**tension**. Like any rope, these ropes cannot pull with an infinite amount of force. Sort of like people, the ropes break when they experience too much tension. The amount of tension force with which they pull is limited by their

**breaking strength**. The breaking strength is the maximum amount of tension with which the ropes can withstand. We assume in this activity that a rope breaks when the tension in it exceeds its breaking strength.

**Be On the Lookout for the Following**

There are slight differences from level to level. You need to be on the lookout for the differences. Success on this interactive requires that you pay attention to these differences. Specifically, pay attention to the following three level features:

**The number of ropes.**

The simplest case is when there is one rope. That one rope is providing all the support. If there are two or more ropes, then the weight of the bag and added gold coins is distributed equally among the multiple ropes.

**The angle at which the ropes are oriented.**

The simplest case is when the rope(s) are oriented vertically. The rope just pulls upward and all the force in the rope is used to support the weight of the bag and added gold coins. When the rope is at an angle, then it pulls both upward and horizontally. The angle must be used to determine the proportion of the total force that is directed upward in order to support the weight of the bag and coins.

**Whether the Mass or the Weight is given.**

The simplest case is when the weight of the bag and a coin is given. If the mass is given, then you must calculate the weight. See the section after the next.

**Accounting for the Bag**

The ropes must support the downward force of gravity on the bag and the added coins. The total mass is equal to the mass of the bag and the mass of __all__ the coins. Think of it this way ...

**m _{total} = N*m_{coin} + m_{bag}**

where N is the number of coins/

There is just one bag so you simply add it on to the mass of all the coins. Since there are likely more than one coin, the coin mass is determined by taking the mass of one coin and multiplying it by the number of coins (**N**). So each time you add a coin, the total mass increased because of the additional coin. That is, N increases by 1 each time you tap on the Add Coin button.

But what if the weight is given and not the mass? Nor sweat! It's the same rule ... only applied to weight. That is ...

_{Wtotal} = N*W_{coin} + W_{bag}

where N is the number of coins.

**Mass and Weight**

Mass and weight are often confused, so do be careful. Weight is a force; it's the force of gravity pulling downward on an object. And that's what we need to know. The weight of the bag and coins (in the unit Newton, abbreviated N) can be determined from their mass (in the unit kilogram, abbreviated kg). To do so, use the equation ...

**Weight = mass*g**

where **g** = gravitational field strength (in N/kg)

The value of g is either 9.8 N/kg or 10.0 N/kg. We left the decision up to you when you started the Interactive. Don't remember which value you selected? No worries. It's not going to affect the outcome of the level so just use the easier-to-work-with 10.0 N/kg. If that goes against your religion, then just use 9.8 N/kg and feel like you're a little *purer* than those estimators. Either value you use is going to lead to the same value for the maximum number of coins that can be added to the bag.

**It's All About the Vertical**

There must be enough up force to balance the down force. The down force is gravity. The up force is provided by the tension force in the rope. If the rope if vertical, then all the tension in the rope is directed upward. Since you know the breaking strength, then you know the maximum amount of upward pull for situations in which the rope is vertical. You just must make sure you don't add too many coins such that the total down force (from the weight of the coins and bag) does not exceed this maximum amount of upward pull. If it doesn't, then *you lose*.

If the rope is at an angle (not vertical), then it does more than just pull upward. It also pulls horizontally. The tension is directed along the angle ... that is, along the line that the rope makes. And so you must account of the angle by using it to relate the upward pull to the tension that is diagonal. A triangle usually helps with this. So does the next section ...

**Accounting for the Angle**

When the rope is *angled* (i.e., not completely vertical), then right-angle trigonometry must be used to relate the upward pull of the rope to the total pull of the rope. As mentioned in the previous section, an angled rope pulls both vertically and horizontally, The vertical pull is what balances the downward force of gravity. The horizontal pull doesn't really matter. It's just there because angled ropes pull that direction. To relate the angled pull (tension) to the vertical component of the pull (the part that balances the weight of the bag and coins, use the sine function. The sine function relates the side opposite the angle to the hypotenuse of the triangle. As demonstrated in the graphic at the right, the relationship is ...

**sine(Θ) = vertical force/tension force**

As mentioned, the upper limit on tension is the breaking strength. So it reasons that the upper limit on the vertical force the breaking strength multiplied by the sine of the angle

**F _{y-max} = (breaking strength)•sine(Θ)**

**Accounting for the Number of Ropes**

The ropes supply the vertical force to balance the downward force of gravity. But if there are more than one rope, then this vertical pull is distributed evenly among the ropes. So the amount of vertical pull in a single rope in a 2-rope situation is one-half the weight of the bag and gold coins. Similarly, the amount of vertical pull in a single rope in a 3-rope situations is one-third the weight of the bag and gold coins. And since the tension in any single rope is related to the vertical pull by the sine of the angle (see previous section), then increasing the number of ropes decreases the tension force by 1/R factor where **R** is the number of ropes.

**Putting the Parts Together (a.k.a., a Pep Talk)**

A complex problem is difficult because there are many parts to the problem to account for. Each part may be simple enough to understand and if there was a problem with just that one part, most students could *nail it*. For instance, most students can determine the weight of the bag from its mass. That's just one of the parts to this complex problem. When those other parts are added in, sometimes students fail to accurately determine (or forget to determine) the weight from the mass. So in this problem, you will have to pay attention to all parts, especially as you progress to the higher difficulty levels.

The parts to this problem are described in the sections above. Take some time to digest each of these section. Reflect on what you might not be accounting for. Take your time as you do your analysis. Get out a sheet of paper and write things down. Diagram. Discuss matters with a friend (if you have smart friends) but don't have them do your work for you. Make sure they listen to your thinking and reasoning.

**"Ain't There An Equation for this Stuff?" (a.k.a. Short cuts)**

Question: Who doesn't like short cuts? Answer: teachers don't. It's not that teachers don't want you (the student) to have to work for a long time. It's just that short cuts can often be a means of accomplishing a task without learning the *ingredients* to completing the task. The task is to determine the maximum number of coins that can be added to the bag without breaking the rope. The task exists in order for you to learn some physics concepts. If you were handed an equation, the task would become a plug-and-chug math problem and not a *learn and use some Physics problem*.

The fact is that there is an equation for this stuff. In fact, most Physics types would even call it a pretty cool equation. But it's only cool if you can derive it yourself. So go ahead and give it a try. When you're done, you have your short cut - the equation for all this stuff. And you will be admitted into the "club" of Physics types. But don't tell your friends. Its there job to derive it for themselves. That's the club rules.

Try this link to The Physics Classroom Tutorial for more help:

Equilibrium

Our video on the topic is found here: https://youtu.be/kYrZfkOyIfU