A 2 635-kg sailboat experiences an eastward force of 2 710 N by the ocean tide and a wind force against its sails with magnitude of 5 748 N directed toward the northwest (45° N of W). What is the direction of the resultant acceleration?
71.57 degrees West of North
To find the direction of the resultant acceleration, we need to add the vectors representing the forces acting on the sailboat.
First, let's draw a diagram to visualize the given forces. The force due to the ocean tide is directed eastward, and we can represent it as a vector pointing to the right. The wind force is directed toward the northwest, 45° north of west. We can represent it as a vector pointing in that direction.
Now, let's break down the wind force vector into its components. The magnitude of the wind force is given as 5,748 N. To find the x-component and y-component of the wind force, we can use trigonometry.
The x-component of the wind force is given by:
x = magnitude * cos(angle)
x = 5,748 N * cos(45°)
x = 5,748 N * 0.7071
x ≈ 4,062 N
The y-component of the wind force is given by:
y = magnitude * sin(angle)
y = 5,748 N * sin(45°)
y = 5,748 N * 0.7071
y ≈ 4,062 N
So, we have the x- and y-components of the wind force as approximately 4,062 N each.
Now let's add the forces together. Since forces are vectors, we add them by combining their components separately.
Horizontal force component = Ocean tide force + x-component of wind force
Horizontal force component = 2,710 N + 4,062 N
Horizontal force component = 6,772 N
Vertical force component = y-component of wind force
Vertical force component = 4,062 N
Using these components, we can find the magnitude of the resultant acceleration using Pythagorean theorem:
Resultant acceleration = sqrt((Horizontal force component)^2 + (Vertical force component)^2)
Resultant acceleration = sqrt((6,772 N)^2 + (4,062 N)^2)
Resultant acceleration = sqrt(45,961,184 N^2 + 16,499,844 N^2)
Resultant acceleration ≈ sqrt(62,461,028 N^2)
Resultant acceleration ≈ 7,900 N
To find the direction of the resultant acceleration, we can use inverse trigonometry. The direction can be given as an angle relative to the positive x-axis.
Direction = atan(Vertical force component / Horizontal force component)
Direction = atan(4,062 N / 6,772 N)
Direction ≈ 30°
Therefore, the direction of the resultant acceleration is approximately 30° north of east.