Find the magnitude of the force needed to accelerate a 150 g mass with a = -0.250 m/s2 i^+0.675 m/s2 j^. Find the direction of this force?
Need to find F and theta.
To find the magnitude of the force, we can use Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object times its acceleration:
F = m * a
Given:
Mass (m) = 150 g = 0.150 kg
Acceleration (a) = -0.250 m/s^2 i + 0.675 m/s^2 j
Substituting the values into the equation, we have:
F = 0.150 kg * (-0.250 m/s^2 i + 0.675 m/s^2 j)
To calculate the magnitude of the force, we need to find the square root of the sum of the squares of its components:
F = √((-0.150 kg * -0.25 m/s^2)^2 + (0.150 kg * 0.675 m/s^2)^2)
F = √((0.0375 kg·m/s^2)^2 + (0.10125 kg·m/s^2)^2)
F = √(0.00140625 kg^2·m^2/s^4 + 0.0102565625 kg^2·m^2/s^4)
F = √0.0116628125 kg^2·m^2/s^4
F = 0.108 kg·m/s^2
Therefore, the magnitude of the force needed to accelerate the 150 g mass is 0.108 N.
Now let's find the direction (θ) of the force. We can use the inverse tangent (arctan) function to calculate the angle:
θ = arctan(y-component of the acceleration / x-component of the acceleration)
Given:
Acceleration (a) = -0.250 m/s^2 i + 0.675 m/s^2 j
θ = arctan(0.675 m/s^2 / -0.250 m/s^2)
θ = arctan(-2.7)
Since the arctan function can only give values between -π/2 and π/2 (-90° to 90°), we need to consider the signs of the components.
Since the y-component of the acceleration is positive and the x-component is negative, the angle will be in the second quadrant. Therefore, we need to add 180° to the angle obtained from the arctan function:
θ = arctan(-2.7) + 180°
Using a calculator, we find:
θ ≈ 116.565°
Thus, the direction of the force is approximately 116.565°.