Example #1: Energy in Curved Paths

Example #2: Energy in Curved Paths

Example #3: Energy in Curved Paths

Concept #1: More Rollercoaster Problems

Concept #2: Gravitational Energy is Relative (Pendulums)

Practice: A 60 kg surfer is moving with 3 m/s at a certain point in a wave. Later on, he is moving with 8 m/s at a second point, 2 meters lower. Calculate the work done by the wave on the surfer.

Example #4: More Pendulum Problems

Practice: A pendulum is built from a 3 kg bob and a 4 m-long light rope. It is attached to the ceiling and pulled from its equilibrium position until it makes an angle of 53° with the vertical. It is then given an initial speed of 2 m/s directed down.

(a) Calculate the maximum speed that the pendulum will attain.

(b) Calculate the maximum angle that the pendulum will make with the vertical on the other side.

Concept #3: Energy Problems with Bumps (Part A)

Concept #4: Energy Problems with Bumps (Part B)

Concept #5: Energy Problems with Bumps (Part C)

A pendulum is released from rest at an angle of 40 o. If the pendulum is made of a light, 1 m string supporting a 2.5 kg bowling ball, what is the speed of the bowling ball at its lowest point in the motion? What is the tension in the string at the lowest point?

A 5 kg object undergoes uniform circular motion. If the object has a tangential speed of 15 m/s in a 0.7 m orbit and undergoes a 1/4 revolution, how much work was done on the object by the centripetal force?

A 500 kg car carrying 200 kg of passengers on a rollercoaster encounters a stop on the track shown in the figure below. How high does the car have to start in order to traverse the loop without any passengers being in danger of falling out if their harness broke? Note that the car starts from rest at the top of the slope and that there is no friction between the car and the track.

A roller coaster car and its occupants are pulled up an initial incline by a chain to point A, and from then on it rolls without friction on the tracks. It descends the first hill and then enters the inside of a loop at point B. An FSU student has, celeverly, brought along a bathroom scale to test out some theories from her Physics class. The scale normally reads 65 kg when she stands on it. She sits on it in the coaster so that her entire weight rests on the scale when the car is level. You can assume that the initial incline is high enough that the car stays on the tracks at point C.

A block of mass m slides down a frictionless track, then around the inside of a circular loop-the-loop of radius exttip{R}{R}.From what minimum height h must the block start to make it around the loop without falling off?
Give your answer as a multiple of R.

A car in an amusement park ride rolls without friction around a track
. The car starts from rest at point A at a height h above the bottom of the loop. Treat the car as a particle.What is the minimum value of h (in terms of R) such that the car moves around the loop without falling off at the top (point B)?If the car starts at height
h=
4.10
R
and the radius is
exttip{R_{
m 1}}{R_1} = 15.0 m , compute the speed of the passengers when the car is at point C, which is at the end of a horizontal diameter.Compute the radial acceleration of the passengers when the car is at point C, which is at the end of a horizontal diameter.Compute the tangential acceleration of the passengers when the car is at point C, which is at the end of a horizontal diameter.

A roller-coaster car shown in the figure is pulled up to point 1 where it is released from rest. Assuming no friction, calculate the speed at point 2.Assuming no friction, calculate the speed at point 3.Assuming no friction, calculate the speed at point 4.

Suppose the roller-coaster car in the figure passes point 1 with a speed of 3.10 m/s . If the average force of friction is equal to 0.23 of its weight, with what speed will it reach point 2? The distance traveled is 55.0 m .

The small mass m sliding without friction along the looped track shown in the figure is to remain on the track at all times, even at the very top of the loop of radius r.If the actual release height is 4
h,
calculate the normal force exerted by the track at the bottom of the loop.If the actual release height is 4
h,
calculate the normal force exerted by the track at the top of the loop.If the actual release height is 6
h,
calculate the normal force exerted by the track after the block exits the loop onto the flat section.Determine the minimum release height h.

A 350 kg roller coaster starts from rest at point exttip{A}{A} and slides down the frictionless loop-the-loop shown in the accompanying figure.How fast is this roller coaster moving at point exttip{B}{B}?How hard does it press against the track at point exttip{B}{B}?

You are testing a new amusement park roller coaster with an empty car with a mass of
120 kg. One part of the track is a vertical loop with a radius of
12.0 m. At the bottom of the loop (point A) the car has a speed of 25.0 m/s and at the top of the loop (point B) it has speed of
8.00 m/s.
You may want to review (Pages 203 - 212).For related problemsolving
tips and strategies, you may
want to view a Video Tutor Solution of A vertical circle with friction.
As the car rolls from point A to point B, how much work is done by friction?

A pendulum is formed from a small ball of mass exttip{m}{m} on a string of length exttip{L}{L}. As the figure shows, a peg is height
exttip{h}{h}
=L/3
above the pendulums lowest point. From what minimum angle exttip{ heta }{theta} must the pendulum be released in order for the ball to go over the top of the peg without the string going slack?

A
95.0-kg
mail bag hangs by a vertical rope
3.9 m
long. A postal worker then displaces the bag to a position
2.9 m
sideways from its original position, always keeping the rope taut.What horizontal force is necessary to hold the bag in the new position?As the bag is moved to this position, how much work is done by the rope?As the bag is moved to this position, how much work is done by the worker?

A pendulum is made by tying a 510 g ball to a
47.0 cm -long string. The pendulum is pulled 24.0 to one side, then released.
You may want to review (Pages 234 - 238).What is the balls speed at the lowest point of its trajectory?To what angle does the pendulum swing on the other side?

A 20 kg child is on a swing that hangs from 3.2-m-long chains.You may want to review (Pages 234 - 238).For general problem-solving tips and strategies for this topic, you may want to view a Video Tutor Solution of Car rolling down a hill.What is her maximum speed if she swings out to a 46 angle?

Tarzan, in one tree, sights Jane in another tree. He grabs the end of a vine with length 20 m that makes an angle of 45 with the vertical, steps off his tree limb, and swings down and then up to Janes open arms. When he arrives, his vine makes an angle of 30 ^circ with the vertical.Calculate Tarzans speed just before he reaches Jane. You can ignore air resistance and the mass of the vine.

One end of a 6.00 m long rope is tied to the ceiling. A small rock with mass 0.500 kg is tied to the other end of the rope. The rock is released from rest at point A, where the rope makes an angle of 53° with the vertical. Point B is where the rock is at its lowest point and the rope is vertical. As the rock swings on the end of the rope from A to B, the work done on the rock by the tension force is
A) zero
B) +11.7 J
C) -11.7 J
D) +29.4 J
E) -29.4 J
F) None of the above answers

A ball is attached to a horizontal cord of length l whose other end is fixed .If the ball is released, what will be its speed at the lowest point of its path?A peg is located a distance h directly below the point of attachment of the cord. If h = 0.80l, what will be the speed of the ball when it reaches the top of its circular path about the peg?

A 0.325 kg potato is tied to a string with length 2.40 m , and the other end of the string is tied to a rigid support. The potato is held straight out horizontally from the point of support, with the string pulled taut, and is then released.What is the speed of the potato at the lowest point of its motion?What is the tension in the string at this point?

A small rock with mass 0.12 kg is fastened to a massless string with length 0.80 m to form a pendulum. The pendulum is swinging so as to make a maximum angle of 45 with the vertical. Air resistance is negligible.What is the speed of the rock when the string passes through the vertical position?What is the tension in the string when it makes an angle of 45 45 with the vertical?What is the tension in the string as it passes through the vertical?

On end of a 2.00 m long rope is tied to the ceiling. A small rock is tied to the other end of the rope. The rock is released from rest with the rope horizontal. As the rock is swinging through its lowest point, where the rope is vertical, the tension in the rope is T = 14.7 N. What is the mass of the rock?
A) 0.50 kg
B) 0.75 kg
C) 1.50 kg
D) 2.25 kg
E) 4.50 kg
F) None of the above answers

A 25.0 kg child plays on a swing having support ropes that are 2.10 m long. A friend pulls her back until the ropes are 43.0 from the vertical and releases her from rest.What is the potential energy for the child just as she is released, compared with the potential energy at the bottom of the swing?How fast will she be moving at the bottom of the swing?How much work does the tension in the ropes do as the child swings from the initial position to the bottom?

A sled starts from rest at the top of the frictionless, hemispherical, snow-covered hill shown in the figure. Find an expression for the sleds speed when it is at angle exttip{phi }{phi}.Use Newtons laws to find the maximum speed the sled can have at angle exttip{phi }{phi} without leaving the surface.At what angle exttip{phi _{
m max}}{phi_max} does the sled "fly off" the hill?

A 56.0 kg skateboarder wants to just make it to the upper edge of a "quarter pipe," a track that is one-quarter of a circle with a radius of 3.10 m .What speed does he need at the bottom?

A small rock of mass m is attached to a strong string and whirled in a vertical circle of radius R. When the rock is at the lowest point in its path, the tension in the string is five times the weight of the rock. At this point the speed of the rock isA) √2gRB) √3gRC) 2√gRD) 3√gRE) √5gRF) √6gRG) None of the above answers

A 240 kg roller coaster car starts from rest at point A and slides down the frictionless loop-the-loop shown in the accompanying figure. Point A is 25.0 m above the ground and point B is 12.0 m above the ground. The height of other points on the track are shown in the diagram.How fast is this roller coaster moving at point B?

A 240 kg roller coaster car starts from rest at point A and slides down the frictionless loop-the-loop shown in the accompanying figure. Point A is 25.0 m above the ground and point B is 12.0 m above the ground. The height of other points on the track are shown in the diagram.What is the magnitude of the normal force that the track exerts on it at point B?

A 5.00 m long light rope is tied to the ceiling. A steel ball with mass 2.00 kg is attached to the lower end of the rope. The ball is pulled to one side and released, and swings back and forth as a pendulum. As the ball passes through its lowest point, with the rope vertical, its speed is 6.00 m/s. As the ball swings through this point, what is the tension in the rope?(a) 34.0 N(b) 26.8 N(c) 19.6 N(d) 14.4 N(e) 12.4 N(f) 5.2 N(g) none of the above answers

One end of a 6.00 m long rope is tied to the ceiling. A small rock with mass 0.500 kg is tied to the other end of the rope. The rock is released from rest with the rope horizontal. What is the tension in the rope when the rock is swinging through its lowest point, where the rope is vertical?A) zeroB) 4.9 NC) 9.8 ND) 14.7 NE) 19.6 NF) None of the above answers

A roller-coaster car shown in Figure P8.72 is released from rest from a height h and then moves freely with negligible friction. The roller-coaster track includes a circular loop of radius R in a vertical plane.(a) First suppose the car barely makes it around the loop; at the top of the loop, the riders are upside down and feel weightless. Find the required height h of the release point above the bottom of the loop in terms of R.(b) Now assume the release point is at or above the minimum required height. Show that the normal force on the car at the bottom of the loop exceeds the normal force at the top of the loop by six times the car’s weight. The normal force on each rider follows the same rule. Such a large normal force is dangerous and very uncomfortable for the riders. Roller coasters are therefore not built with circular loops in vertical planes. Figure P6.17 (page 170) shows an actual design.

A block of mass m = 5.00 kg is released from point A and slides on the frictionless track shown in Figure P8.6. Determine(a) the block’s speed at points B and C and(b) the net work done by the gravitational force on the block as it moves from point A to point C.

A 1 000-kg roller coaster car is initially at the top of a rise, at point Ⓐ. It then moves 135 ft, at an angle of 40.0° below the horizontal, to a lower point Ⓑ. (a) Choose the car at point Ⓑ to be the zero configuration for gravitational potential energy of the roller coaster– Earth system. Find the potential energy of the system when the car is at points Ⓐ and Ⓑ, and the change in potential energy as the car moves between these points. (b) Repeat part (a), setting the zero configuration with the car at point Ⓐ.

A pendulum, comprising a light string of length L and a small sphere, swings in the vertical plane. The string hits a peg located a distance d below the point of suspension (Fig. P8.68).(a) Show that if the sphere is released from a height below that of the peg, it will return to this height after the string strikes the peg.(b) Show that if the pendulum is released from rest at the horizontal position (θ = 90°) and is to swing in a complete circle centered on the peg, the minimum value of d must be 3L/5.

Jane, whose mass is 50.0 kg, needs to swing across a river (having width D) filled with person-eating crocodiles to save Tarzan from danger. She must swing into a wind exerting constant horizontal force , on a vine having length L and initially making an angle θ with the vertical (Fig. P8.81). Take D = 50.0 m, F = 110 N, L = 40.0 m, and θ = 50.0°.(a) With what minimum speed must Jane begin her swing to just make it to the other side?(b) Once the rescue is complete, Tarzan and Jane must swing back across the river. With what minimum speed must they begin their swing? Assume Tarzan has a mass of 80.0 kg.

A ball of mass m = 300 g is connected by a strong string of length L = 80.0 cm to a pivot and held in place with the string vertical. A wind exerts constant force F to the right on the ball as shown in Figure P8.82. The ball is released from rest. The wind makes it swing up to attain maximum height H above its starting point before it swings down again.(a) Find H as a function of F. Evaluate H for(b) F = 1.00 N and(c) F = 10.0 N. How does H behave(d) as F approaches zero and(e) as F approaches infinity?(f) Now consider the equilibrium height of the ball with the wind blowing. Determine it as a function of F. Evaluate the equilibrium height for(g) F = 10 N and(h) F going to infinity.

A 400-N child is in a swing that is attached to a pair of ropes 2.00 m long. Find the gravitational potential energy of the child–Earth system relative to the child’s lowest position when (a) the ropes are horizontal, (b) the ropes make a 30.0° angle with the vertical, and (c) the child is at the bottom of the circular arc.

A skateboarder with his board can be modeled as a particle of mass 76.0 kg, located at his center of mass (which we will study in Chapter 9). As shown in Figure P8.49, the skateboarder starts from rest in a crouching position at one lip of a half-pipe (point A). The half-pipe is one half of a cylinder of radius 6.80 m with its axis horizontal. On his descent, the skateboarder moves without friction so that his center of mass moves through one-quarter of a circle of radius 6.30 m.(a) Find his speed at the bottom of the half-pipe (point B).(b) Immediately after passing point B, he stands up and raises his arms, lifting his center of mass from 0.500 m to 0.950 m above the concrete (point C). Next, the skateboarder glides upward with his center of mass moving in a quarter circle of radius 5.85 m. His body is horizontal when he passes point D, the far lip of the half-pipe. As he passes through point D, the speed of the skateboarder is 5.14 m/s. How much chemical potential energy in the body of the skateboarder was converted to mechanical energy in the skateboarder–Earth system when he stood up at point B?(c) How high above point D does he rise? Caution: Do not try this stunt yourself without the required skill and protective equipment.

A light, rigid rod is 77.0 cm long. Its top end is pivoted on a frictionless, horizontal axle. The rod hangs straight down at rest with a small, massive ball attached to its bottom end. You strike the ball, suddenly giving it a horizontal velocity so that it swings around in a full circle. What minimum speed at the bottom is required to make the ball go over the top of the circle?

A ball whirls around in a vertical circle at the end of a string. The other end of the string is fixed at the center of the circle. Assuming the total energy of the ball– Earth system remains constant, show that the tension in the string at the bottom is greater than the tension at the top by six times the ball’s weight.

A small block of mass m = 200 g is released from rest at point A along the horizontal diameter on the inside of a frictionless, hemispherical bowl of radius R = 30.0 cm (Fig. P8.43). Calculate(a) the gravitational potential energy of the block–Earth system when the block is at point A relative to point B,(b) the kinetic energy of the block at point B,(c) its speed at point B, and(d) its kinetic energy and the potential energy when the block is at point C