An Atwood’s machine that has a frictionless massless pulley and massless strings has a 2.00-kg object hanging from one side and 4.00-kg object hanging from the other side.

$(a)$ What is the speed of each object 1.50 s after they are simultaneously released from rest?

Two parts of a machine are held together by bolts, each of which carries a static tensile load of 3100 N.

(a) What size of class 5.8 coarse-thread metric bolt is required using a safety factor of 4 (based on proof strength)?

(b) What is the least number of threads that must be engaged for the thread shear strength to be equal to the bolt tensile strength if the nuts are made of steel whose yield and proof strengths are 70 percent those of the bolt steel?

You are standing at the very rear of a 6.0-m-long, 120-kg raft that is at rest in a lake with its prow only $0.50 \mathrm{~m}$ from the end of the pier (Figure 5-87). Your mass is $60 \mathrm{~kg}$. Neglect frictional forces between the raft and the water.

$(b)$ You walk to the front of the raft and then stop. How far from the end of the pier is the center of mass now?

A spaceship travels at a constant speed from earth to a planet orbiting another star. When the spacecraft arrives, 12 years have elapsed on earth, and 9.2 years have elapsed on board the ship. How far away (in meters ) is the planet, according to observers on earth?

A countershaft carrying two V-belt pulleys is shown in the figure. Pulley A receives power from a motor through a belt with the belt tension. The power is transmitted through the shaft and delivered to the belt on pulley B. Assume the belt tension on the loose side at B is 15 percent of the tension on the tight side.

a.) Determine the tensions in the belt on pulley B, assuming the shaft is running at a constant speed.

b.) Find the magnitudes of the bearing reaction forces, assuming the bearings act as simple supports.

c.) Draw shear-force and bending-moment diagrams for the shaft. If needed, make one set for the horizontal plane and another set for the vertical plane.

d.) At the point of maximum bending moment, determine the bending stress and the torsional shear stress.

e.) At the point of maximum bending moment, determine the principal stresses and the maximum shear stress.

You are standing at the very rear of a 6.0-m-long, 120-kg raft that is at rest in a lake with its prow only $0.50 \mathrm{~m}$ from the end of the pier (Figure 5-87). Your mass is $60 \mathrm{~kg}$. Neglect frictional forces between the raft and the water.

$(a)$ How far from the end of the pier is the center of mass of the you-raft system?

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?

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 $59 \mathrm{~cm}$.)

$(c)$ What is the net external force on the baton while in the air?

A lamina occupies the region inside the circle x^2+y^2=2y but outside the circle x^2+y^2=1. Find the center of mass if the density at any point is inversely proportional to its distance from the origin.

A lamina occupies the part of the disk x^2+y^2<=1 in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.

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 $59 \mathrm{~cm}$.)

$(b)$ If this baton is tossed straight up (but spinning) so that its initial center of mass speed is $10.0 \mathrm{~m} / \mathrm{s}$, what is the velocity of the center of mass $1.5 \mathrm{~s}$ later?

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 $59 \mathrm{~cm}$.)

$(a)$ Where, relative to the center of the light sphere, is the center of mass of this baton?

(a) Calculate the rate of heat conduction through house walls that are 13.0 cm thick and have an average thermal conductivity twice that of glass wool. Assume there are no windows or doors. The walls’ surface area is $120 \mathrm { m } ^ { 2 }$ and their inside surface is at $18.0 ^ { \circ } \mathrm { C }$, while their outside surface is at $5.00 ^ { \circ } \mathrm { C }$. (b) How many 1-kW room heaters would be needed to balance the heat transfer due to conduction?

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.