(Q1) 2 forces act on a point object as follows: 100 at 170 and 100 at 50.Find the resultant force (a)110N at 50 (b)110N at 100 (c)100N at 110 (d) 100N at 50 (Q2) given 3 vectors a=-i-4j+2k,b=3i+2j-2k,c=2i-3j+k,Calculate a-(b-c) (a)-6(b)6(d)9(d)-9 (Q3)The resultant of vectors A and B has a magnitude of 20 units.A has a magnitude of 8 units,and the angle between A andB is 40. Calculate the magnitude of B. (a)12.6(b)16.2(c)14.8(d)18.4
F1 = 100[170o]
F2 = 100[50o]
X = 100*cos170+100*cos50o = -34.2 N.
Y = 100*sin170+100*sin50 = 94 N.
tan Ar = Y/X = 94/-34.2 = -2.74838
Ar = -70o = Reference angle.
A = -70+180 = 110o
Fr = X/cosA = -34.2/cos110 = 100N.[110o]
(Q1) To find the resultant force of two forces acting on a point object, you need to use vector addition.
Given the magnitudes and angles of the forces:
Force 1: 100 N at 170°
Force 2: 100 N at 50°
Step 1: Convert each force into vector form using their magnitudes and angles:
Force 1 vector: 100N * cos(170°) i + 100N * sin(170°) j
Force 2 vector: 100N * cos(50°) i + 100N * sin(50°) j
Step 2: Add the two vectors together to get the resultant force vector:
Resultant force vector = Force 1 vector + Force 2 vector
Step 3: Calculate the magnitude and angle of the resultant force vector:
Magnitude of resultant force = sqrt((x-component of resultant force)^2 + (y-component of resultant force)^2)
Angle of resultant force = arctan((y-component of resultant force) / (x-component of resultant force))
Step 4: Compare the magnitude and angle of the resultant force with the given options to find the correct answer.
(Q2) To calculate a-(b-c), you need to perform vector subtraction.
Given the vectors:
a = -i - 4j + 2k
b = 3i + 2j - 2k
c = 2i - 3j + k
Step 1: Negate vector c by changing the sign of its components:
-c = -2i + 3j - k
Step 2: Subtract b-c:
a - (b-c) = a - b + c
Step 3: Perform vector addition with the corresponding components:
x-component: -1 - 3 + 2 = -2
y-component: -4 - 2 + 3 = -3
z-component: 2 + 2 - (-1) = 5
The resulting vector is -2i - 3j + 5k.
Step 4: Compare the resulting vector with the given options to find the correct answer.
(Q3) To calculate the magnitude of vector B, given the magnitude and angle between vectors A and B, you can use the cosine law.
Given:
Magnitude of A (|A|) = 8 units
Magnitude of resultant (|R|) = 20 units
Angle between A and B (θ) = 40°
Step 1: Apply the cosine law: |R|^2 = |A|^2 + |B|^2 - 2 * |A| * |B| * cos(θ)
Step 2: Substitute the known values and solve for |B|:
|B|^2 = |R|^2 - |A|^2 + 2 * |A| * |B| * cos(θ)
|B|^2 - 2 * |A| * |B| * cos(θ) = |R|^2 - |A|^2
|B|^2 * (1 - 2 * |A| * cos(θ)) = |R|^2 - |A|^2
|B|^2 = (|R|^2 - |A|^2) / (1 - 2 * |A| * cos(θ))
|B| = sqrt((|R|^2 - |A|^2) / (1 - 2 * |A| * cos(θ)))
Step 3: Substitute the values and calculate the magnitude of B:
|B| = sqrt((20^2 - 8^2) / (1 - 2 * 8 * cos(40°)))
Step 4: Evaluate the expression to find the magnitude of B and compare it with the given options to find the correct answer.