 1 N = 1 kg m/s2. Determine the dimensions and the SI units of the momentum in terms of newtons.d) Newton’s law of universal gravitation is F =Gm1m2/r2 , where F is the magnitude of the gravitational force exerted by one object on another, m1 and m2 are the masses of the objects, and r is the distance between them. Determine the dimensions and the SI units of the proportionality constant G.

An Olympic-size swimming pool is 50 m long, 25 m wide, and 2 m deep. In terms of volume, this pool, when full, holds how many kilogallons of water? Use the following equivalences: 1 gallon = 3.8 L     1 kilogallon = 103 gallon      1 L = 103 mL     for water: 1 mL = 1 cm3      1 m = 100 cm   [Numerical Problem: Just write the numerical answer without units. Express your answer to 2 significant figures in standard notation]

The extraterrestrials in an exoplanet in a distant galaxy have the following units to measure distances.1 dray = 12 elim; 1 toof = 24 dray; 1 chin = 1250 toof. a. Convert 2.4 chin to elim. __________________ elim b. The alien grows vegetables in 16.0 square dray (i.e., dray2) patch. Calculate the area of the patch insquare elim (i.e., elim2).____________ clim2

Consider the physical quantities m, s, v, a, t, and r with dimensions [m] = M, [s] = L, [v] = LT–1, [a] = LT–2, [t] = T, and [r] = L. Assuming each of the following equations is dimensionally consistent, find the dimension of the quantity on the left-hand side of the equation: (a) F = ma; (b) K = 0.5mv2; (c) p = mv; (d) W = mas; (e) L = mvr.