A box with weight $w$ is pulled at constant speed along a level floor by a force $\overrightarrow{\boldsymbol{F}}$ that is at an angle $\theta$ above the horizontal. The coefficient of kinetic friction between the floor and box is $\mu_{\mathrm{k}}$. $(a)$ In terms of $\theta, \mu_{\mathrm{k}}$, and $w$, calculate $F$. $(b)$ For $w=400 \mathrm{~N}$ and $\mu_{\mathrm{k}}=0.25$, calculate $F$ for $\theta$ ranging from $0^{\circ}$ to $90^{\circ}$ in increments of $10^{\circ}$. Graph $F$ versus $\theta$. $(c)$ From the general expression in part $(a)$, calculate the value of $\theta$ for which the value of $F$, required to maintain constant speed, is a minimum. (Hint: At a point where a function is minimum, what are the first and second derivatives of the function? Here $F$ is a function of $\theta$.) For the special case of $w=400 \mathrm{~N}$ and $\mu_{\mathrm{k}}=0.25$, evaluate this optimal $\theta$ and compare your result to the graph you constructed in part $(b)$.

A physics major is working to pay her college tuition by performing in a traveling carnival. She rides a motorcycle inside a hollow, transparent plastic sphere. After gaining sufficient speed, she travels in a vertical circle with radius $13.0$ m. She has mass $70.0$ $\mathrm{kg}$, and her motorcycle has mass $40.0$ $\mathrm{kg}$.
($b$) At the bottom of the circle, her speed is twice the value calculated in part (a). What is the magnitude of the normal force exerted on the motorcycle by the sphere at this point?

A physics major is working to pay her college tuition by performing in a traveling carnival. She rides a motorcycle inside a hollow, transparent plastic sphere. After gaining sufficient speed, she travels in a vertical circle with radius 13.0 m. She has mass 70.0 kg, and her motorcycle has mass 40.0 kg. (a) What minimum speed must she have at the top of the circle for the motorcycle tires to remain in contact with the sphere? (b) At the bottom of the circle, her speed is twice the value calculated in part (a). What is the magnitude of the normal force exerted on the motorcycle by the sphere at this point?

You are driving a classic $1954$ Nash Ambassador with a friend who is sitting to your right on the passenger side of the front seat. The Ambassador has flat bench seats. You would like to be closer to your friend and decide to use physics to achieve your romantic goal by making a quick turn. $(a)$ Which way (to the left or to the right) should you turn the car to get your friend to slide closer to you? $(b)$ If the coefficient of static friction between your friend and the car seat is $0.35$, and you keep driving at a constant speed of $20 \mathrm{~m} / \mathrm{s}$, what is the maximum radius you could make your turn and still have your friend slide your way?

On the ride "Spindletop" at the amusement park Six Flags Over Texas, people stood against the inner wall of a hollow vertical cylinder with radius $2.5$ m. The cylinder started to rotate, and when it reached a constant rotation rate of $0.60$ rev/s, the floor dropped about $0.5$ m. The people remained pinned against the wall without touching the floor.
($c$) Does your answer in part (b) depend on the person's mass? ($\textit{Note}$: When such a ride is over, the cylinder is slowly brought to rest. As it slows down, people slide down the walls to the floor.)

On the ride "Spindletop" at the amusement park Six Flags Over Texas, people stood against the inner wall of a hollow vertical cylinder with radius $2.5$ m. The cylinder started to rotate, and when it reached a constant rotation rate of $0.60$ rev/s, the floor dropped about $0.5$ m. The people remained pinned against the wall without touching the floor.
(b) What mini- mum coefficient of static friction was required for the person not to slide downward to the new position of the floor?