Solved: A 1.5-kg block rests on top of a 7.5-kg block

Why does a child in a wagon seem to fall backward when you give the wagon a sharp pull forward?

A box rests on the (frictionless) bed of a truck. The truck driver starts the truck and accelerates forward. The box immediately starts to slide toward the rear of the truck bed. Discuss the motion of the box, in terms of Newtons laws, as seen (a) by Andrea standing on the ground beside the truck, and (b) by Jim who is riding on the truck (Fig. 4-27).

If the acceleration of an object is zero, are no forces acting on it? Explain.

If an object is moving, is it possible for the net force acting on it to be zero?

Only one force acts on an object. Can the object have zero acceleration? Can it have zero velocity? Explain.

When a golf ball is dropped to the pavement, it bounces back up. (a) Is a force needed to make it bounce back up? (b) If so, what exerts the force?

If you walk along a log floating on a lake, why does the log move in the opposite direction?

Why might your foot hurt if you kick a heavy desk or a wall?

When you are running and want to stop quickly, you must decelerate quickly, (a) What is the origin of the force that causes you to stop? (b) Estimate (using your own experience) the maximum rate of deceleration of a person running at top speed to come to rest.

(a) Why do you push down harder on the pedals of a bicycle when first starting out than when moving at constant speed? (b) Why do you need to pedal at all when cycling at constant speed?

A father and his young daughter are ice skating. They face each other at rest and push each other, moving in opposite directions. Which one has the greater final speed?

Suppose that you are standing on a cardboard carton that just barely supports you. What would happen to it if you jumped up into the air? It would (a) collapse; (b) be unaffected; (c) spring upward a bit; (d) move sideways.

A stone hangs by a fine thread from the ceiling, and a section of the same thread dangles from the bottom of the stone (Fig. 4-28). If a person gives a sharp pull on the dangling thread, where is the thread likely to break: below the stone or above it? What if the person gives a slow and steady pull? Explain your answers.

The force of gravity on a 2-kg rock is twice as great as that on a 1-kg rock. Why then doesnt the heavier rock fall faster?

Would a spring scale carried to the Moon give accurate results if the scale had been calibrated on Earth, (a) in pounds, or (b) in kilograms?

You pull a box with a constant force across a frictionless table using an attached rope held horizontally. If you now pull the rope with the same force at an angle to the horizontal (with the box remaining flat on the table), does the acceleration of the box (a) remain the same, (b) increase, or (c) decrease? Explain.

When an object falls freely under the influence of gravity there is a net force mg exerted on it by the Earth. Yet by Newtons third law the object exerts an equal and opposite force on the Earth. Does the Earth move?

Compare the effort (or force) needed to lift a 10-kg object when you are on the Moon with the force needed to lift it on Earth. Compare the force needed to throw a 2-kg object horizontally with a given speed on the Moon and on Earth.

Which of the following objects weighs about 1 N: (a) an apple, (b) a mosquito, (c) this book, (d) you?

According to Newtons third law, each team in a tug of war (Fig. 4-29) pulls with equal force on the other team. What, then, determines which team will win?

When you stand still on the ground, how large a force does the ground exert on you? Why doesnt this force make you rise up into the air?

Whiplash sometimes results from an automobile accident when the victims car is struck violently from the rear. Explain why the head of the victim seems to be thrown backward in this situation. Is it really?

Mary exerts an upward force of 40 N to hold a bag of groceries. Describe the reaction force (Newtons third law) by stating (a) its magnitude, (b) its direction, (c) on what object it is exerted, and (d) by what object it is exerted.

A bear sling, Fig. 4-30, is used in some national parks for placing backpackers food out of the reach of bears. Explain why the force needed to pull the backpack up increases as the backpack gets higher and higher. Is it possible to pull the rope hard enough so that it doesnt sag at all?

(Ill) The 100-m dash can be run by the best sprinters in 10.0 s. A 66-kg sprinter accelerates uniformly for the first 45 m to reach top speed, which he maintains for the remaining 55 m. (a) What is the average horizontal component of force exerted on his feet by the ground during acceleration? (b) What is the speed of the sprinter over the last 55 m of the race (i.e., his top speed)?

(Ill) A person jumps from the roof of a house 3.9-m high. When he strikes the ground below, he bends his knees so that his torso decelerates over an approximate distance of 0.70 m. If the mass of his torso (excluding legs) is 42 kg, find (a) his velocity just before his feet strike the ground, and (b) the average force exerted on his torso by his legs during deceleration.

(I) A box weighing 77.0 N rests on a table. A rope tied to the box runs vertically upward over a pulley and a weight is hung from the other end (Fig. 4-33). Determine the force that the table exerts on the box if the weight hanging on the other side of the pulley weighs (a) 30.0 N, (b) 60.0 N, and (c) 90.0 N.

(I) Draw the free-body diagram for a basketball player (a) just before leaving the ground on a jump, and (b) while in the air. See Fig. 4-34.

(I) Sketch the free-body diagram of a baseball (a) at the moment it is hit by the bat, and again (b) after it has left the bat and is flying toward the outfield.

(I) A 650-N force acts in a northwesterly direction. A second 650-N force must be exerted in what direction so that the resultant of the two forces points westward? Illustrate your answer with a vector diagram.

(II) Christian is making a Tyrolean traverse as shown in Fig. 4-35. That is, he traverses a chasm by stringing a rope between a tree on one side of the chasm and a tree on the opposite side, 25 m away. The rope must sag sufficiently so it wont break. Assume the rope can provide a tension force of up to 29 kN before breaking, and use a safety factor of 10 (that is, the rope should only be required to undergo a tension force of 2.9 kN) at the center of the Tyrolean traverse, (a) Determine the distance x that the rope must sag if it is to be within its recommended safety range and Christians mass is 72.0 kg. (b) If the Tyrolean traverse is incorrectly set up so that the rope sags by only one-fourth the distance found in (a), determine the tension force in the rope. Will the rope break?

(II) A window washer pulls herself upward using the bucket-pulley apparatus shown in Fig. 4-36. (a) How hard must she pull downward to raise herself slowly at constant speed? (b) If she increases this force by 15%, what will her acceleration be? The mass of the person plus the bucket is 72 kg.

(II) One 3.2-kg paint bucket is hanging by a massless cord from another 3.2-kg paint bucket, also hanging by a massless cord, as shown in Fig. 4-37. (a) If the buckets are at rest, what is the tension in each cord? (b) If the two buckets are pulled upward with an acceleration of 1.25 m /s2 by the upper cord, calculate the tension in each cord

(II) The cords accelerating the buckets in Problem 33b, Fig. 4-37, each has a weight of 2.0 N. Determine the tension in each cord at the three points of attachment.

(II) Two snowcats in Antarctica are towing a housing unit to a new location, as shown in Fig. 4-38. The sum of the forces Fa and Fb exerted on the unit by the horizontal cables is parallel to the line L, and F& = 4500 N. Determine FB and the magnitude of Fa + FB.

(II) A train locomotive is pulling two cars of the same mass behind it, Fig. 4-39. Determine the ratio of the tension in the coupling (think of it as a cord) between the locomotive and the first car (Fxl), to that between the first car and the second car (FT2), for any nonzero acceleration of the train.

(II) The two forces Fi and F2 shown in Fig. 4-40a and b (looking down) act on a 18.5-kg object on a frictionless tabletop. If F\ = 10.2 N and F2 = 16.0 N, find the net force on the object and its acceleration for (a) and (b).

(II) At the instant a race began, a 65-kg sprinter exerted a force of 720 N on the starting block at a 22 angle with respect to the ground, (a) What was the horizontal acceleration of the sprinter? (b) If the force was exerted for 0.32 s, with what speed did the sprinter leave the starting block?

(II) A mass m is at rest on a horizontal frictionless surface at t = 0. Then a constant force F0 acts on it for a time t0. Suddenly the force doubles to 2F0 and remains constant until t = 2t0. Determine the total distance traveled from t = 0 to t = 210.

(II) A 3.0-kg object has the following two forces acting on it: Fi = ( l6i + 12j) N F2 = (101 + 22j) N If the object is initially at rest, determine its velocity v at t = 3.0 s.

(II) Uphill escape ramps are sometimes provided to the side of steep downhill highways for trucks with overheated brakes. For a simple 11 upward ramp, what length would be needed for a runaway truck traveling 140 km/h? Note the large size of your calculated length. (If sand is used for the bed of the ramp, its length can be reduced by a factor of about 2.)

(II) A child on a sled reaches the bottom of a hill with a velocity of 10.0 m/s and travels 25.0 m along a horizontal straightaway to a stop. If the child and sled together have a mass of 60.0 kg, what is the average retarding force on the sled on the horizontal straightaway?

II) A skateboarder, with an initial speed of 2.0 m/s, rolls virtually friction free down a straight incline of length 18 m in 3.3 s. At what angle 6 is the incline oriented above the horizontal?

(II) As shown in Fig. 4-41, five balls (masses 2.00, 2.05, 2.10, 2.15, 2.20 kg) hang from a crossbar. Each mass is supported by 5-lb test fishing line which will break when its tension force exceeds 22.2 N (= 5 lb). When this device is placed in an elevator, which accelerates upward, only the lines attached [! a to the 2.05 and 2.00 kg masses do not break. Within what range is the elevators acceleration?

(II) A 27-kg chandelier hangs from a ceiling on a vertical 4.0-m-long wire, (a) What horizontal force would be necessary to displace its position 0.15 m to one side? (b) What will be the tension in the wire?

(II) Three blocks on a frictionless horizontal surface are in contact with each other as shown in Fig. 4-42. A force F is applied to block A (mass mA). (a) Draw a free-body diagram for each block. Determine (b) the acceleration of the system (in terms of mA, raB, and rac), (c) the net force on each block, and (d) the force of contact that each block exerts on its neighbor. (e) If mA = mB = mc = 10.0 kg and F = 96.0 N, give numerical answers to (b), (c), and (d). Explain how your answers make sense intuitively.

(II) Redo Example 4-13 but (a) set up the equations so that the direction of the acceleration a of each object is in the direction of motion of that object. (In Example 4-13, we took a as positive upward for both masses.) (b) Solve the equations to obtain the same answers as in Example 4-13.

(II) The block shown in Fig. 4-43 has mass m = 7.0 kg and lies on a fixed smooth frictionless plane tilted at an angle 0 = 22.0 to the horizontal, (a) Determine the acceleration of the block as it slides down the plane, (b) If the block starts from rest 12.0 m up the plane from its base, what will be the blocks speed when it reaches the bottom of the incline?

(II) A block is given an initial speed of 4.5 m/s up the 22 plane shown in Fig. 4-43. (a) How far up the plane will it go? (b) How much time elapses before it returns to its starting point? Ignore friction.

(II) An object is hanging by a string from your rearview mirror. While you are accelerating at a constant rate from rest to 28 m/s in 6.0 s, what angle 6 does the v string make with the vertical? See Fig. 4-44.

(II) Figure 4-45 shows a block (mass mA) on a smooth horizontal surface, connected by a thin cord that passes over a pulley to a second block (raB), which hangs vertically, (a) Draw a free-body diagram for each block, showing the force of gravity on each, the force (tension) exerted by the cord, and any normal force. (b) Apply Newtons second law to find formulas for the acceleration of the system and for the tension in the cord. Ignore friction and the masses of the pulley and cord.

(II) (a) If mA = 13.0 kg and mB = 5.0 kg in Fig. 4-45, determine the acceleration of each block. (b) If initially mA is at rest 1.250 m from the edge of the table, how long does it take to reach the edge of the table if the system is allowed to move freely? (c) If mB = 1.0 kg, how large must mA be if the acceleration of the system is to be kept at XJ0 g?

(Ill) Determine a formula for the acceleration of the system shown in Fig. 4-45 (see Problem 51) if the cord has a non-negligible mass rac . Specify in terms of iA and B, the lengths of cord from the respective masses to the pulley. (The total cord length is I = A + fB.)

(Ill) Suppose the pulley in Fig. 4-46 is suspended by a cord C. Determine the tension in this cord after the masses are released and before one hits the ground. Ignore the mass of the pulley and cords.

(Ill) A small block of mass m rests on the sloping side of a triangular block of mass M which itself rests on a horizontal table as shown in Fig. 4-47. Assuming all surfaces are frictionless, determine the magnitude of the force F that must be applied to M so that m remains in a fixed position relative to M (that is, m doesnt move on the incline). [Hint: Take x and y axes horizontal and vertical.!

(Ill) The double Atwood machine shown in Fig. 4-48 has frictionless, massless pulleys and cords. Determ mine (a) the acceleration of masses mA, mB, and mc , and (b) the tensions FTA and FJC in the cords.

(Ill) Suppose two boxes on a frictionless table are connected by a heavy cord of mass 1.0 kg. Calculate the acceleration of each box and the tension at each end of the cord, using the free-body diagrams shown in Fig. 4-49. Assume FP = 35.0 N, and ignore sagging of the cord. Compare your results to Example 4-12 and Fig. 4-22.

(Ill) The two masses shown in Fig. 4-50 are each initially 1.8 m above the ground, and the massless frictionless pulley is 4.8 m above the ground. What maximum height does the lighter object reach after the system is released? [Hint: First determine the acceleration of the lighter mass and then its velocity at the moment the heavier one hits the ground. This is its launch speed. Assume the mass doesnt hit the pulley. Ignore the mass of the cord.l

(Ill) Determine a formula for the magnitude of the force F exerted on the large block (rac) in Fig. 4-51 so that the mass mA does not move relative to mc . Ignore all friction. Assume raB does not make contact with mc .

(Ill) A particle of mass m, initially at rest at x = 0, is accelerated by a force that increases in time as F = Ct2. Determine its velocity v and position x as a function of time.

(Ill) A heavy steel cable of length i and mass M passes over a small massless, frictionless pulley, (a) If a length y hangs on one side of the pulley (so - y hangs on the other side), calculate the acceleration of the cable as a function of y. (b) Assuming the cable starts from rest with length y0 on one side of the pulley, determine the velocity Vf at the moment the whole cable has fallen from the pulley, (c) Evaluate Vf for y0 = \L [Hint: Use the chain rule, dv /dt = (dv / dy)(dy / dt), and integrate.]

A person has a reasonable chance of surviving an automobile crash if the deceleration is no more than 30 gs. Calculate the force on a 65-kg person accelerating at this rate. What distance is traveled if brought to rest at this rate from 95 km/h?

A 2.0-kg purse is dropped 58 m from the top of the Leaning Tower of Pisa and falls 55 m before reaching the ground with a speed of 27m/s. What was the average force of air resistance?

Toms hang glider supports his weight using the six ropes shown in Fig. 4-52. Each rope is designed to support an equal fraction of Toms weight. Toms mass is 74.0 kg. What is the tension in each of the support ropes?

A wet bar of soap (m = 150 g) slides freely down a ramp 3.0 m long inclined at 8.5. How long does it take to reach the bottom? How would this change if the soaps mass were 300 g?

A cranes trolley at point P in Fig. 4-53 moves for a few seconds to the right with constant acceleration, and the 870-kg load hangs at a 5.0 angle to the vertical as shown. What is the acceleration of the trolley and load?

A block (mass raA) lying on a fixed frictionless inclined plane is connected to a mass mB by a cord passing over a pulley, as shown in Fig. 4-54. (a) Determine a formula for the acceleration of the system in terms of mA, raB, 0, and g. (b) What conditions apply to masses mA and mB for the acceleration to be in one direction (say, mA down the plane), or in the opposite direction? Ignore the mass of the cord and pulley.

(a) In Fig. 4-54, if mA = mB = 1.00 kg and 6 = 33.0, what will be the acceleration of the system? (b) If mA = 1.00 kg and the system remains at rest, what must the mass mB be? (c) Calculate the tension in the cord for (a) and (b).

The masses mA and mB slide on the smooth (frictionless) inclines fixed as shown in Fig. 4-55. (a) Determine a formula for the acceleration of the system in terms of raA, raB ,dA ,dB, and g. (b) If 0A = 32, 0B = 23, and raA = 5.0 kg, what value of raB would keep the system at rest? What would be the tension in the cord (negligible mass) in this case? (c) What ratio, mA/mB, would allow the masses to move at constant speed along their ramps in either direction?

A 75.0-kg person stands on a scale in an elevator. What does the scale read (in N and in kg) when (a) the elevator is at rest, (b) the elevator is climbing at a constant speed of 3.0 m/s, (c) the elevator is descending at 3.0 m/s, (d) the elevator is accelerating upward at 3.0 m/s2, (e) the elevator is accelerating downward at 3.0 m/s2?

A city planner is working on the redesign of a hilly portion of a city. An important consideration is how steep the roads can be so that even low-powered cars can get up the hills without slowing down. A particular small car, with a mass of 920 kg, can accelerate on a level road from rest to 21 m/s (75 km/h) in 12.5 s. Using these data, calculate the maximum steepness of a hill.

If a bicyclist of mass 65 kg (including the bicycle) can coast down a 6.5 hill at a steady speed of 6.0 km/h because of air resistance, how much force must be applied to climb the hill at the same speed (and the same air resistance)?

A bicyclist can coast down a 5.0 hill at a constant speed of 6.0 km/h. If the force of air resistance is proportional to the speed v so that Fair = cv, calculate (a) the value of the constant c, and (b) the average force that must be applied in order to descend the hill at 18.0 km/h. The mass of the cyclist plus bicycle is 80.0 kg.

Francesca dangles her watch from a thin piece of string while the jetliner she is in accelerates for takeoff, which takes about 16 s. Estimate the takeoff speed of the aircraft if the string makes an angle of 25 with respect to the vertical, Fig. 4-56.

(a) What minimum force F is needed to lift the piano (mass M) using the pulley apparatus shown in Fig. 4-57? (b) Determine the tension in each section of rope: FT1, FT2, FT3, and Fx4 .

In the design of a supermarket, there are to be several ramps connecting different parts of the store. Customers will have to push grocery carts up the ramps and it is obviously desirable that this not be too difficult. The engineer has done a survey and found that almost no one complains if the force required is no more than 18 N. Ignoring friction, at what maximum angle 0 should the ramps be built, assuming a full 25-kg grocery cart?

A jet aircraft is accelerating at 3.8 m/s2 as it climbs at an angle of 18 above the horizontal (Fig. 4-58). What is the total force that the cockpit seat exerts on the 75-kg pilot?

A 7650-kg helicopter accelerates upward at 0.80 m/s2 while lifting a 1250-kg frame at a construction site, Fig. 4-59. (a) What is the lift force exerted by the air on the helicopter rotors? (b) What is the tension in the cable (ignore its mass) that connects the frame to the helicopter? (c) What force does the cable exert on the helicopter?

A super high-speed 14-car Italian train has a mass of 640 metric tons (640,000 kg). It can exert a maximum force of 400 kN horizontally against the tracks, whereas at maximum constant velocity (300 km/h), it exerts a force of about 150 kN. Calculate (a) its maximum acceleration, and (b) estimate the force of friction and air resistance at top speed.

A fisherman in a boat is using a 10-lb test fishing line. This means that the line can exert a force of 45 N without breaking (1 lb = 4.45 N). (a) How heavy a fish can the fisherman land if he pulls the fish up vertically at constant speed? (b) If he accelerates the fish upward at 2.0m/s2, what maximum weight fish can he land? (c) Is it possible to land a 15-lb trout on 10-lb test line? Why or why not?

An elevator in a tall building is allowed to reach a maximum speed of 3.5 m/s going down. What must the tension be in the cable to stop this elevator over a distance of 2.6 m if the elevator has a mass of 1450 kg including occupants?

Two rock climbers, Bill and Karen, use safety ropes of similar length. Karens rope is more elastic, called a dynamic rope by climbers. Bill has a static rope, not recommended for safety purposes in pro climbing, (a) Karen falls freely about 2.0 m and then the rope stops her over a distance of 1.0 m (Fig. 4-60). Estimate how large a force (assume constant) she will feel from the rope. (Express the result in multiples of her weight.) (b) In a similar fall, Bills rope stretches by only 30 cm. How many times his weight will the rope pull on him? Which climber is more likely to be hurt?

Three mountain climbers who are roped together in a line are ascending an icefield inclined at 31.0 to the horizontal (Fig. 4-61). The last climber slips, pulling the second climber off his feet. The first climber is able to hold them both. If each climber has a mass of 75 kg, calculate the tension in each of the two sections of rope between the three climbers. Ignore friction between the ice and the fallen climbers.

A doomsday asteroid with a mass of 1.0 X 1010kg is hurtling through space. Unless the asteroids speed is changed by about 0.20 cm/s, it will collide with Earth and cause tremendous damage. Researchers suggest that a small space tug sent to the asteroids surface could exert a gentle constant force of 2.5 N. For how long must this force act?

A 450-kg piano is being unloaded from a truck by rolling it down a ramp inclined at 22. There is negligible friction and the ramp is 11.5 m long. Two workers slow the rate at which the piano moves by pushing with a combined force of 1420 N parallel to the ramp. If the piano starts from rest, how fast is it moving at the bottom?

Consider the system shown in Fig. 4-62 with raA = 9.5 kg and mB = 11.5 kg. The angles 0A = 59 and 0B = 32. (a) In the absence of friction, what force F would be required to pull the masses at a constant velocity up the fixed inclines? (b) The force F is now removed. What p is the magnitude and direction of the acceleration of m i| the two blocks? (c) In the absence of F, what is the tension in the string?

A 1.5-kg block rests on top of a 7.5-kg block (Fig. 4-63). The cord and pulley have negligible mass, and there is no significant friction anywhere, (a) What force F must be applied to the bottom block so the top block accelerates to the right at 2.5 m/s2? (b) What is the tension in the connecting cord?

You are driving home in your 750-kg car at 15 m/s. At a point 45 m from the beginning of an intersection, you see a green traffic light change to yellow, which you expect will last 4.0 s, and the distance to the far side of the intersection is 65 m (Fig. 4-64). (a) If you choose to accelerate, your cars engine will furnish a forward force of 1200 N. Will you make it completely through the intersection before the light turns red? (b) If you decide to panic stop, your brakes will provide a force of 1800 N. Will you stop before entering the intersection?

(II) A large crate of mass 1500 kg starts sliding from rest along a frictionless ramp, whose length is t and whose inclination with the horizontal is 0. (a) Determine as a function of 6: (i) the acceleration a of the crate as it goes downhill, (ii) the time t to reach the bottom of the incline, (iii) the final velocity v of the crate when it reaches the bottom of the ramp, and (iv) the normal force FN on the crate, (b) Now assume i = 100 m. Use a spreadsheet to calculate and graph a, t, v, and FN as functions of 6 from 0 = 0 to 90 in 1 steps. Are your results consistent with the known result for the limiting cases 0 = 0 and 6 = 90?