A box is going to be placed on a slanted surface. How can I calculate whether the box will remain stationary or slide down the surface?
In this problem we will use Newton's third law [for every action there is an opposite reaction]. We will draw an axis with x running parallel to the slanted surface, and the y axis being perpendicular to the surface. To calculate whether the box will remain stationary, we will draw a force body diagram, breaking up each force into two components, x and y. 1. We will calculate the force of gravity on the box. 2. We will calculate the normal force of the surface on the box. 3. We will calculate the static friction force the surface will exert on the box. Once these calculations have been completed, we will determine and then solve the box's equilibrium equations in the x axis and the y axis. The equilibrium equation of the y axis will be zero. In this axis there are only two forces, gravity and the normal force. The normal force from the slanted surface is an equal and opposite force to the force of gravity, therefore they cancel each other out. The result of the equation of forces in the x axis will determine whether the box will move.
Prove or disprove: Let B be a non-invertible matrix in R(nxn); then there exists a non-zero matrix A in R(nxn) such that B*A=0.
This statement is TRUE. B is non-invertible if and only if the dimension of the ker(B) does not equal zero. Therefore there exists solutions to the equation B*A = 0. The solution space has dim(ker(B)) > 0, so there exists a non-zero matrix AeR(nxn) such that B*A = 0. QED.
Please prove the following: sin(x) - (sin(x))^(3) = tan(x)*(cos(x))^(3)
sin(x) - (sin(x))^3 = tan(x)*(cos(x))^3 sin(x) * (1 - (sin(x))^2) = tan(x)*(cos(x))^3 [left side: pulled out common factor sin(x)] sin(x) * (1 - (sin(x))^2) = (sin(x)/cos(x)) * (cos(x))^3 [right side: wrote tan(x) as sin(x)/cos(x)] sin(x) * (cos(x))^2 = (sin(x)/cos(x)) * (cos(x))^3 [left side: used identity 1-(sin(x))^2 = (cos(x))^2] sin(x) * (cos(x))^2 = sin(x)*(cos(x))^2 [right side: simplified cos(x)] QED