Forces of 11.1N north, 19.9N east, and 14.0N south are simultaneously applied to a 3.62kg mass as it rests on an air table. What is the magnitude of its acceleration?
b) What is the direction of the acceleration in degrees? (Take east to be 0 degrees and counterclockwise to be positive. Enter an angle between -180 degrees and +180 degrees.)
F(x) =F1(x)+F2(x)+F3(x) =0+19.9 +0 =19.9 N
F(y) =F1(y)+F2(y)+F3(y) =11.1+0-14=-3 N
F=sqrt{F(x)^2 + F(y)^2}
a=F/m
tan alpha=F(y)/F(x)
unknown angle =appha +90 degrees
To find the magnitude of acceleration, we can use Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
1. Calculate the net force: To find the net force, we need to add all the forces together. Since they are given in different directions, we need to break them down into their horizontal and vertical components.
- The force of 11.1N north has no horizontal component and only a vertical component of -11.1N.
- The force of 19.9N east has a horizontal component of +19.9N and no vertical component.
- The force of 14.0N south has no horizontal component and a vertical component of +14.0N.
Add up the horizontal and vertical components separately:
Horizontal component: 19.9N
Vertical component: -11.1N + 14.0N = 2.9N
So, the net force acting on the mass is the vector sum of the horizontal and vertical components:
Net force = sqrt((19.9N)^2 + (2.9N)^2) = sqrt(396.01N^2 + 8.41N^2) = sqrt(404.42N^2) = 20.11N
2. Find the magnitude of acceleration: Now, we can use Newton's second law to find the magnitude of acceleration. The equation is: F = ma, where F is the net force and m is the mass.
Rearranging the equation to solve for acceleration:
a = F / m = 20.11N / 3.62kg = 5.54 m/s^2
Therefore, the magnitude of the acceleration is 5.54 m/s^2.
To find the direction of the acceleration in degrees, we need to consider the components of the net force along the x-axis and y-axis and then convert it into an angle.
3. Determine the direction of the acceleration: We have already calculated the horizontal (x-axis) and vertical (y-axis) components of the net force:
Horizontal component: 19.9N
Vertical component: 2.9N
The direction of the acceleration can be determined using the tangent function:
angle = arctan(vertical component / horizontal component) = arctan(2.9N / 19.9N)
Using a calculator, find the arctan of 2.9N / 19.9N, which gives approximately 8.1 degrees.
To determine the direction, we need to consider that east is 0 degrees and counterclockwise is positive. Since we calculated the angle to be positive, we can take the eastward direction as the reference, giving an angle of 8.1 degrees.
Therefore, the direction of the acceleration is approximately 8.1 degrees counterclockwise from east.