A mass of 8 kg lies on a horizontal, frictionless floor. A force of 72 Newtons pushes to the left (negative x direction) with a force of 72 Newtons. Another force of unknown magnitude pushes the mass in a direction of 21.4 degrees above the positive x axis. The mass is originally at rest before these forces are applied and 2.4 seconds after the forces have been applied, the mass has moved to the left a distance of 11.1 meters. What is the magnitude of the unknown force in Newtons?

d = Vo*t + 0.5at^2 = 11.1m,

0 + 0.5a*(2.4)^2 = 11.1,
2.88a = 11.1,
a = 3.85m/s^2.

Fn = 72 - fcos21.4 = ma,
-72 - Fcos21.4 = 8*3.85 = 30.6,
-Fcos21.4 = 30.6 + 72 = 102.6,
-F =110.2N
F = -110.2N. t0 the left.

NOTE: The sine of 21.4 was not used,
because it does not contribute to hor.
movement.

To find the magnitude of the unknown force, we need to use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration.

First, let's calculate the acceleration of the mass. We can use the equation:

acceleration = (change in velocity) / (time taken)

Since the mass is initially at rest, the change in velocity is equal to the final velocity. The velocity can be calculated using the equation:

velocity = (change in position) / (time taken)

Given that the mass has moved to the left a distance of 11.1 meters in 2.4 seconds, we can calculate the velocity:

velocity = 11.1 meters / 2.4 seconds

Next, we need to calculate the horizontal and vertical components of the unknown force.

The horizontal component of the unknown force can be found using the equation:

horizontal force = force * cos(theta)

Where theta is the angle between the unknown force and the positive x-axis. In this case, theta is 21.4 degrees.

Similarly, the vertical component of the unknown force can be found using the equation:

vertical force = force * sin(theta)

Now, let's calculate the horizontal and vertical components of the unknown force:

horizontal force = unknown force * cos(21.4 degrees)
vertical force = unknown force * sin(21.4 degrees)

Next, we can sum the forces in the horizontal direction to find the net force. Since the floor is frictionless, the only horizontal force acting on the mass is the force pushing to the left.

net force in the horizontal direction = force pushing to the left

Finally, we can apply Newton's second law of motion:

net force = mass * acceleration

By substituting the values we have obtained, we can solve for the unknown force:

force pushing to the left = mass * acceleration
force pushing to the left = 8 kg * (11.1 meters / 2.4 seconds)
force pushing to the left = 36.67 N

Since the net force in the horizontal direction is equal to the unknown horizontal force, the magnitude of the unknown force is 36.67 N.