Mechanics Problem 2: Friction of Box on an Accelerating Truck

A flatbed truck carries a load on its flat bed as shown in the picture above. The load is tied to the back of the truck cab with a steel cable to prevent it from sliding. Schematically it would look like below.

If we identify all forces acting on the load, then the free body diagram, whose purpose is to show these forces, would look like below.

This problem asks what should be the maximum acceleration of the truck as it accelerates from rest if we want to prevent the load from sliding, given that the load is being held by the friction force μ_s N, where N is the normal force and is equal to the load's weight m g, and the cable tension T.

1. Let us scrutinize this free body diagram. The first question is why the static friction force points to the right. The answer is conservation of momentum. Momentum p is equal to m v, where m is the object's mass and v is its velocity. The initial momentum is zero since it is initially at rest (v = 0). As the truck accelerates, it wants to conserve the zero momentum,

0 = m_truck v_truck + m_load v_load.

Since the truck velocity v_truck points to the right due to the acceleration a to the right (see the above diagram), the load tends to move to the left, i.e., v_load points to the left. Friction opposes the motion, i.e., the velocity; therefore, the friction force points to the right. The cable tension T also points to the right since it serves to prevent the load from sliding.

2. The sliding motion of the load represents a relative motion the load has with respect to the flat bed. The load accelerates with respect to the truck which is also accelerating. We say that the load moves in a noninertial frame. The Newton's second law, F = m a, does not work in a noninertial frame. F = m a has to be anchored in an inertial frame. The inertial frame is obtained when we regard the acceleration of the load is equal to the sum of the acceleration of the sliding motion and the acceleration of the truck,

a_load = a_truck + a_sliding.

Thus, the Newton's second law now reads

T + μ_s m g = m (a_truck + a_sliding).

3. We do not want the load to slide; thus, a_sliding = 0. The maximum acceleration is equal to

a_truck = T/m + μ_s g,

where T is maximum when it is equal to the tensile strength of the steel cable, which is about 400 MPa. For a cable with a cross section area of 0.25 cm², we get a maximum T of 10,000 N. Typical value of static friction coefficient is about 0.5, so that μ_s g = 4.9 m/s². For a 1000 kg load, we get T/m = 10 m/s². Hence, the maximum acceleration for these data is 14.9 m/s². 100 km/h = 27.8 m/s, so 14.9 m/s² is thus quite a high acceleration. But for a really heavy load, say a 10,000 kg load, the cable can now only bear 1 m/s² acceleration and in this case, the friction force becomes a lot more important. As a rule, therefore, the heavier the load is, the more it has to rely on friction to prevent the load from sliding.

Mechanics Problem 1: Pulley System

I want to start a blog series which focuses on applications of mechanics. I have been giving a math-physics workshop to high school teachers in the past two months. It is a problem-driven workshop, where concepts are discussed through problems. I am turning the concept of learning around. Usually, concepts are laid out first - laboriously - and then examples are given.

I find through this workshop that by working out problems my students - the high school teachers - understand the concepts better. I want to share this idea with you. If you find this blog series interesting, give me a shout. Thank you.

The diagram above shows a sytem of two pulleys, which is used to lift block B. The person who uses this pulley system exerts a force P to lift the block B. The two pulleys are connected by the blue cable that runs from A to C to D ... E, F and all the way to G.

Through mechanics analysis, you will find that this pulley system makes lifting B easier. P is only 50% of the weight of B. Directly lifting weight B without pulleys would mean requiring exerting a 100% of the weight B.

Our analysis relies on a formula called Newton's laws of motion, in particular the so-called Newton's second law. It states that an acceleration a of a body is equal to the sum of all forces T acting on the body divided by its mass m. That is, T = m a.

1. Let us apply T = m a to our problem. The blue cable and pulleys are assumed massless and rigid, so that the force P is transferred to the block B without dissipation. The cord that hangs the block B to the lower pulley transfers a force 2P since the cable wraps around the lower pulley and each cable section carries a force P. T = m a for the block B thus gives

2P - mg = m a.

The weight of block B, i.e., m g, is directed downward, while the force 2P is directed upward. This is the reason why they have opposite signs in the previous equation. Thus,

P = m (a + g)/2.

When the block B starts to be lifted, its acceleration a is not zero. But when it moves upward with a constant speed, a = 0. At this situation, we have P = m g/2. This shows that P equals 50% of the B's weight.

2. The pulley system thus provides a mechanical advantage. To lift B, one pulls only 50% of its weight. How is that possible? Is there a catch? The answer is yes. The block B moves 200% slower than it would if it were lifted directly without pulleys. You will spend the same amount of energy to lift the block over a certain distance regardless whether you exert a force 50% of its weight, or a 100%. The energy you spend per second is lower due to the slower upward motion of B, which makes the lifting work easier.

The key to understand this aspect is that the cable portion connected to the force P travels twice as far as the portion connected to the block. That is, x_A = -2x_B, where the negative sign means A and B move in opposite directions. This fact comes straight from the fact that the length of the cable is constant

x_A + 2x_B = constant,

so that by differentiating with time we get their velocity relation,

v_A + 2v_B = 0,

and another differentiation yields their acceleration relation,

a_A + 2a_B = 0.

These kinematic relations are constraints to the motion and arise independently of the dynamics dictated by the Newton's second law.

3. The work done is equal to force times distance travelled. Thus, when the block B moves at a constant speed

Work = P x_A = (m g/2)(2 x_B) = m g x_B.

This shows that the work needed to lift the block B remains the same: m g x_B. The conservation of energy is still obeyed. The beauty of the pulley system, however, is that it makes lifting it easier since per second the force needed is only 50%. The trade off is the distance travelled at A becomes twice as long.

Canadian Cost of Living

There are things I miss from Calgary: the snow-capped Rockies (I really miss backpacking in Kananaskis, Yoho, and Banff hiking trails!), clear water of Bow River and its running/bike paths, and libraries. My son would add Five Guys burgers and Banzai. It is fair to say I feel homesick while already at home now.

But there are things I don't miss as well. Cellphone rates are way too high in Canada. I pay $20/month for my smartphone in Jakarta, and yes, that includes everything you can imagine getting from a smartphone package. I paid $100/month in Calgary for an identical package, not including additional airtime charges.

House heating cost is not cheap in Calgary and Canadian winter is cold and long. It can cost $300/month to heat a house. Utilities - such as electricity, water, garbage removal, and sewage charge - add another $300/month. These costs get reduced when we moved to an apartment though as they got absorbed into a rental fee. In fact, I like the simplicity and frugality of living in a small dwelling space.

Housing cost in Canada is very expensive compared to Indonesia's. Roughly, the ratio of average house price to average salary is twice as high in Canada. We get what we pay. The road and residential infrastructure in Canada is indeed a lot better. The question is simply whether one is willing to pay for it.

The food cost in Canada is a lot more expensive than Indonesia's. It is about 500% more expensive. In my previous blog, I talked about paying $6 lunch for 6 people in Jakarta, while in Calgary I had to pay $10 for lunch. I believe this huge difference comes from stricter food and health regulations, building code regulations, and more expensive labour and transport cost.  And again, the question is simply whether it is too expensive for its benefits. We get what we pay, but in Canada this huge price increase does not translate into varieties of food choices. Indonesia simply has a lot more food choices due to its tropical climate.

If you factor in all these costs, Canadian cost of living is about 4-5 times more expensive than Indonesian. I have been asked before by Indonesians who planned to work overseas and I always advise them to not just care about salary. They have to pay attention to cost as well.

Are costs of living limited only to those I listed already? There are health and risk costs. I used to think that developed countries win hands down here, but there are many health and insurance companies - including Canadian - that operate in Indonesia now. Indonesians pay reasonable insurance rates for health and accident coverages. Health costs due to air pollution are difficult to assess, but Canada is clearly miles ahead here.

Car prices in Indonesia are more expensive than in Canada. Gasoline is, however, subsidized in Indonesia, unless you drive a fancy car that requires a premium gasoline. Roughly, transportation costs are similar, although Canada is better since roads are less polluted and less congested.

Canada though has one clear advantage: very good public school and university system. They are not free though. University tuition fees of reputable Indonesian universities are about a third of those in Canada. The quality, however, can be lower than that.

Bike-to-Work Challenges in Jakarta

Yesterday I decided to bike 14 km to meet a colleague in downtown Jakarta at nine-thirty in the morning. It took me 40 minutes through a huge traffic jam that always occurs every morning along Jakarta's main throroughfares. Had I driven a car, it would have been a lot longer. Biking during Jakarta's mad rush hours is a good alternative: cheap, healthy, and fast.

One big challenge of bike to work in Jakarta, however, is the hot weather. It makes me sweat profusely. My shirt was wet. Luckily, when I arrived at Wisma Mulia, the security staff gave me a run-around to park my folding bike and get a visitor tag; my shirt got dry during this delay. The motorbike and bike parking area is tucked way at the back of the office tower and is not easily found. (Office towers in downtown Jakarta are friendly to cars but not to pedestrians and bikes.)

The sweat issue is what concerns me when I want to bike to meet friends and colleagues. I have to pick what I wear carefully so that the sweat does not produce smell. A synthetic material wicks sweat fast but produces smell, while cotton can reduce smell if not too thick. A blend of the two seems the perfect combination so long as it is not thick. It is a good thing my head is bald; otherwise, my hair would still be wet despite the delay.

Another big challenge is the lung-clogging exhaust fume from cars and motorcycles. This problem worsens during a stand-still traffic jam since heat and fume conspire to choke a cyclist. I have resisted so far to wear a bike mask, but I have been thinking about it more and more since I bike everyday to work.

Yet another challenge is the rain season where roads will be wet and slippery. Not to mention messy and dirty clothing from dirt and muck splashes. My backpack has a rain cover and I have a thin raincoat, but I still have to get mud fenders for my bike. The prospect of mixing sweat, rain, and mud is not as exciting as mixing snow and sweat.

As it turns out, biking in a hot weather is more challenging than in a temperate weather around 0-20 °C. No wonder there are not many people biking unless you are physically fit to deal with the aerobic/agility demands or a daredevil - such as yours truly :-). The challenges of biking in Jakarta can be lessened by having dedicated bike lanes. Bike to work is actually a viable, healthy, and cheap alternative to rapid mass transit to reduce traffic congestion and improve air quality. During morning weekend, biking in Jakarta is very pleasant due to much less exhaust and less congestion. But the Jakarta government has done little to make the bike lane idea a reality.

$6 lunch for 6 people

We went to an ordinary food stall - aka warteg, a shorthand for 'war(ung) teg(al)', which means Tegal food stall, with Tegal being a city in Central Java where a lot of food stall vendors hail from - yesterday for lunch. It cost $6 for 6 of us. We had six full plates with sweet drinks to top them off. Go figure.

The cost boggled my mind. How come it cost only 6 bucks? $6 got me a decent Asian food lunch at the U of Calgary food court ... for one person, and I had tap water to drink. The food are similar in taste and use similar ingredients. Where does the price difference come from? I like to think about these problems. They always intrigue me.

The $1 lunch/person at the warteg could only happen if the vendor's cost of living is low or her sales volume is huge. The latter is not true, so her living cost is low. Where does the low living cost come from? Well, from the low food price to begin with ... and we are in an infinite loop now, trying to unravel it.

But the truth is that IT IS NOT AN INFINITE LOOP. The owner of the food court vendor drives a BMW X3 while the owner of the warteg lives in a small hut and rides a motorcycle. The BMW-riding owner is smarter and wealthier than the warteg owner. The former also has a higher expectation than the latter, and thus demands a higher rate of return from his investment.

My wife and I had thought previously of opening a small fast food place in Calgary, so I am not alien to its costs. A healthy revenue would be at least $12,000/month since the rent and utility would cost at least $2000. Two full-time staff would cost $4,000. It is reasonable to expect at least $4,000 monthly revenue. It is a hard work and actually does not make a lot of money. In fact, the food court vendor has more than 1 food outlet in Calgary.

The high cost of lunch in Calgary comes from high living cost of business and owner. The high standard of building causes the high cost and comes from continuous building code improvements over the years. The high standard of health and food regulations also make operating a food outlet expensive; I only saw an LPG stove and simple food utensils - no stainless steel friers and all - at the warteg. The continuing improvement process causes the living cost high since employees cannot be paid lower than the costs to rent a decent apartment and to buy food.

It is what I call the karma of technology. As technologies improve and get implemented, they create additional costs that banks will provide at additional costs. This increased cost gets rolled in to the next generation. When the current generation fumble and are not successful as the previous, things can get really difficult, which I think underlie the current recession in North America and Europe. Not a bad thought from having a $6 lunch for 6 people!