An unbalanced baton consists of a 50-cm-long uniform rod of mass $200 \mathrm{~g}$. At one end there is a 10 -cm-diameter uniform solid sphere of mass $500 \mathrm{~g}$, and at the other end there is a $8.0-\mathrm{cm}-$ diameter uniform solid sphere of mass $750 \mathrm{~g}$. (The center-to-center distance between the spheres is $59 \mathrm{~cm}$.)

$(d)$ What is the acceleration of the baton's center of mass $1.5 \mathrm{~s}$ following its release?

A circular plate of radius $R$ has a circular hole of radius $R / 2$ cut out of it (Figure 5-86). Find the center of mass of the plate after the hole has been cut. Hint: The plate can be modeled as two disks superimposed, with the hole modeled as a disk negative mass.

Find the location of the center of mass of the Earth-moon system relative to the center of Earth. Is it inside or outside the surface of Earth?

After a parachutist jumps from an airplane (but before he pulls the rip cord to open his parachute), a downward speed of up to $180 \mathrm{~km} / \mathrm{h}$ can be reached. When the parachute is finally opened, the drag force is increased by about a factor of 10, and this can create a large jolt on the jumper. Suppose this jumper falls at $180 \mathrm{~km} / \mathrm{h}$ before opening his chute.

$(b)$ If rapid accelerations greater than $5.0 \mathrm{~g}$ can harm the structure of the human body, is this a safe practice?

After a parachutist jumps from an airplane (but before he pulls the rip cord to open his parachute), a downward speed of up to $180 \mathrm{~km} / \mathrm{h}$ can be reached. When the parachute is finally opened, the drag force is increased by about a factor of 10, and this can create a large jolt on the jumper. Suppose this jumper falls at $180 \mathrm{~km} / \mathrm{h}$ before opening his chute.

$(a)$ Determine the parachutist's acceleration when the chute is just opened, assuming his mass is $60 \mathrm{~kg}$.

Sally claims flying squirrels do not really fly; they jump and use the folds of skin that connect their forelegs and their back legs like a parachute to allow them to glide from tree to tree. Liz decides to test Sally’s hypothesis by calculating the terminal speed of a falling outstretched flying squirrel. If the constant b in the drag force is proportional to the area of the object facing the air flow, use the results of Example 5-12 and some assumptions about the size of the squirrel to estimate its terminal (downward) speed. Is Sally’s claim supported by Liz’s calculation?

An block of mass $m_1$ is on a horizontal table. The block is attached to a $2.5-\mathrm{kg}$ block $\left(m_2\right)$ by a light string that passes over a pulley at the edge of the table. The block of mass $m_2$ dangles $1.5 \mathrm{~m}$ above the ground (Figure 5-85). The string that connects them passes over a frictionless, massless pulley. This system is released from rest at $t=0$ and the $2.5-\mathrm{kg}$ block strikes the ground at $t=0.82 \mathrm{~s}$. The system is now placed in its initial configuration and a $1.2-\mathrm{kg}$ block is placed on top of the block of mass $m_1$. Released from rest, the $2.5-\mathrm{kg}$ block now strikes the ground $1.3 \mathrm{~s}$ later. Determine the mass $m_1$ and the coefficient of kinetic friction between the block whose mass is $m_1$ and the table.

You are on an amusement park ride with your back against the wall of a spinning vertical cylinder. The floor falls away and you are held up by static friction. Assume your mass is 75 kg.

$(c)$ If the radius of the cylinder is 4.0 m and the coefficient of static friction between you and the wall is 0.55. What is the minimum number of revolutions per minute necessary to prevent you from sliding down the wall? Does this answer hold only for you? Will other, more massive, patrons fall downward? Explain.

You are on an amusement park ride with your back against the wall of a spinning vertical cylinder. The floor falls away and you are held up by static friction. Assume your mass is 75 kg.

$(b)$ Use this diagram with Newton’s laws to determine the force of friction on you.

You are on an amusement park ride with your back against the wall of a spinning vertical cylinder. The floor falls away and you are held up by static friction. Assume your mass is 75 kg.

$(a)$ Draw a free-body diagram of yourself.

The position of a particle of mass $m=0.80 \mathrm{~kg}$ as a function of time is given by $\vec{r}=x \hat{i}+y \hat{j}=(R \sin \omega t) \hat{i}+(R \cos \omega t) \hat{j}$, where $R=4.0 \mathrm{~m}$ and $\omega=2 \pi \mathrm{s}^{-1}$.

$(d)$ Find the magnitude and direction of the net force acting on the particle.

The position of a particle of mass $m=0.80 \mathrm{~kg}$ as a function of time is given by $\vec{r}=x \hat{i}+y \hat{j}=(R \sin \omega t) \hat{i}+(R \cos \omega t) \hat{j}$, where $R=4.0 \mathrm{~m}$ and $\omega=2 \pi \mathrm{s}^{-1}$.

$(c)$ Compute the acceleration vector and show that it is directed toward the origin and has the magnitude $v^2 / R$.

The position of a particle of mass $m=0.80 \mathrm{~kg}$ as a function of time is given by $\vec{r}=x \hat{i}+y \hat{j}=(R \sin \omega t) \hat{i}+(R \cos \omega t) \hat{j}$, where $R=4.0 \mathrm{~m}$ and $\omega=2 \pi \mathrm{s}^{-1}$.

$(b)$ Compute the velocity vector. Show that $v_x / v_y=-y / x$.

The position of a particle of mass $m=0.80 \mathrm{~kg}$ as a function of time is given by $\vec{r}=x \hat{i}+y \hat{j}=(R \sin \omega t) \hat{i}+(R \cos \omega t) \hat{j}$, where $R=4.0 \mathrm{~m}$ and $\omega=2 \pi \mathrm{s}^{-1}$.

$(a)$ Show that the path of this particle is a circle of radius $R$, with its center at the origin of the $x y$ plane.