A 435-kg sailboat has an acceleration of 0.80 m/s2 at an angle of 50° north of east. Find the magnitude and direction of the net force that acts on the sailboat
Net force= mass*acceleration
To find the magnitude and direction of the net force acting on the sailboat, we can use Newton's second law of motion, which states that the net force on an object is equal to the product of its mass and acceleration. In this case, the given mass of the sailboat is 435 kg, and the given acceleration is 0.80 m/s².
First, we need to find the horizontal and vertical components of the acceleration. The angle given (50° north of east) indicates that the acceleration has both horizontal and vertical components.
To find the horizontal component, we can use trigonometry. The horizontal component (or the component in the east direction) can be found by multiplying the magnitude of the acceleration (0.80 m/s²) by the cosine of the angle (50°).
Horizontal component of acceleration = 0.80 m/s² * cos(50°)
To find the vertical component, we use the sine of the angle:
Vertical component of acceleration = 0.80 m/s² * sin(50°)
Now, let's calculate these components:
Horizontal component of acceleration = 0.80 m/s² * cos(50°) = 0.80 m/s² * 0.6428 ≈ 0.514 m/s²
Vertical component of acceleration = 0.80 m/s² * sin(50°) = 0.80 m/s² * 0.766 ≈ 0.613 m/s²
Next, we can determine the horizontal and vertical forces acting on the sailboat using the equation:
Force = mass * acceleration
Horizontal component of force = 435 kg * 0.514 m/s²
Vertical component of force = 435 kg * 0.613 m/s²
Now, let's calculate these forces:
Horizontal component of force = 435 kg * 0.514 m/s² ≈ 223.71 N
Vertical component of force = 435 kg * 0.613 m/s² ≈ 266.61 N
Finally, we can use these horizontal and vertical components to find the magnitude and direction of the net force. The net force is simply the vector sum of the horizontal and vertical forces.
Using the Pythagorean theorem, the magnitude of the net force is:
Magnitude of the net force = √[(horizontal component of force)² + (vertical component of force)²]
Magnitude of the net force = √[223.71 N² + 266.61 N²] ≈ √(49952.5041 N²) ≈ 223.72 N
To find the direction of the net force, we can use trigonometry. The angle of the net force (θ) can be found using the inverse tangent of the vertical component divided by the horizontal component:
θ = arctan(vertical component of force / horizontal component of force)
θ = arctan(266.61 N / 223.71 N)
Now, let's calculate this angle:
θ ≈ arctan(1.19) ≈ 50.39°
Therefore, the magnitude of the net force acting on the sailboat is approximately 223.72 N, and the direction (angle) of the net force is approximately 50.39° north of east.