PHYS101 Vectors Spring 2014

Transcription

1 Vectors Before we start with the tutorials, we should state the following summar for the calculation of the angles (direction) Let us consider the following situations In general we can summarise If 1 > 0 and 1 > 0, then θ 1 = θ calculator = tan 1 > 0. If 2 < 0 and 2 > 0, then θ calculator = tan 1 < 0. To get the correct angle of the vector against the positive -ais we calculate θ 2 = θ calculator If 3 < 0 and 3 < 0, then θ calculator = tan 1 > 0. To get the correct angle of the vector against the positive -ais we calculate θ 3 = θ calculator If 4 > 0 and 4 < 0, then θ calculator = tan 1 < 0. To get the correct angle of the vector against the positive -ais we calculate θ 4 = θ calculator c 2014 Department of Phsics, Eastern Mediterranean Universit Page 1 of 5

2 Tutorial Questions 1. What are the component and the component of a vector a in the plane if its direction is 250 counterclockwise from the positive direction of the ais and its magnitude is 7.3m? Solutions (a) The component of a is given b a = a cos(θ) = (7.3 m) cos 250 = 2.50 m (b) Similarl, the component is given b a = a sin(θ) = (7.3m) sin 250 = 6.86 m 2. The component of vector A is 25.0 m and the component is m. (a) What is the magnitude of A? (b) What is the angle between the direction of A and the positive direction of? Solutions (a) The magnitude of A is A = A 2 + A 2 = ( 25.0 m) 2 + (40.0 m) 2 = 47.2 m (b) The angle between the direction of A and the positive direction of is 3. Two vectors are given b In unit vector notation find (a) a + b (b) a b θ = tan 1 A = tan m A 25.0 m = 122 a = (4.0 m)î (3.0 m)ĵ + (1.0 m) ˆk and b = ( 1.0 m)î + (1.0 m)ĵ + (4.0 m) ˆk. (c) a third vector c such that a b + c = 0. Solutions c 2014 Department of Phsics, Eastern Mediterranean Universit Page 2 of 5

3 (a) (b) a + b = (4.0 m 1.0 m) î + ( 3.0 m m) ĵ + (1.0 m m) ˆk = (3.0 î 2.0 ĵ ĵ) m a b = (4.0 m m) î + ( 3.0 m 1.0 m) ĵ + (1.0 m 4.0 m) ˆk = (5.0 î 4.0 ĵ 3.0 ˆk) m (c) The requirement a b + c = 0 leads to c = b a which we note is the opposite of what we found in part (b). Thus, c = ( 5.0 î ĵ ˆk) m. 4. Given are the vectors a = (4.0 m)î + (3.0 m)ĵ, and b = ( 13.0 m)î + (7.0 m)ĵ (a) What is the sum a + b? (b) What is the magnitude of a + b? (c) What is the direction of a + b? Solution (a) The sum a + b = 9.0 m î + 10 m ĵ (b) The magnitude of a + b = ( 9.0 m) 2 + (10 m) 2 = 13 m (c) The direction of a + b is θ = tan 1 10 m 9.0 m = Note The calculator displas 48 as the range of the tan 1 function is between 90 and 90. As the actual angle θ > 90, we can get the real angle as θ = = 132. c 2014 Department of Phsics, Eastern Mediterranean Universit Page 3 of 5

4 5. A car is driven east for a distance of 50 km, then north for 30 km, and then in a direction 30 east of north for 25 km. Sketch the vector diagram and determine (a) the magnitude and (b) the angle of the car s total displacement from its starting point. Solutions (a) Given is A = 50 î km, B = 30 î km, and C = 25 km cos(60 ) î + 25 km sin(60 ) ĵ The total displacement r of the car from its initial position is then r has the magnitude r = A + B + C = 63 km î + 52 km ĵ. r = (63 km) 2 + (52 km) 2 = 82 km (b) The direction of the displacement vector r is then ( ) 51.7 km tan 1 = km hallida_c03_ hr.qd Page 55 against the horizontal. 6. The two vectors a and b in the figure below have equal magnitudes of 10.0 m and the angles are θ 1 = 30 and θ 2 = Find the (a) and (b) components of their vector sum r, (c) the magnitude of r, and (d) the angle r makes with the positive direc- tion of the ais. 16 For the displacement vectors a (3.0 m)î (4.0 m)ĵ and b (5.0 m)î ( 2.0 m)ĵ,give a b in a (a) unit-vector notation, and as (b) a θ 1 magnitude and (c) an angle (relative to ). Now give b O î a in (d) Fig Problem 15. unit-vector notation, and as (e) a magnitude and (f) an angle. 17 ILW Three vectors a, b, and c each have a magnitude of 50 m and lie in an plane. Their directions relative to the positive direction of the ais are 30, 195, and 315, respectivel.what are b θ 2 PROBLEMS 25 Oasis B is 25 km due east of oasis A. Starting fro A, a camel walks 24km in a direction 15 south of east an walks 8.0 km due north. How far is the camel then from oasi 26 What is the sum of the following four vectors in (a) u tor notation, and as (b) a magnitude and (c) an angle? A (2.00 m)î (3.00 m)ĵ B 4.00 m, at 65.0 C ( 4.00 m)î ( 6.00 m)ĵ D 5.00 m, at If d1 d 2 5d 3, d 1 d 2 3d 3, and 3 2î 4 what are, in unit-vector notation, (a) d1 and (b) d2? 28 Two beetles run across flat sand, starting at the same Beetle 1 runs 0.50 m due east, then 0.80 m at 30 north of d Beetle 2 also makes two runs; the first is 1.6 m at 40 east north. What must be (a) the magnitude and (b) the directio second run if it is to end up at the new location of beetle 1? c 2014 Department of Phsics, Eastern Mediterranean Universit Page 4 of 5 d

5 If r is the sum r = a + b. Find the (a) the component of r, (b) the component of r, (c) the magnitude of r, (d) the angle r makes with the positive direction of the -ais. Solutions (a) r = 10.0 m cos(30 ) m cos( ) = 1.59 m (b) r = 10.0 m sin(30 ) m sin( ) = 12.1 m (c) r = (1.59 m) 2 + (12.1 m) 2 = 12.2 m (d) θ = tan 1 ( 12.1 m 1.59 m ) = 82.5 c 2014 Department of Phsics, Eastern Mediterranean Universit Page 5 of 5