|| FIGURE P7.26 shows a 6.0 N force pushing two gliders along an air track. The 200 g spring between the gliders is compressed. How much force does the spring exert on (a) glider A and (b) glider B? The spring is firmly attached to the gliders, and it does not sag.

FIGURE P7.26

|| FIGURE P7.27 shows two strong magnets on opposite sides of a small table. The long-range attractive force between the magnets keeps the lower magnet in place.

1. Draw an interaction diagram and draw free-body diagrams for both magnets and the table. Use dashed lines to connect the members of an action/reaction pair.

2. The lower magnet is being pulled upward against the bottom of the table. Suppose that each magnet’s weight is 2.0 N and that the magnetic force of the lower magnet on the upper magnet is 6.0 N. How hard does the lower magnet push against the table?

FIGURE P7.27

|| A rope of length L and mass m is suspended from the ceiling. Find an expression for the tension in the rope at position y, measured upward from the free end of the rope.

|| An 85 kg cheerleader stands on a scale that reads in kg.

1. What does the scale read if the 85 kg cheerleader has a 50 kg cheerleader standing on his shoulders?

2. What does the scale read if the 85 kg cheerleader lifts the 50 kg cheerleader upward with an acceleration of $2.0{\text{m/s}}^2?$

| BIO Your forehead can withstand a force of about 6.0 kN before fracturing, while your cheekbone can withstand only about 1.3 kN. Suppose a 140 g baseball traveling at 30 m/s strikes your head and stops in 1.5 ms.

1. What is the magnitude of the force that stops the baseball?

2. What force does the baseball exert on your head? Explain.

3. Are you in danger of a fracture if the ball hits you in the forehead? On the cheek?

|| A Federation starship $(2.0\times {10}^6\text{kg})$ uses its tractor beam to pull a shuttlecraft $(2.0\times {10}^4\text{kg})$ aboard from a distance of 10 km away. The tractor beam exerts a constant force of $4.0\times {10}^4\text{N}$ on the shuttlecraft. Both spacecraft are initially at rest. How far does the starship move as it pulls the shuttlecraft aboard?

|| A 75 kg archer on ice skates is standing at rest on very smooth ice. He shoots a 450 g arrow horizontally. When released, the arrow reaches a speed of 110 m/s in 0.25 s. Assume that the force of the bow string on the arrow is constant.

1. What is the force of the arrow on the bow that the archer is holding?

2. What is the archer’s recoil speed?

||| Two packages at UPS start sliding down the $20\text{°}$ ramp shown in FIGURE P7.33. Package A has a mass of 5.0 kg and a coefficient of friction of 0.20. Package B has a mass of 10 kg and a coefficient of friction of 0.15. How long does it take package A to reach the bottom?

FIGURE P7.33

|| The two blocks in FIGURE P7.34 are sliding down the incline. What is the tension in the massless string?

FIGURE P7.34

|| The coefficient of static friction is 0.60 between the two blocks in FIGURE P7.35. The coefficient of kinetic friction between the lower block and the floor is 0.20. Force $\vec{F}$ causes both blocks to cross a distance of 5.0 m, starting from rest. What is the least amount of time in which this motion can be completed without the top block sliding on the lower block?

FIGURE P7.35

1. ||| The block of mass M in FIGURE P7.36 slides on a frictionless surface. Find an expression for the tension in the string.

   FIGURE P7.36

   

   FIGURE P7.36 Full Alternative Text

2. A 2.0 kg block on a horizontal, frictionless surface is connected by a massless spring and a massless, frictionless pulley to a hanging mass. For what value of the hanging mass does the block accelerate at $1.5{\text{m/s}}^2?$

|| The coefficient of kinetic friction between the 2.0 kg block in FIGURE P7.37 and the table is 0.30. What is the acceleration of the 2.0 kg block?

FIGURE P7.37

|| The 10.2 kg block in FIGURE P7.38 is held in place by a force applied to a rope passing over two massless, frictionless pulleys. Find the tensions $T_1$ to $T_5$ and the magnitude of force $\vec{F}$.

FIGURE P7.38

||| FIGURE P7.39 shows a block of mass m resting on a $20\text{°}$ slope. The block has coefficients of friction ${\mu }_{\text{s}}=0.80$ and ${\mu }_{\text{k}}=0.50$ with the surface. It is connected via a massless string over a massless, frictionless pulley to a hanging block of mass 2.0 kg.

1. What is the minimum mass m that will stick and not slip?

2. If this minimum mass is nudged ever so slightly, it will start being pulled up the incline. What acceleration will it have?

FIGURE P7.39

|| The 1.0 kg physics book in FIGURE P7.40 is connected by a string to a 500 g coffee cup. The book is given a push up the slope and released with a speed of $3.0\text{m}\text{/}\text{s}$. The coefficients of friction are ${\mu }_{\text{s}}=0.50$ and ${\mu }_{\text{k}}=0.20$.

1. How far does the book slide?

2. At the highest point, does the book stick to the slope, or does it slide back down?

FIGURE P7.40

|| A 4.0 kg box is on a frictionless $35\text{°}$ slope and is connected via a massless string over a massless, frictionless pulley to a hanging 2.0 kg weight. The picture for this situation is similar to FIGURE P7.39.

1. What is the tension in the string if the 4.0 kg box is held in place, so that it cannot move?

2. If the box is then released, which way will it move on the slope?

3. What is the tension in the string once the box begins to move?

|| The static and kinetic friction coefficients of the surface in FIGURE P7.42 are 0.90 and 0.50, respectively. The hanging mass is increased to the point where the two stacked masses are on the verge of sliding. If the top mass m is then removed, what is the acceleration of mass 2m? The rope is massless, and the pulley is massless and frictionless.

FIGURE P7.42

|| The century-old ascensores in Valparaiso, Chile, are picturesque cable cars built on stilts to keep the passenger compartments level as they go up and down the steep hillsides. As FIGURE P7.43 shows, one car ascends as the other descends. The cars use a two-cable arrangement to compensate for friction; one cable passing around a large pulley connects the cars, the second is pulled by a small motor. Suppose the mass of each car (with passengers) is 1500 kg, the coefficient of rolling friction is 0.020, and the cars move at constant speed. What is the tension in (a) the connecting cable and (b) the cable to the motor?

FIGURE P7.43

|| A 50 g rubber ball A is traveling to the right at 5.0 m/s. It collides with a 25 g rubber ball B that is initially at rest. FIGURE P7.44 shows how the velocity of A changes during the collision until it rebounds with a speed of 4.0 m/s. Consider only horizontal motion.

1. What is the magnitude of the force of A on B during the collision?

2. What is the speed of ball B after the collision?

FIGURE P7.44

|| A house painter uses the chair-and-pulley arrangement of FIGURE P7.45 to lift himself up the side of a house. The painter’s mass is 70 kg and the chair’s mass is 10 kg. With what force must he pull down on the rope in order to accelerate upward at $0.20\text{m}\text{/}{\text{s}}^2?$

FIGURE P7.45

||| A long, 1.0 kg rope hangs from a support that breaks, causing the rope to fall, if a downward pull on the rope exceeds 40 N. A student team has built a 2.0 kg robot “mouse” that runs up and down the rope. What maximum acceleration can the robot have—both magnitude and direction—without the rope falling?

||| FIGURE P7.47 shows a 200 g hamster sitting on an 800 g wedge-shaped block. The block, in turn, rests on a spring scale. An extra-fine lubricating oil having ${\mu }_{\text{s}}={\mu }_{\text{k}}=0$ is sprayed on the top surface of the block, causing the hamster to slide down. Friction between the block and the scale is large enough that the block does not slip on the scale. What does the scale read, in grams, as the hamster slides down?

FIGURE P7.47

|| A 2.5 kg cart with a vertical front is pushed from behind across a horizontal surface by force $\vec{F}$. Rolling friction is negligible. A 100 g button is placed against the front of the cart. What minimum magnitude must the pushing force have to prevent the button from sliding down? The coefficient of static friction between the button and the cart is 0.75.

|| Find an expression for the magnitude of the horizontal force $\vec{F}$ in FIGURE P7.49 for which $m_1$ does not slip either up or down along the wedge. All surfaces are frictionless.

FIGURE P7.49

|| A rocket burns fuel at a rate of $5.0\text{kg}\text{/}\text{s,}$ expelling the exhaust gases at a speed of $4.0\text{km}\text{/}\text{s}$ relative to the rocket. We would like to find the thrust of the rocket engine.

1. Model the fuel burning as a steady ejection of small pellets, each with the small mass $\text{Δ}m$. Suppose it takes a short time $\text{Δ}t$ to accelerate a pellet (at constant acceleration) to the exhaust speed $v_{\text{ex}}$. Further, suppose the rocket is clamped down so that it can’t recoil. Find an expression for the magnitude of the force that one pellet exerts on the rocket during the short time while the pellet is being expelled.

2. If the rocket is moving, $v_{\text{ex}}$ is no longer the pellet’s speed through space but it is still the pellet’s speed relative to the rocket. By considering the limiting case of $\text{Δ}m$ and $\text{Δ}t$ approaching zero, in which case the rocket is now burning fuel continuously, calculate the rocket thrust for the values given above.

||| The lower block in FIGURE CP7.53 is pulled on by a rope with a tension force of 20 N. The coefficient of kinetic friction between the lower block and the surface is 0.30. The coefficient of kinetic friction between the lower block and the upper block is also 0.30. What is the acceleration of the 2.0 kg block?

FIGURE CP7.53

||| In FIGURE CP7.54, find an expression for the acceleration of $m_1$. The pulleys are massless and frictionless.

Hint: Think carefully about the acceleration constraint.

FIGURE CP7.54

||| What is the acceleration of the 3.0 kg block in FIGURE CP7.55 across the frictionless table?

Hint: Think carefully about the acceleration constraint.

FIGURE CP7.55

||| The 500 g mass in FIGURE CP7.56 is pushed across the 2.0-m-long cart by a 3.0 N force, starting at the left side. The coefficient of kinetic friction between the mass and the cart is 0.30. The cart’s coefficient of rolling friction is 0.010. How far has the cart rolled when the mass falls off the right side?

FIGURE CP7.56

||| A 50-cm-diameter, 400 g beach ball is dropped with a 4.0 mg ant riding on the top. The ball experiences air resistance, but the ant does not. What is the magnitude of the normal force exerted on the ant when the ball’s speed is 2.0 m/s?

||| FIGURE CP7.58 shows three hanging masses connected by massless strings over two massless, frictionless pulleys.

1. Find the acceleration constraint for this system. It is a single equation relating $a_{1y}$, $a_{2y}$, and $a_{3y}$.

   Hint: $y_{\text{A}}$ isn’t constant.

2. Find an expression for the tension in string A.

   Hint: You should be able to write four second-law equations. These, plus the acceleration constraint, are five equations in five unknowns.

3. Suppose: $m_1=2.5\text{kg,}$ $m_2=1.5\text{kg,}$ and $m_3=4.0\text{kg}$. Find the acceleration of each.

4. The 4.0 kg mass would appear to be in equilibrium. Explain why it accelerates.

FIGURE CP7.58

||| CALC FIGURE CP7.59 shows a small block of mass m sliding with velocity $v_x$ across a very long plate of mass M. The plate itself is sliding across a frictionless surface with velocity $V_x$. The plate has a viscous upper surface, so the retarding force on the block is not the usual kinetic friction but, instead, a linear drag force $(F_{\text{drag}}){}_x=-b{v}_{\text{rel}}$, where $v_{\text{rel}}=v_x-V_x$ is the velocity of the block relative to the plate. Initially the plate is at rest and the block, having just been hit with a hammer, has velocity $v_{0x}=v_0$. Find an expression for the time $t_{1/2}$ at which the relative velocity $v_{\text{rel}}$ has been reduced to half its initial value.

FIGURE CP7.59