A mass of 5 kg lies on a frictionless, horizontal floor. A force of 135 Newtons is applied to the mass at an angle above the positive x axis and a force of 14 Newtons is applied to the mass at an angle of 22.8 degrees below the negative x direction (see sketch). If the angle for 135 Newtons is increased until the object just starts to leave the surface, what is the magnitude of the acceleration on the mass in m/s2 at this point?
To find the magnitude of acceleration on the mass when it just starts to leave the surface, we need to consider the forces acting on the mass.
Given:
Mass of the object, m = 5 kg
Force applied at an angle above the positive x-axis, F1 = 135 N
Force applied at an angle of 22.8 degrees below the negative x-axis, F2 = 14 N
Let's break down the forces into their x and y components:
For the force F1:
The x-component of F1 can be found using the formula: F1x = F1 * cos(theta1)
where theta1 is the angle above the positive x-axis.
Similarly, the y-component of F1 can be found using the formula: F1y = F1 * sin(theta1)
For the force F2:
The x-component of F2 can be found using the formula: F2x = F2 * cos(theta2)
where theta2 is the angle below the negative x-axis.
Similarly, the y-component of F2 can be found using the formula: F2y = F2 * sin(theta2)
Now, since the mass lies on a frictionless floor, the only forces acting on it are the vertical forces (perpendicular to the surface) which contribute to the normal force, and the horizontal forces (parallel to the surface) which contribute to acceleration.
In the y-direction, the net vertical force should be zero for the mass to remain on the surface. Therefore:
F1y - F2y - N = 0
The normal force N is equal to the gravitational force acting on the mass, which can be calculated using the formula: N = m * g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Solving the equation for N, we get:
N = F1y - F2y = F1 * sin(theta1) - F2 * sin(theta2)
Now, in the x-direction, the net horizontal force should provide the acceleration to the mass. Therefore:
F1x + F2x = m * a
where a is the magnitude of the acceleration we need to find.
Substituting the values we have:
F1x = F1 * cos(theta1)
F2x = F2 * cos(theta2)
a = (F1 * cos(theta1) + F2 * cos(theta2)) / m
Now we can calculate the magnitude of the acceleration on the mass by plugging in the known values for F1, theta1, F2, theta2, and m:
a = (135 N * cos(theta1) + 14 N * cos(theta2)) / 5 kg
The final step is to calculate the value of theta1 for which the object just starts to leave the surface. To do this, you may need additional information or assumptions, such as the angle of the inclined plane or the coefficient of friction.