Only two forces act on an object (mass = 4.20 kg), as in the drawing. (F = 73.0 N.) Find the magnitude and direction (relative to the x axis) of the acceleration of the object.
magnitude m/s2
direction ° counterclockwise from the +x axis
The angle from the block is pointing 45 degrees north of east. the force in the x direction going east is 40.0N
What is the mqagnitude and direction of
each of the 2 forces?
To find the magnitude and direction of the acceleration of the object, we need to resolve the forces into their x and y components and apply Newton's second law equation.
1. Resolve the forces into their components:
Since the force of 73.0 N is given at an angle 45 degrees north of east, we can find its x and y components using trigonometry.
The x component of the force (F_x) is given by:
F_x = F * cos(theta)
F_x = 73.0 N * cos(45°)
F_x = 51.62 N
The y component of the force (F_y) is given by:
F_y = F * sin(theta)
F_y = 73.0 N * sin(45°)
F_y = 51.62 N
2. Calculate the net force in each direction:
The net force in the x-direction will be the sum of the forces in that direction:
Net Force in x-direction = F_x (from the given 40.0 N force in the east direction)
Net Force in x-direction = 40.0 N + 51.62 N
Net Force in x-direction = 91.62 N
The net force in the y-direction will be the sum of the forces in that direction:
Net Force in y-direction = F_y
Net Force in y-direction = 51.62 N
3. Calculate the acceleration:
Using Newton's second law equation, we can find the acceleration:
Net Force = mass * acceleration
In the x-direction:
91.62 N = 4.20 kg * acceleration_x
acceleration_x = 91.62 N / 4.20 kg
acceleration_x ≈ 21.82 m/s^2
In the y-direction:
51.62 N = 4.20 kg * acceleration_y
acceleration_y = 51.62 N / 4.20 kg
acceleration_y ≈ 12.29 m/s^2
4. Find the magnitude of the acceleration:
The magnitude of the acceleration is given by the Pythagorean theorem:
acceleration = sqrt(acceleration_x^2 + acceleration_y^2)
acceleration = sqrt((21.82 m/s^2)^2 + (12.29 m/s^2)^2)
acceleration ≈ 24.94 m/s^2
5. Find the direction of the acceleration:
The direction of the acceleration can be found using trigonometry:
direction = atan(acceleration_y / acceleration_x)
direction = atan(12.29 m/s^2 / 21.82 m/s^2)
direction ≈ 30.5°
Since the object is moving counterclockwise from the +x axis, the direction of the acceleration is approximately 30.5° counterclockwise from the +x axis.