What Forces Keep the Book Stable on an Inclined Bookshelf?

[vanitymdl]

Homework Statement

A 0.407 kg book rests at an angle against one side of a bookshelf. The magnitude and direction of the total force exerted on the book by the left side of the bookshelf are given by:

Force Left = 2.17 Newtons and θ=59 degrees

What must the magnitude and direction of the total force exerted on the book by the bottom of the bookshelf be in order for the book to remain in this position?

Force book = ___?___Newtons
θ = ___?___degrees

Homework Equations

The Attempt at a Solution

Sum of forces in x:

2.17sin(59) - Fcos(θ) = 0 => Fcos(θ) = 1.86

Sum of forces in y

2.17cos(59) + Fsin(θ) = 0 => Fsin(θ) = -1.1176

tan(θ) = (-1.1176)/(1.86) => θ = 211 degrees

F = 2.17 N

this is wrong though

 

[rcgldr]

The sum of the forces in y must counter the force exerted by gravity, which is m g.

 

[vanitymdl]

So I was in the right track?
I just need to change my formula to 2.17cos(59) + Fsin(θ) = mg

 

[rcgldr]

Is the 59 θ the angle from vertical or the angle from horizontal?

 

[Nickie]

so idk what the right answer is

Thank you for providing the homework statement and your attempt at a solution. It seems like you have correctly set up the equations for the sum of forces in the x and y directions. However, there are a few errors in your calculations.

Firstly, the value of sin(59) is 0.857 and not 0.86. Secondly, when solving for Fcos(θ), you have incorrectly used the value of 1.86 instead of 2.17. This leads to an incorrect value of Fcos(θ) = 1.86.

Similarly, when solving for Fsin(θ), you have used the incorrect value of 1.1176 instead of 1.1178. This leads to an incorrect value of Fsin(θ) = -1.1176.

Using the correct values, we get Fcos(θ) = 1.86 and Fsin(θ) = -1.1178. Solving for θ using the equation tan(θ) = (-1.1178)/(1.86), we get θ = -33.9 degrees.

Therefore, the magnitude and direction of the total force exerted on the book by the bottom of the bookshelf should be:

Force book = 2.17 N
θ = -33.9 degrees

I hope this helps and please let me know if you have any further questions.