A small box is held in place against a rough vertical wall by someone pushing on it with a force directed upward at 25o above the horizontal. The coefficients of static friction and kinetic friction between the box and wall are 0.21 and 0.51, respectively. The box slides down unless the applied force has magnitude 43-N. What is the mass (in kg) of the box?

normal force=F*cos25

friction force= .51*F*cos25
downward force= mg-F*sin25

setting downward force= friction force
mg-41*sin25=.51*41*cos25
now solve for mass m.

To find the mass of the box, we can use the concept of friction and Newton's second law of motion. Here's how we can approach this problem step-by-step:

Step 1: Draw a free-body diagram for the box. The forces acting on the box include:
- The normal force N, perpendicular to the wall
- The gravitational force (Weight) mg, pointing downward
- The applied force F, making an angle of 25o above the horizontal
- The frictional force f, opposing the motion of the box

Step 2: Write the equations of motion in the vertical and horizontal directions. Since the box is not accelerating in the vertical direction, the sum of the vertical forces equals zero:

N - mg = 0 (Equation 1)

In the horizontal direction, the equation of motion is:

F - f = ma (Equation 2)

where a is the acceleration of the box.

Step 3: Determine the values of the forces:

- The gravitational force can be calculated as mg, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
- The normal force N is equal in magnitude and opposite in direction to the gravitational force, so N = mg.
- The applied force F is given as 43 N.

Step 4: Calculate the frictional force f. The frictional force can be expressed in terms of the coefficient of static friction (μs) and the normal force (N). The formula is f = μs * N:

f = 0.21 * N

Substitute N = mg into the equation:

f = 0.21 * mg

Step 5: Substitute the values of the forces into Equation 2:

F - f = ma

43 - 0.21 * mg = ma

Step 6: Substitute mg from Equation 1 into the Equation 2:

43 - 0.21 * (N) = ma

43 - 0.21 * (mg) = ma

43 - 0.21 * (mg) = m * a

Step 7: Notice that mass (m) appears on both sides of the equation. To solve for it, let's substitute a = g:

43 - 0.21 * (mg) = m * g

43 = 0.21 * (mg) + m * g

Step 8: Factor out m from the right side of the equation:

43 = (0.21 + 1) * mg

43 = 1.21 * mg

Divide both sides of the equation by 1.21g:

43 / (1.21 * g) = mg

Step 9: Finally, solve for m:

m = (43 / (1.21 * g))

Substitute the value of g (approximately 9.8 m/s^2) into the equation and evaluate m to find the mass of the box in kg.