Two vectors P and Q of magnitude 5N and 3N respectively are inclined at an angle of 30 degree to each other. Calculate the components of the resultant (i) in the driection of Q (ii) perpendicular to the direction of Q (iii) Hence, calculate the magnitude of the resultant and the angle it makes with the direction of Q.
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Two vectors P and Q f magnitude 5N and 3N respectively are inclined at an angle of 30° to each other. Calculate the component of the resultant.
To calculate the components of the resultant vector, we can use trigonometry. Here's how:
Step 1: Find the horizontal components of vector P and Q.
The horizontal component of vector P is given by P * cos(angle), where angle is the angle between vector P and the x-axis (30 degrees in this case).
Horizontal component of P = 5N * cos(30°) = 5N * √(3)/2 = (15/2)N.
The horizontal component of vector Q is given by Q * cos(angle), where angle is the angle between vector Q and the x-axis (30 degrees in this case).
Horizontal component of Q = 3N * cos(30°) = 3N * √(3)/2 = (9/2)N.
(i) In the direction of Q:
The component of the resultant vector in the direction of Q is equal to the sum of the horizontal components of P and Q.
Resultant in the direction of Q = Horizontal component of P + Horizontal component of Q
Resultant in the direction of Q = (15/2)N + (9/2)N = (24/2)N = 12N.
Step 2: Find the vertical components of vector P and Q.
The vertical component of vector P is given by P * sin(angle), where angle is the angle between vector P and the x-axis (30 degrees in this case).
Vertical component of P = 5N * sin(30°) = 5N * 1/2 = 5/2 N.
The vertical component of vector Q is given by Q * sin(angle), where angle is the angle between vector Q and the x-axis (30 degrees in this case).
Vertical component of Q = 3N * sin(30°) = 3N * 1/2 = 3/2 N.
(ii) Perpendicular to the direction of Q:
The component of the resultant vector perpendicular to the direction of Q is equal to the difference of the vertical components of P and Q.
Resultant perpendicular to the direction of Q = Vertical component of P - Vertical component of Q
Resultant perpendicular to the direction of Q = (5/2)N - (3/2)N = (2/2)N = 1N.
Step 3: Calculate the magnitude of the resultant vector:
The magnitude of the resultant vector is given by the square root of the sum of the squares of its components.
Magnitude of the resultant = √[(Resultant in the direction of Q)^2 + (Resultant perpendicular to the direction of Q)^2]
Magnitude of the resultant = √[(12N)^2 + (1N)^2] = √[144N^2 + 1N^2] = √145N^2 = 12.04N (approximately).
Step 4: Calculate the angle between the resultant vector and the direction of Q:
The angle can be calculated using the inverse tangent function.
Angle = atan(Resultant perpendicular to the direction of Q / Resultant in the direction of Q)
Angle = atan(1N / 12N) = atan(1/12) = 4.76 degrees (approximately).
Therefore, the magnitude of the resultant vector is approximately 12.04N, and it makes an angle of 4.76 degrees with the direction of Q.
To calculate the components of the resultant vector, we can use trigonometry. Let's break down the problem step-by-step:
Step 1: Calculate the components of vector P in the x and y direction.
- The magnitude of vector P is 5 N.
- Since the vectors are inclined at an angle of 30 degrees to each other, the x-component of vector P can be calculated using cosine:
Px = P * cos(30°)
Px = 5 N * cos(30°)
Px = 5 N * (√3/2)
Px = 5√3/2 N
- The y-component of vector P can be calculated using sine:
Py = P * sin(30°)
Py = 5 N * sin(30°)
Py = 5 N * (1/2)
Py = 5/2 N
Step 2: Calculate the components of vector Q in the x and y direction.
- The magnitude of vector Q is 3 N.
- Since the vectors are inclined at an angle of 30 degrees to each other, the x-component of vector Q can be calculated using cosine:
Qx = Q * cos(30°)
Qx = 3 N * cos(30°)
Qx = 3 N * (√3/2)
Qx = 3√3/2 N
- The y-component of vector Q can be calculated using sine:
Qy = Q * sin(30°)
Qy = 3 N * sin(30°)
Qy = 3 N * (1/2)
Qy = 3/2 N
Step 3: Calculate the components of the resultant vector.
(i) Component in the direction of Q:
The x-component of the resultant vector in the direction of Q can be obtained by adding the x-components of P and Q:
Resultant_x = Px + Qx
Resultant_x = 5√3/2 N + 3√3/2 N
Resultant_x = (5√3 + 3√3)/2 N
Resultant_x = 8√3/2 N
The y-component of the resultant vector in the direction of Q can be obtained by adding the y-components of P and Q:
Resultant_y = Py + Qy
Resultant_y = 5/2 N + 3/2 N
Resultant_y = 8/2 N
Resultant_y = 4 N
(ii) Component perpendicular to the direction of Q:
The x-component of the resultant vector perpendicular to the direction of Q can be obtained by subtracting the x-component of Q from the x-component of P:
Resultant_perpendicular_x = Px - Qx
Resultant_perpendicular_x = 5√3/2 N - 3√3/2 N
Resultant_perpendicular_x = (5√3 - 3√3)/2 N
Resultant_perpendicular_x = 2√3/2 N
Resultant_perpendicular_x = √3 N
The y-component of the resultant vector perpendicular to the direction of Q remains the same as Qy:
Resultant_perpendicular_y = Qy
Resultant_perpendicular_y = 3/2 N
Step 4: Calculate the magnitude of the resultant vector (R) and the angle it makes with the direction of Q.
- The magnitude of a vector (R) can be calculated using the Pythagorean theorem:
R^2 = (Resultant_x)^2 + (Resultant_y)^2
R^2 = (8√3/2 N)^2 + (4 N)^2
R^2 = (4√3 N)^2 + (4 N)^2
R^2 = 16 * 3 N^2 + 16 N^2
R^2 = 48 N^2 + 16 N^2
R^2 = 64 N^2
R = √(64 N^2)
R = 8 N
- The angle (θ) that the resultant vector makes with the direction of Q can be found using tangent:
tan(θ) = (Resultant_perpendicular_x) / (Resultant_x)
tan(θ) = (√3 N) / (8√3/2 N)
tan(θ) = (√3 N) / (4√3 N)
tan(θ) = (1/√3)
θ = tan^(-1)(1/√3)
θ ≈ 30 degrees
So, the components of the resultant vector are:
(i) In the direction of Q: 8√3/2 N in the x-direction and 4 N in the y-direction.
(ii) Perpendicular to the direction of Q: √3 N in the x-direction and 3/2 N in the y-direction.
The magnitude of the resultant vector is 8 N, and it makes an angle of approximately 30 degrees with the direction of Q.