A 300 kg sailboat accelerates at 0.5 m/s^2 at an angle of 25degrees North of East. Find the magnitude and direction of the force responsible for this accelereation.
W.k.t f=ma.and then use resultant and direction formula.
To find the magnitude and direction of the force responsible for the acceleration of the sailboat, we can break down the given information and use Newton's second law of motion.
Newton's second law of motion states that the net force acting on an object is equal to the product of its mass and acceleration (F = ma). In this case, the net force is the force responsible for the acceleration of the sailboat.
Given:
Mass of the sailboat (m) = 300 kg
Acceleration of the sailboat (a) = 0.5 m/s^2
Angle of acceleration (θ) = 25 degrees North of East
Step 1: Resolve the acceleration into its components.
The angle of acceleration is given as 25 degrees North of East. This means that we need to break down the acceleration into its East (horizontal) and North (vertical) components.
Vertical Component (a_North) = a * sin(θ)
a_North = 0.5 m/s^2 * sin(25 degrees)
Horizontal Component (a_East) = a * cos(θ)
a_East = 0.5 m/s^2 * cos(25 degrees)
Step 2: Calculate the force in each component.
Using Newton's second law, we can now calculate the force in each component.
Force in the North direction (F_North) = mass * acceleration_North
F_North = m * a_North
Force in the East direction (F_East) = mass * acceleration_East
F_East = m * a_East
Step 3: Determine the magnitude and direction of the force.
Using the Pythagorean theorem, we can find the magnitude of the force:
Magnitude of the force (F) = √(F_North^2 + F_East^2)
To find the direction of the force, we can use trigonometric functions:
Direction of the force (θ_Force) = arctan(F_North / F_East)
Now, let's substitute the given values and solve the equations:
a_North = 0.5 m/s^2 * sin(25 degrees) ≈ 0.21 m/s^2
a_East = 0.5 m/s^2 * cos(25 degrees) ≈ 0.45 m/s^2
F_North = 300 kg * 0.21 m/s^2 ≈ 63 N
F_East = 300 kg * 0.45 m/s^2 ≈ 135 N
F = √(63 N^2 + 135 N^2) ≈ 148.5 N
θ_Force = arctan(63 N / 135 N) ≈ 25.62 degrees
Therefore, the magnitude of the force responsible for the acceleration of the sailboat is approximately 148.5 N, and its direction is about 25.62 degrees North of East.