The force exerted by the wind on the sails of a sailboat is 370 N north. The water exerts a force of 210 N east. If the boat (including its crew) has a mass of 270 kg, what are the magnitude and direction of its acceleration?
(Acceleration vector) = Resultant force)/mass
The resultant force on the boat is 424.4 Newtons
To find the magnitude and direction of the sailboat's acceleration, we need to use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration.
Given:
- Force exerted by the wind on the sails (Fwind) = 370 N north
- Force exerted by the water (Fwater) = 210 N east
- Mass of the boat (m) = 270 kg
First, let's find the net force acting on the boat. Since force is a vector quantity, we'll need to resolve the forces into their horizontal and vertical components.
The wind force has a magnitude of 370 N and acts north. Therefore, its horizontal component is 0 N (since it acts in the north-south direction) and its vertical component is 370 N.
The water force has a magnitude of 210 N and acts east. Therefore, its horizontal component is 210 N and its vertical component is 0 N (since it acts in the east-west direction).
Now, let's find the net horizontal and vertical forces by adding up the respective components:
Horizontal force (Fhorizontal) = Fwater (horizontal) = 210 N east
Vertical force (Fvertical) = Fwind (vertical) = 370 N north
Next, we can calculate the net force using the Pythagorean theorem:
Net force (Fnet) = sqrt(Fhorizontal^2 + Fvertical^2)
Fnet = sqrt((210 N)^2 + (370 N)^2)
Fnet = sqrt(44100 N^2 + 136900 N^2)
Fnet = sqrt(181000 N^2)
Fnet ≈ 425.36 N
Now that we have the net force, we can find the acceleration using Newton's second law:
Fnet = m * a
425.36 N = 270 kg * a
a = 425.36 N / 270 kg
a ≈ 1.57 m/s²
So, the magnitude of the sailboat's acceleration is approximately 1.57 m/s².
To determine the direction of the acceleration, we need to consider the signs of the forces involved. The wind force acts north (positive direction), and the water force acts east (positive direction).
Since both forces are positive, the resulting acceleration will have a positive direction, which is north-east.