Chapter 2: Force, motion, and gravity (C6008135)

 #toc { border: 1px solid #bba; background-color: #f7f8ff; padding: 1em; font-size: 90%; text-align: center; } #toc-header { display: inline; padding: 0; font-size: 100%; font-weight: bold; } #toc ul { list-style-type: none; margin-left: 0; padding-left: 0; text-align: left; } .toc2 { margin-left: 1em; } .toc3 { margin-left: 2em; } .toc4 { margin-left: 3em; } Last modified: 799d agoWord count: 6,410 wordsLegend: Key principles // Storyline

1 Center of mass

Center of mass is the point at which mass is uniformly weighted. It can be thought of as the point, which can be held, and the object able to balance. This point [if pushed] has no moment [of inertia] (aka rotational inertia, refer ), as there is no â€œspinâ€ generated.

â€œIn physics, we try to simplify things,â€ Jamie said.

â€œI like easy ,â€ Mandy answered.

"Yeah you are ," Emma snided.

â€œIt is therefore convenient to consider any mass as a single particle, with a mass concentrated at a single point, known as the center of mass,â€ Jamie replied.

It can be found even in asymmetrical objects, and can be at a point where there is no mass [outside an object].

Force applied at the center of mass will cause translational motion, without causing rotation/moment. If a mass has a constant density, its center of mass coincides with its geometric center.

â€œFor instance, your center of mass, Mandy, is not your geometric center, because different body parts weigh differently,â€ Jamie said, â€œrather, itâ€™s a point inside your body, that is closer to your heart than your waist, because your upper body is more packed.â€

â€œIt really brings a new perspective to â€˜finding your center,â€™â€ Mandy giggled.

Center of gravity is where gravity is also considered. If there is uniform gravity, the center of gravity and center of mass coincide.

 Frequently asked questions What is center of mass?The point at which the mass can be balanced at. That's because at that point, there is no moment.What is moment?Moment is spin. So it's a force applied to an object that is not at the center of mass.How doeds moment differ from torque?It doesn't. They're the same thing.What was inertia again? And how does this relate to moment?Inertia will soon be discussed. It is Newton's 1st law, which is that moments in motion tend to remain in motion. Moment is a rotational type of inertia. That is, it is that an object currently rotating, is likely to continue to rotate (unless opposed).Is there a center of mass for items that aren't symmetrical? Like everyday items, that aren't theoretical cubes and spheres?Yes. Think about it. Otherwise, how could steadicams work? Weights are used to balance the video camera on a ball hinge, to provide smooth motion.Is there a center of mass if there's nothing there? Like in a hollow sphere?Yes. Although we visualize center of mass at a place where you can hold on to using your finger, this can be theoretically where there is nothing.Why do we say that force at the center of mass causes translational motion? I thought it causes moment?No. Force at the center causes the whole thing to move. That's translational motion. Force NOT at the center of mass causes the thing to spin. That's moment. Think about it. You push someone at their chest, and they fall backwards. But if you push someone at the shoulder, they spin.Is the geometric center always the center of mass? It seems like it.Usually, unless the object does not have an equal density throughout.What is the difference between center of gravity and center of mass? Once again, they seem like the same thing.Again, they are usually identical. It is only different if there is unequal gravity throughout.

2 Newton's 1st law

Newtonâ€™s 1st law is the law of inertia, that an object tends to remain at its present speed/state of motion [or non-motion], unless acted on by a force.

â€œâ€˜By holy marriage: when and where and how; We met, we wooâ€™d and made exchange of vow,â€™ Jamie leapt to say, â€˜Iâ€™ll tell thee as we pass; but this I pray, That thou consent to marry us to-day.â€™â€

â€œ-Obviously that Shakespearean moment didnâ€™t occur.â€

â€œThatâ€™s sort of like Newtonâ€™s 1st law,â€ Mandy continued, â€œyou and the Sophster were free falling, and brother, did it continue in that present state of motion .â€

Translational inertia is inertia other than rotational inertia, which is the tendency to continue rotating/spinning at its present angular velocity.

Mass is a measure of an objectâ€™s translational inertia. For example, a $100kg$ person is twice as resistant to changes in velocity, than a $50kg$ person. Moment/torque, which is $M=f.d$, measures rotational inertia. This means increases in mass [and thus force] or distance, increases rotational inertia. Whereas translational motion relates to movement of an object in its entirety, rotational motion involves the spinning of the object.

â€œIf you push someone against their center of mass, at their chest, theyâ€™ll be pushed back,â€ Mandy said, â€œhowever, if you push them against their arm, theyâ€™re more likely to spin.â€

 Frequently asked questions What is Newton's 1st law?It is the law of inertia.What's the law of inertia?Inertia is the tendency of an object to remain in its present state of motion [or lack of motion], unless opposed.So as we said before, there's translational and rotational inertia. Come again, what's the difference between them?Translational inertia is the tendency to move in a straight line. Rotational inertia is the tendency to rotate.And rotational inertia is a synonym for moment and torque. They mean the same thing?Yep, you got it !How is translational inertia measured?Mass.Well that makes sense, the heavier the weight, the harder it is to start it moving, but once in motion, the harder it is to stop.How about rotational inertia? How is that measured?Moment.I know you said moment is the tendency to rotate, but how is it calculated?With the formula $M=f.d$, where f is force, and d is distance.It's quite obvious increasing force will increase the tendency to rotate. How about distance?Distance is NOT how far the point itself is shifted. [This is automatically factored into the "force".] It is measured as the distance away from the center [from which the point is pushed].

3 Newton's 2nd law

Newtonâ€™s 2nd law is $F=m.a$, which explains what occurs when a force is applied to an object. It also helps mathematically depict Newtonâ€™s 1st law, which is that where there is no force, $F=0$, meaning $a=0$, meaning that there is no acceleration, and therefore the object will neither speed up nor slow down. Rather, it stays at rest. When acted on by a force, an object will accelerate at a rate inversely proportional to its mass, such that lighter objects will accelerate more, and vice versa.

 Frequently asked questions What is Newton's 2nd law?It's $F=ma$.Uh, that's a formula... What does it mean??It explains what happens when a force is applied to an object.How does Newton's 2nd law relate to Newton's 1st law?Newton's 1st law was that unless opposed, an object continues in its present state of motion. Since $F=ma$, if $F$ is 0, so is $a$. So again, unless acted on by a force, acceleration (i.e. change of motion) does not occur.What else does F=ma tell us?[Other than the fact that acceleration is proportional to force,] the formula also tells us that acceleration is inversely proportional to mass. This means that lighter masses accelerate more easily, and heavier masses find it more difficult to accelerate (i.e. change the speed of their motion). That also makes sense.

4 Newton's 3rd law

Newtonâ€™s 3rd law is that every force has an equal but opposite force [and those forces act on opposite bodies].

â€œJust one problem ,â€ Mandy raised, â€œif every force has an equal-opposite force, how does any work get done?â€

â€œFor example, if you push a soda can, the soda can pushes against you with an equal but opposite force.â€

â€œThe answer lies in the system considered,â€ Jamie replied, â€œconsidering your hand as a system, the soda can does put force against your hand, but in turn, is supported by forces in the arm, etc.â€

â€œOh!â€ Mandy interjected, â€œbut considering the soda can as a system, there is only one force acting on it, which causes it to move .â€

 Frequently asked questions What is Newton's 3rd law?Every force has an equal and opposite force.So when someone punches you on the arm, they could say that your arm punched their fist?From a theoretical standpoint, I guess so.

5 Concept of a field

Field is a concept describing force (per unit of its underlying reason, whether it be due to gravitational or electric force). Field lines can be used to show the direction a test particle (mass for gravity, and a positively charged particle for electricity) will move, due to the force exerted by the field at the relevant given points. Note that field lines cannot cross, as force (at any given point) is only in one direction. Closer field lines indicate stronger fields.

 Frequently asked questions What is a field?A theoretical region which has forces (of a particular type, such as gravity, electricity, or magnetic) operating within this region.What are field lines?They show what happens to a test particle when acted on by the relevant force (so once again, gravity, electricity, or magnetic).What is this particle?Depends on what sort of field you're talking about. For gravity, a test mass. For electricity, a test positive charge.Field lines can't cross. Why?If a test particle is pushed slightly to the left, and slightly to the right... The "net" push is straight forward. So field lines that cross will interact. That's why they can't cross. A test particle can't go in 2 separate directions.How do field lines work?They come out at 90 degrees (as they can't cross).Also, closer field lines indicate a stronger field. That's just something to know, by definition.

6 Law of gravitation

The law of gravitation is $F=\dfrac{GMm}{r^2}$. This states that gravitational force is directly proportional to the masses of either object, and inversely proportional to the distance between their centers, squared. This gravitational force is a mutual force between both objects. $G$ is the Gravitational constant, which is $6.67*10^{-11}$.

The gravitational law is a type of inverse square law, which is that gravitational force is inversely proportional to distance squared.

â€œJamie, I feel your gravitational pull,â€ Mandy giggled.

â€œStepping back,â€ Jamie winked, â€œwhat happens to force?â€

â€œBecause radius is doubled, the related effect will be quadrupled,â€ Mandy replied, â€œhowever, since force and radius are inversely related, rather than quadrupled, gravitational force is quartered !â€

â€œOh poor diddums,â€ Blaire remarked.

Since $W=F.d$ (introduced ), $W=\dfrac{GMm}{r^2}.r$, or simplifying, $E=W=\dfrac{GMm}{r}$. Because work has the same units as energy (introduced ), this is gravitational potential energy.

 Frequently asked questions What is the law of gravity?It is $F=\dfrac{GMm}{r^2}$.Why is it called an inverse square law?The distance between the two objects (r) is not proportional to force. Rather, it is inversely proportional - but even then, it's not inversely proportional per se. It needs to be squared.Questions commonly like to ask, what happens to force when you double distance. Because it's an inverse square law, you need to understand that force isn't doubled. "Inverse" means that it's at least halved. However, because there's a "square", it's not only halved, it's actually $\dfrac{1}{2}^2=\dfrac{1}{4}$, quartered.Do I need to memorize that?It's so straightforward, that most people do. It tells you a lot. It tells you what happens to force, when you alter the masses of either gravitational objects, or the distance between them.Is gravitational force a scalar or a vector?It's a vector, but the direction, is towards each another. That's why we say it's a "mutual force". So the force of one on the other, is in the opposite direction as the force of the other on the former.What is "G"?The universal gravitational constant. It's unique because it's the same throughout the universe.What is gravitational potential energy?It is the potential energy that is caused by raising an object away from the earth's surface, to a certain distance.

Gravitational fields explain how particles distort space time via their mass. Gravitational field is defined as $g=\dfrac{F}{m}$, where $F$ is gravitational force, $m$ is the mass of a test particle, and gravitational field has the units $m/s^2$ or $N/kg$. Since $F=\dfrac{GMm}{r^2}$, in-substituting, $g=\dfrac{GMm}{r^2}.\dfrac{1}{m}=\dfrac{GM}{r^2}$. Note therefore that although it is commonly said that $g=9.8$, this is only an approximation of gravity close to the surface of earth. Given that we know constants $G=6.67*10^{-11}$, $M=5.97 \times 10^24 kg$, and $r=6.37 \times 10^6$, we can calculate $g \approx 9.8m/s^2$ using the gravitational field equation. Gravitational field is visualized by forces that would act on a test mass particle. Accordingly, force lines spring in all directions towards (the center of) a mass. This shows that a mass travels towards the center of another mass, due to the latterâ€™s gravitational force.

â€œUnlike biology, physics does not have a fine tuning argument,â€ Mandy started, â€œso the Teleological argument: God makes sense of the extraordinarily narrow range of life-permitting values the universe is tuned for.â€

â€œThere are constants in nature, like the Gravitational constant, which aren't determined by the laws of nature,â€ Blaire continued, â€œAdditionally, there are variables, like the amount of entropy or the balance between matter and anti-matter, which have been arbitrarily inserted as initial conditions.â€

â€œNot only must each quantity be exquisitely fine-tuned, but so must their ratios to one another,â€ Mandy replied, â€œThe remarkable fine tuning has forced the M-theory to create around 10^500 multiverses, to render the observed values even reasonably attainable.â€

â€œHave you ever noticed the unreasonable effectiveness of mathematics?â€ Mandy continued, â€œFor example, Peter Higgs predicted the fundamental particle Higgs boson through mathematics, 30 years before it was experimentally observable. Itâ€™s the language of nature!â€

 Frequently asked questions What is a gravitational field?A theoretical region which has gravitational force operating within the region.So gravity is $\dfrac{GM}{r^2}$?Yes.So it's not 9.8?No. That's actually only an approximation at a certain "distance" away from the earth's center.So how are gravitational field lines drawn?Visualize how gravity will impact on a test mass.So that's towards the center of the larger object?Yep, you got it . So on earth, that's down to the center of earth. So on the opposite side of the earth, gravity acts in the opposite direction!

7 Uniform circular motion

Uniform circular motion is motion in a circular path. Although speed is constant, velocity isnâ€™t. This is because although the magnitude of speed doesnâ€™t change, the direction of velocity does [as weâ€™re going round and round in circles]. When travelling in a circle, the direction in which an object travels on top of the circle, is evidently going to be in the opposite direction as at the bottom of the circle. As there is change in velocity, there is acceleration, known as centripetal acceleration. This acceleration points towards the center of the path. Thus, the velocity and acceleration act at $90^\{circ}$ to each other. Radius of curvature is the radius of the circle that would be created if an object moving on a curve, were to continue turning. Thus, a sharper turn has a smaller radius of curvature.

Objects orbiting earth are really in continuous free fall, thereby making all contents within the station seem to float. Orbiting objects never hit earth, because their high velocity (perpendicular to gravitational pull) causes it to achieve circular motion with a curvature similar to the curvature of the earth.

 Frequently asked questions What is uniform circular motion?Motion in a circular path.But why call it "uniform" motion? What exactly is "uniform"?Speed. Speed is uniform.So there's no acceleration?No. That's where it becomes complicated! Acceleration is NOT change in "speed". It's change in "velocity". Although there is no change in speed, there is change in velocity.But how? There's no change in speed...?That's where it's useful to remember the distinction between speed and velocity. Velocity has direction, whereas speed doesn't. So although speed is constant, there is change in direction. Moving in a circle, means that at any instance, the direction at which the "arrow" [of where the object is moving] is constantly changing. For instance, if moving clockwise, at the top, the arrow may be to the right; but at the bottom, the arrow will be to the left.So there is acceleration. That's... what centripetal acceleration is?Yep. That's centripetal acceleration. Acceleration that causes... centripetal motion!What is radius of curvature?Radius of the circle that would be created if the curve were to continue turning.Why do you say "if" it were to continue turning? Doesn't it?In cases like a satellite, it does. But sometimes it doesn't, and in places you might not expect. Think about a car making a corner turn. If you extend this turn, you could create a theoretical circle.I see, so that's why a sharper turn would create a...Smaller circle. Try it! Complete the circle.So objects in orbit "keep missing earth". HUH? What does this mean?So going back to our example of the satellite, you have motion at right angles with the center of earth, and the centripetal force of earth's gravity pulling towards the center of the earth. But the object "keeps missing earth". Although it's constantly falling due to a pull downwards (a little under 9.8m/s^2, because of the great distance away from earth's center), it continually misses, because of it's fast speeds moving at "right angles" with relation to the earth's center.

8 Centripetal force

As there is centripetal acceleration in circular motion, there is a related force, known as centripetal force, based upon Newtonâ€™s 2nd law $F=ma$. Because centripetal acceleration is defined as $a=\dfrac{v^2}{r}$, based on Newtonâ€™s 2nd law $F=ma$, substituting, $F=\dfrac{mv^2}{r}$.

For planets orbiting, gravitational force is $F=\dfrac{GMm}{r^2}$. As centripetal force is caused by gravity, gravitational force can be equated with centripetal force, meaning $\dfrac{GMm}{r^2}=\dfrac{mv^2}{r}$, or reducing, $v^2=\dfrac{GM}{r}$. This means that the velocity of an orbiting planet is independent of the mass of orbiting planet itself.

 Frequently asked questions What is centripetal force?Through $F=ma$, we know that every acceleration has an associated force. Centripetal force acts towards the center of the circle. This makes sense, because, if whilst travelling at right angles to the center of the circle, BUT, whilst that occurs, there is a "push" towards the center of the circle, this will cause circular motion!That's hard to visualize...Not really. Think about satellites. They have motion at right angles to earth. Gravity is it's centripetal force. Or, a ball on a string. The ball moves at right angles to where you hold on to the string. And the string is providing a centripetal force.So is why is centripetal force $\dfrac{mv^2}{r}$? Isn't it $\dfrac{ma}$?It's both. They're the same thing. The thing that's linking them is centripetal acceleration, which is $a=\dfrac{v^2}{r}$

9 Weight

Weight is the force of a mass as a result of gravity/gravitational acceleration. As $F=ma$, weight is the force, and acceleration is due to gravity, this formula can be rewritten as $W=mg$. Although mass of an object is the same on whatever planet, as weight is dependent on gravitational acceleration which is different on different planets, weight is different on different planets.

â€œSounds like the perfect solution for those weight-loss obsessed girls,â€ Mandy giggled, â€œSend them to Mars .â€

â€œFor example, Selena Gomez is 50kg, meaning her mass $m=50$. Her weight on Earth is thus $F=50 \times 10=500N down$.â€

â€œIn contrast, Gravitational acceleration on Mars is $a=3.7m/s^2$, meaning Selenaâ€™s weight on Mars is thus $F=50 \times 3.7=185N down$, a lot less than on Earth!â€

 Frequently asked questions What is weight?Weight is a force. It is the force that results from gravitational acceleration on a mass.Oh, so weight and mass is different?!Yeah, it is. Your mass is the same on any planet. Your weight however, differs.So in fact a so-called "weight loss" program, could include going to a planet with less gravity ?Yep... But your mass would still be the same - and it's that that affects appearance

10 Friction

Forces due to two contiguous (contacting) surfaces, can be either friction or normal force. Friction is the contact force resisting sliding between contiguous surfaces, and is parallel to the contiguous surface. Friction is not always in the opposite direction of motion (per se). For example, when sliding a mobile phone away from you, motion of the phone is away, and friction is also away. This is because friction resists sliding between your hand and the phone, thereby moving the phone away. There are two types, including:

The maximum static friction is usually greater than the kinetic friction, which makes sense because it usually takes more force to get an object to start moving (from static), than it is whilst moving (whilst kinetic).

 Frequently asked questions What is friction?The rubbing of 2 surfaces against another.Â

11 Inclined plane

Machines are devices that provide leverage. Leverage reduces force required. Keeping in mind $W=F \cdot d$ (see ), machines do not reduce work. In fact, non-ideal machines increase work, as they are not entirely efficient. Given work is constant, force is inversely proportional with displacement, meaning reducing force requires increasing displacement. Some classical machines include:

Inclined plane is a surface that lies at an angle relative to the ground, used to aid raising or lowering a load. For example, lifting a $5kg$ load $2m$ high, $F=m.a=5 \times 10=50$, and $W=F.d=50 \times 2=100J$, so 100J of work must be done. Remember that machines do not change work. So, if the same load is lifted up a ramp that is 4m long, the force required is $F=\dfrac{W}{d}=\dfrac{100}{4}=25N$. This makes sense because as force and distance is inversely proportional, doubling the distance, will result in half the force required.

Even though Sophie was only an acquaintance of Jamieâ€™s, he was madly in love with her. He already played out how awesome her first name with his last name, â€˜Sophie Swiftâ€™, would sound. Alliteration, right? In his mind, he already photoshopped an image of how suave his emo black hair would look offsetting her blonde. And letâ€™s not forget about the kids .

But to say that Jamie would need to hike up the hill to find this girl was an understatement. She never came back to Pacific Coast Baptist. And he could sit in the same seat she sat in, but she was all but memories. And at that, increasing aching memories. Jamie would require, quite literally, a miracle.

â€œWho knows Godâ€™s in the business of miracles !!â€ Mandy declared. And that is exactly what happened.

A diagram can also be drawn for an inclined plane. Gravity is the force that acts towards the center of earth. Normal force is the contact force perpendicular to the contiguous surface. Inclined plane questions are usually assume frictionless.

Assuming there is no friction: as centripetal force is the force that directs towards the center of curvature, as the normal force is perpendicular, the only force that can contribute towards centripetal force is a portion of gravitational force. This means centripetal force is $F=mg.sin(\theta)$. Thus, centripetal acceleration is $a=\dfrac{F}{m}=g.sin(\theta)$. In the example , the force down the inclined plane $F=mg.sin(\theta)$ can in fact be used, noting that $sin(\theta)=\dfrac{O}{H}=\dfrac{2}{4}$. Therefore, $F=5\times 10\times \dfrac{2}{4}=25N$. This is the same figure calculated  as the force required to lift up the ramp, quod erat demonstrandum.

 Formative learning activity Maps to RK2.11 What is an inclined plane?

12 Pulley systems

Pulley involves using a wheel on an axle to support movement of a single continuous rope to transmit tension forces, and hence lift loads. A wheel is used to reduce the effects of friction. Note that the single continuous rope has the same tension throughout. Tension is a pulling force exerted by a rope on another object. The rope is assumed to have negligible mass, meaning that acceleration $a=\dfrac{F}{m}=\dfrac{F}{0}$ is infinite. This means that the tension on either end of a massless string must be equal. For example, a mass $m$ hanging from a massless string, will have the tension $mg$. Analogously, the ceiling will also apply the force $mg$ on to the rope. This demonstrates that the rope transfers the force $mg$  from the ceiling to the mass, and vice versa.

For example, if a rope has both ends tied to a roof (one which can be untied so that it can later be raised), and at the loop of the rope (closest to the floor), there is a wheel, which is in turn attached to a force, because the ropes are in equilibrium, the two ends of the rope (added together) must be equal and opposite to the force of the mass. Remember from  that the mass has force $F=mgh$. Notice that each end of the rope therefore has force $\dfrac{1}{2}.F$, as they both add up to $F$ (to balance out the equal and opposite force of gravity, down). Therefore, if one end of the rope were to be lifted, only the force $\dfrac{1}{2}.F$, or half the force, is required. Notice also that if the mass were to be lifted distance $\dfrac{d}{2}$  (half the distance of the rope), because the wheel is going to roll along the rope, the raising distance required to lift the mass that distance, will be $d$.

Little did Jamie and Sophie know that, whilst each were studying full time (law and medicine), they were both studying part time bible college. But they soon did, as they landed themselves in the same classroom.

â€œHey,â€ Jamie remarked, as he grabbed a seat next to Sophie, â€œits Sophie, right?â€ Jamie continued, before the lecturer proceeded to quieten the class.

â€œHELLO,â€ Sophie scribbled on a piece of paper.

Even whilst the lecturer was talking, the two at times rattled quietly, before switching back to pen and paper.

â€œDo you think Iâ€™m hot?â€ Jamie eventually wrote.

â€œYes,â€ Sophie wrote without much thought. Then, thinking tactfully, she scribbled out â€˜hotâ€™, causing a temporary feeling of angst in Jamie, before drawing two arrows to two new phrases, â€˜Good personalityâ€™, and â€˜Good looksâ€™.

â€œDo you believe God created the world ?â€ Jamie asked, as the pair stepped out to lunch break.

â€œHmm,â€ Sophie replied, â€œThatâ€™s a good question, because I think there are some people who think that the world was created in seven days, and others who think that-â€œ

â€œSophie !!â€ Jamie interrupted, â€œIâ€™m trying to tell you a joke ! -Do you believe God created the world?â€

â€œWell â€¦ yes,â€ Sophie replied.

â€œâ€™cause He was a massive show off creating you !!â€ Jamie giggled, Sophie smiling awkwardly .

â€œYou two clearly hit it off with a bang,â€ Mandy giggled.

Hookeâ€™s law is $F=-k\Delta x$, which states that the displacement of a spring (that doesn't exceed the material's elastic limit) is directly related to the load applied to it, for materials that obey this law. The constant $k$ is the spring constant, which is related to the spring material. $\Delta x$ is the extent of deformity, for example how far a spring is compressed or stretched from its rest position. The negative indicates the force exerted by the spring is in the opposite direction of displacement. For example, a spring compressed will bounce back out, and vice versa. It thus restores the system to equilibrium.

 Formative learning activity Maps to RK2.12 What is a pulley system?

13 Force

Based on Newtonâ€™s 1st law, force is an influence that moves an object. Based on Newtonâ€™s 2nd law $F=ma$ , force is directly proportional to mass and acceleration. As there is force when there is acceleration, any phenomena that causes a mass to accelerate is a force, including gravitational force, electrical force or magnetic force. Tension, friction, air resistance and normal forces are magnetic forces acting at an atomic level between contacting surfaces.

 Formative learning activity Maps to RK2.13 What is force? What are examples of force?

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