Two forces are applied to a car in an effort to move it, as shown in the following figure, where F1 = 433 N and F2 = 354 N. (Assume up and to the right as positive directions.)
F1 is 10 degrees west of the y axis, F2 is 30 degrees east of the y axis.
(a)What is the direction ° to the right of the forward direction?
(b) If the car has a mass of 3,000 kg, what acceleration does it have? Ignore friction.
3 m/s2
To solve this problem, we can break down the forces into horizontal and vertical components. Let's start with force F1.
(a) To determine the direction to the right of the forward direction, we need to find the horizontal component of force F1.
The horizontal component of F1 can be calculated using the formula:
F1_horizontal = F1 * cos(angle)
where the angle is measured from the y-axis (vertical direction).
Given that F1 = 433 N and the angle is 10 degrees west of the y-axis, we need to convert this angle to be measured from the positive x-axis. We can do this by subtracting the angle from 90 degrees.
New angle = 90 degrees - 10 degrees = 80 degrees
Now, we can calculate the horizontal component of F1 using the cosine function:
F1_horizontal = 433 N * cos(80 degrees)
Using a calculator, we find that F1_horizontal is approximately 97.71 N.
To find the direction to the right of the forward direction, we can take the inverse cosine of the ratio of the horizontal component to the magnitude of F1.
Direction = acos(F1_horizontal / F1)
Direction = acos(97.71 N / 433 N)
Using a calculator, we find that the direction to the right of the forward direction is approximately 75.4 degrees.
Therefore, the answer to part (a) is that the direction to the right of the forward direction is approximately 75.4 degrees.
(b) To find the acceleration of the car, we can use Newton's second law:
Force = mass * acceleration
Since we want to calculate the acceleration, we can rearrange the equation as:
Acceleration = Force / mass
We need to find the net force acting on the car. To do this, we need to calculate the horizontal and vertical components for both forces F1 and F2.
The horizontal component of F1 is already calculated as F1_horizontal = 97.71 N.
Now let's calculate the horizontal and vertical components of F2.
The horizontal component of F2 is given by:
F2_horizontal = F2 * cos(angle)
Given that F2 = 354 N and the angle is 30 degrees east of the y-axis, we can calculate the horizontal component as follows:
F2_horizontal = 354 N * cos(30 degrees)
Using a calculator, we find that F2_horizontal is approximately 306.26 N.
Since the forces F1 and F2 act in opposite directions, we can subtract the horizontal components to find the net horizontal force:
Net_horizontal_force = F1_horizontal - F2_horizontal
Net_horizontal_force = 97.71 N - 306.26 N
Net_horizontal_force is approximately -208.55 N. (Note the negative sign indicates that the net force is acting in the negative x-direction.)
The net vertical force is zero since the vertical components of the forces cancel each other out.
Now we can calculate the acceleration using the net force:
Acceleration = Net_horizontal_force / mass
Acceleration = -208.55 N / 3000 kg
Calculating this, we find that the acceleration is approximately -0.0695 m/s^2.
Therefore, the answer to part (b) is that the car has an acceleration of approximately -0.0695 m/s^2.