In the baggage claim area of an American airport, a particular baggage carousel is shaped like a section of a large cone, steadily rotating about its vertical axis. Its surface slopes downward toward the outside, making an angle of theta$=20\mathrm{.}5\backslash$deg with the horizontal. A travel bag having mass m $=30\mathrm{.}0$ kg is placed on the carousel at a position r $=7\mathrm{.}46$ m measured horizontally from the axis of rotation. The travel bag goes around once in $\backslash$Tau $=41\mathrm{.}5$ s$.$

$($a$.)$ What is the angular speed of the bag?

$($b$.)$ What is the linear speed of the bag?

$($c$.)$ Calculate the force of static friction exerted by the carousel on the travel bag. $($Give the magnitude of your answer in Newtons $[$N$].$ Please work through the steps below on your way to finding the answer.$)$

step $1$: Sketch a large and detailed free$-$body diagram for the bag on an inclined surface. Two of the diagrammed force vectors will need to be split into x$-$y components, so a large, careful picture is useful in keeping track of all vector information.

Consider the force of static friction $($magnitude f$)$ acting on the bag: recall that friction acts in a direction parallel to but opposite the direction of $($potential$)$ motion. What direction does the bag threaten to slip, both due to gravity and the circular motion of the carousel? Draw the friction force vector opposite the direction of potential slippage. Include the direction of acceleration of the bag on this diagram, as it is experiencing circular motion.

step $2$: Establish the frame of reference pictured above on the free$-$body diagram $($positive$-$x points horizontally and to the left, positive$-$y points vertically and up$).$ This orientation of the axes is chosen so that the acceleration is parallel to a single, positive axis. According to this frame of reference, break two of the three force vectors pictured into their x and y components.

step $3$: Apply Newton's $2$nd law for the sum of forces in the y$-$direction and the sum of forces in the x$-$direction. The bag is not accelerating in one of these directions, and is experiencing centripetal acceleration ac in the other.

Check:

$($i$)$ Using the Newton's $2$nd law expression in the y direction, express n $($the magnitude of the normal force acting on the bag$)$ in terms of symbols$/$variables m$,$ g$,$ theta, and unknown friction force f:

$($ii$)$ Using the other Newton's $2$nd law expression, enter the sum of forces in the x direction $($which is equal to mac$)$ in terms of symbols$/$variables n$,$ theta, and f:

step $4$: Given ac $=$ v$2/$r in your Newton's $2$nd law expression, you now have two equations with two unknowns $($normal force n and friction f$).$ The problem is asking for the magnitude of the force of friction f$,$ so plug your expression for n from part $($i$)$ into the equation from part $($ii$).$ This will leave you with an equation for only the unknown friction. Solve for f$.$ Only when you have an equation with f on one side of the equal sign, and all other numbers$/$symbols on the other side, should you enter numerical values, including units.