HW1. N = ρv/m =


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1 1 HW1 1. Two spheres are cut from a certain uniform rock. One has radius r=5.0 cm. The mass of the other is eight times greater. Find its radius R. Since mass radius 3 R = r 8 1/3 = 10cm 2. The mass of a copper atom is m = kg, and the density of copper is ρ = 8920kg/m 3. (a) Determine the number of atoms in V = 1cm 3 of copper. N = ρv/m = (b) Visualize the one cubic centimeter as formed by stacking up identical cubes, with one copper atom at the center of each. Determine the volume of each cube. V/N = m 3 (c) Find the edge dimension of each cube, which represents an estimate for the spacing between atoms. ( ) 1/3 V = m N 3. Newton s law of universal gravitation is represented by F = GMm/r 2 where F is the magnitude of the gravitational force exerted by one small object on another, M and m are the masses of the objects, and r is a distance. Force has the SI units kg m/s 2. What are the SI units of the proportionality constant G? [G] = [F]m 2 /kg 2 = m 3 /kg/s 2 4. Kinetic energy KE (Chapter 5) has dimensions kg m 2 /s 2. It can be written in terms of the momentum p (Chapter 6) and mass m as KE = p 2 /2m (a) Determine the proper units for momentum using dimensional analysis. (Use the following as necessary: kg, m, and s.) [p] = kg m/s (b) Given the units of force, write a simple equation relating a constant force F exerted on an object, an interval of time t during which the force is applied, and the resulting momentum of the object, p. Look for F t α p β with yet unknown α, β. Compare the dimensions: kg m/s 2 s α (kg m/s) β Thus, α = 1, β = 1 and Ft p
2 2 5. a) Assume the equation x = At 3 +Bt describes the motion of a particular object, with x having the dimension of length and t having the dimension of time. Determine the dimensions of the constants A and B. (Use the following as necessary: L and T, where L is the unit of length and T is the unit of time.) [A] = L/T 3, [B] = L/T (b) Determine the dimensions of the derivative dx/dt = 3At 2 + B.(Use the following as necessary: L and T, where L is the unit of length and T is the unit of time.) [dx/dt] = L/T 6. A rectangular building lot has a width of 72.5 ft and a length of 110 ft. Determine the area of this lot in square meters ft 2 = (0.305m) 2 = 740.9m 2 7. A solid piece of lead has a mass of g and a volume of 2.72 cm3. From these data, calculate the density of lead in SI units (kilograms per cubic meter). 1 kg/m (0.001 kg) 2.72 (0.01m) 3 = kg/m 3 8. Suppose your hair grows at the rate 1/31 in. per day. Find the rate at which it grows in nanometers per second. Because the distance between atoms in a molecule is on the order of 0.1 nm, your answer suggests how rapidly layers of atoms are assembled in this protein synthesis 1 in 31 day = nm s = 9.48nm/s 9. Find the order of magnitude of the number of tabletennis balls that would fit into a typicalsize room (without being crushed). (Assume that the dimensions of the room are 4 m by 4 m by 3 m.) 10. a) Compute the order of magnitude of the mass of a bathtub half full of water. (Assume the tub measures 1.3 m by 0.5 m by 0.3 m.) 10 6 m = ρv 10 2 kg (b) Compute the order of magnitude of the mass of a bathtub half full of pennies. (Assume the pennies are made entirely of copper.) m = ρ Cu V 10 3 kg
3 11. A surveyor measures the distance across a straight river by the following method. Starting directly across from a tree on the opposite bank, she walks d = 92 m along the riverbank to establish a baseline. Then she sights across to the tree. The angle from her baseline to the tree is θ = How wide is the river? d tanθ = 55.28m 12. The displacement vectors and shown in the figure below both have magnitudes of L=1.58 m. The direction of vector A is θ =38.8. y B 3 A Θ x a) Find C = A + B First, find components A = (L cos θ, L sin θ), B = (0, L) Then, C x = A x + B x = L cos θ = 1.23, C y = A y + B y = L sin θ + L = 2.57 magnitude Cx 2 + Cy 2 = 2.85 direction arctan(c y /C x ) = 64.4 o (b) Find C = A B C x = A x B x = L cos θ = 1.23 > 0, C y = A y B y = L sin θ L = 0.59 < 0 magnitude Cx 2 + Cy 2 = 1.37 direction arctan(c y /C x ) = (c) Find C = A + B C x = A x + B x = L cos θ < 0, C y = A y + B y = L sin θ + L > 0
4 4 magnitude C 2 x + C 2 y = 1.37 direction arctan(c y /C x ) = o Note: arctan is confined between ±90 o while angle is measured from 0 to 360. Hence the above corrections  need to look at the picture. 13. The polar coordinates of a point are r = 6.00 m and θ = 250. What are the Cartesian coordinates of this point? r = (x,y), x = r cos θ = 2.05m, y = r sin θ = 5.64m 14. Vector A has a magnitude of A=26 units and points in the positive ydirection. When vector B is added to A, the resultant vector C = A + B points in the negative ydirection with a magnitude of C=16 units. Find the magnitude of B? First, write in components A = (0,A), C = (0, C) Then and B = C A = (0 0, C A) = (0, C A) B = A + C = A vector r has an x component of units and a y component of 39.8 units. Find the magnitude and direction of this vector. magnitude r = x 2 + y 2 = direction θ = arctan(y/x) o = o counterclockwise from the +x axis 16. Use the component method to add the vectors and shown in the figure. The length of B is 3.55 m and the angle θ = Length of A is 3.0 m. Express the resultant in unitvector notation. y B A Θ x
5 5 One has A = A cos θ i + A sin θ j = i j, B = B j = 3.55 j Then A + B = A cos θ i + A sin θ j + B j = A cos θ i + (A sin θ + B) j = 2.51 i j 17. Consider the two vectors A = i 3 j and B = i 4 j. (a) Calculate A + B = 0 i 7 j (b) Calculate (c) Calculate (d) Calculate A B = 2 i + j A + B = 7 A B = 5 (e) Calculate the directions of + and . A + B : 270 o (counterclockwise from the +x axis) A B : 26.6 o (counterclockwise from the +x axis)
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