The forces acting on a sailboat are 390 N north and 180 N east. If the boat (including crew) has a mass of 270 kg, what are the magnitude and direction of its acceleration?
Compute the resultant force by vector addition and divide it by mass for the acceleration vector. The vector addition is easy since the two forces are at right angles. Remember Pythagoras?
awdawd
To compute the resultant force acting on the sailboat, we can use the Pythagorean theorem. Since the two forces are at right angles, we can calculate the magnitude of the resultant force using the formula:
Resultant force = √(force north)^2 + (force east)^2
Plugging in the values given, we have:
Resultant force = √(390 N)^2 + (180 N)^2
= √(152100 N^2) + (32400 N^2)
= √(184500 N^2)
≈ 429.37 N
Now, to find the acceleration of the sailboat, we need to divide the magnitude of the resultant force by the mass of the boat:
Acceleration = Resultant force / Mass
= 429.37 N / 270 kg
≈ 1.59 m/s^2
Therefore, the magnitude of the sailboat's acceleration is approximately 1.59 m/s^2.
To determine the direction of the acceleration, we can use trigonometry. Since the force north and force east are perpendicular, the angle of the resultant force with the positive x-axis can be found using:
θ = arctan(force north / force east)
Plugging in the values given, we have:
θ = arctan(390 N / 180 N)
≈ 63.64 degrees
Since the forces are acting north and east, the resultant force is in the northeast direction. Therefore, the direction of the acceleration is 63.64 degrees relative to the positive x-axis, in the northeast direction.
To find the magnitude and direction of the boat's acceleration, we first need to calculate the resultant force by adding the given forces together. Since the two forces are acting at right angles, we can use the Pythagorean theorem to calculate their resultant magnitude.
Let's start with the given forces:
- Force 1: 390 N north
- Force 2: 180 N east
Now, let's break down Force 1 into its north and east components:
- Force 1 north component: 390 N * sin(90°) = 0 N (since the force is acting directly north, there is no east component)
- Force 1 east component: 390 N * cos(90°) = 390 N
Next, let's break down Force 2 into its north and east components:
- Force 2 north component: 180 N * sin(0°) = 0 N (since the force is acting directly east, there is no north component)
- Force 2 east component: 180 N * cos(0°) = 180 N
Now we can add the east components and north components separately:
- East component: 390 N + 180 N = 570 N (directed east)
- North component: 0 N + 0 N = 0 N (there are no north components)
Using the Pythagorean theorem, we can find the magnitude of the resultant force:
Resultant magnitude = sqrt((East component)^2 + (North component)^2)
= sqrt((570 N)^2 + (0 N)^2)
= sqrt(324,900 N^2)
≈ 569.36 N
Thus, the magnitude of the resultant force is approximately 569.36 N.
To find the direction of the resultant force, we can use the trigonometric inverse tangent function (tan^(-1)) to find the angle between the resultant force vector and the east direction. Since the north component is 0 N, the angle will be 90° to the east.
Therefore, the direction of the resultant force is 90° east.
Finally, to find the acceleration, we divide the magnitude of the resultant force by the mass of the sailboat:
Acceleration = Resultant magnitude / Mass
= 569.36 N / 270 kg
≈ 2.11 m/s^2
Thus, the magnitude of the boat's acceleration is approximately 2.11 m/s^2, and its direction is 90° east.