## 5 Steps to a 5: AP Physics 1: Algebra-Based 2017 (2016)

**STEP **__4__

__4__

**Review the Knowledge You Need to Score High**

**CHAPTER 18**

**Extra Drills on Difficult but Frequently Tested Topics**

**IN THIS CHAPTER**

**Summary:** Included in this chapter are problems providing extra practice on frequently tested topics that students often find difficult:

Springs and graphs

Tension

Inclined planes

Motion graphs

Simple circuits

If you have any extra time, spend it sharpening your AP Physics 1 skills working out these problems. Following each question set are detailed explanations that take you step-by-step to the solution.

**How to Use This Chapter**

Practice problems and tests cannot possibly cover every situation that you may be asked to understand in physics. However, some categories of topics come up again and again, so much so that they might be worth some extra review. And that’s exactly what this chapter is for—to give you a focused, intensive review of a few of the most essential physics topics.

We call them “drills” for a reason. They are designed to be skill-building exercises, and as such, they stress repetition and technique. Working through these exercises might remind you of playing scales if you’re a musician or of running laps around the field if you’re an athlete. Not much fun, maybe a little tedious, but very helpful in the long run.

The questions in each drill are all solved essentially the same way. *Don’t* just do one problem after the other … rather, do a couple, check to see that your answers are right,^{ }^{1}^{ }and then, half an hour or a few days later, do a few more, just to remind yourself of the techniques involved.

**Springs and Graphs**

**The Drill**

A 0.50-kg lab cart on a frictionless surface is attached to a spring, as shown in the preceding figure. The rightward direction is considered positive. The spring is neither stretched nor compressed at position *x* = 0. The cart is released from rest at the position *x* = +0.25 m at time *t* = 0.

** 1 .** On the axes below, sketch a graph of the kinetic energy of the cart as a function of position

*x*.

** 2 .** On the axes below, sketch a graph of the total mechanical energy of the cart-spring system as a function of position

*x*.

** 3 .** On the axes below, sketch a graph of the speed of the cart-spring system as a function of position

*x*.

** 4 .** On the axes below, sketch a graph of the force applied by the spring on the cart as a function of position

*x*.

** 5 .** On the axes below, sketch a graph of the force applied by the cart on the spring as a function of position

*x*.

** 6 .** On the axes below, sketch a graph of the spring constant of the spring as a function of position

*x*.

** 7 .** On the axes below, sketch a graph of the acceleration of the cart-spring system as a function of position

*x*.

** 8 .** On the axes below, sketch a graph of the potential energy of the cart-spring system as a function of position

*x*.

**Answers with Explanations**

In this section, the figure is shown first, followed by the explanation.

** 1 .** Kinetic energy is zero at the endpoints, where the cart comes briefly to rest. At the midpoint, the cart-spring system’s potential energy (½

*kx*

^{2}) is zero since

*x*= 0. All the energy is kinetic at the midpoint. The graph is curved because the potential energy graph is curved; the kinetic and potential energy must add to the same value, since friction is negligible.

** 2 .** Total mechanical energy of a system with no nonconservative forces doing work is conserved. Friction is negligible; gravity and the normal force do no work. The only relevant force is the spring force, which is conservative. Mechanical energy cannot change.

** 3 .** The cart comes briefly to rest at both ends, so speed is zero there. At the midpoint, kinetic energy is greatest, so speed is as well:

*KE*= ½

*mv*

^{2}. Speed has no direction, so graph positive values only. The graph is curved—you can know that because the spring force changes, so the acceleration of the cart is not constant. Constant acceleration means a straight velocity-time graph, so this graph must be curved.

^{ }

^{2}^{ }Or, you can know the fact that in simple harmonic motion, a position-time graph and a velocity-time graph will look like curvy sine functions.

** 4 .** The relevant equation relating force of a spring to displacement of the spring is

*F*=

*kx*. At the midpoint where

*x*= 0, the net force is zero as well. The amount of force gets bigger as the distance from the midpoint gets bigger. When the cart is far to the left, the spring pushes the cart to the right; so the force is positive when the distance is negative. When the cart is far to the right, the spring pushes the cart to the left; so the force is negative when the distance is positive. The graph is straight because the

*x*is neither squared nor square rooted, but just to the first power.

** 5 .** By Newton’s third law, the force of the cart on the spring is equal in magnitude and opposite in direction to the force of the spring on the cart. Just flip the direction of the answer to Question 4.

** 6 .** The spring constant of a spring is a property of the spring itself. Since the spring isn’t replaced while the cart oscillates, the spring constant cannot change.

** 7 .** By Newton’s second law, the net force on the cart is equal to

*ma*. The acceleration graph should look exactly like the force of the spring on the cart graph. If we cared about the actual values on the graph, we’d divide the force by the cart’s mass at each position. The problem says “sketch”—just the shape is necessary.

** 8 .** The relevant equation for the potential energy due to a spring is

*PE*= ½

*kx*

^{2}. At

*x*= 0, the potential energy is also zero. The potential energy is largest at the endpoints. And the equation includes an

*x*

^{2}, so the graph is curved.

**Tension**

**How to Do It**

Use the following steps to solve these kinds of problems: (1) Draw a free-body diagram for each block; (2) resolve vectors into their components; (3) write Newton’s second law for each block, being careful to stick to your choice of positive direction; and (4) solve the simultaneous equations for whatever the problem asks for.

**The Drill**

In the diagrams below, assume all pulleys and ropes are massless, and use the following variable definitions:

Find the tension in each rope and the acceleration of the set of masses.

(For a greater challenge, solve in terms of *F, M* , and *μ* instead of plugging in values.)

** 1 .** Frictionless

** 2 .** Frictionless

** 3 .** Frictionless

** 4 .** Coefficient of Friction

*μ*

__5__ .

__6__ .

** 7 .** Frictionless

** 8 .** Frictionless

** 9 .** Frictionless

** 10 .** Coefficient of Friction

*μ*

** 11 .** Coefficient of Friction

*μ*

** 12 .** Frictionless

** 13 .** Frictionless

** 14 .** Coefficient of Friction

*μ*

**The Answers**

__1__ .*a* = 10 m/s^{2}

__2__ .*a* = 3.3 m/s^{2}

*T* = 3.3 N

See step-by-step solution below.

__3__ .*a* = 1.7 m/s^{2}

*T* _{1} = 1.7 N

*T* _{2} = 5.1 N

__4__ .*a* = 1.3 m/s^{2}

*T* = 3.3 N

__5__ .*a* = 3.3 m/s^{2}

*T* = 13 N

See step-by-step solution below.

__6__ .*a* = 7.1 m/s^{2}

*T* _{1} = 17 N

*T* _{2} = 11 N

__7__ .*a* = 3.3 m/s^{2}

*T* = 6.6 N

__8__ .*a* = 6.7 m/s^{2}

*T* _{1} = 13 N

*T* _{2} = 10 N

__9__ .*a* = 1.7 m/s^{2}

*T* _{1} = 5.1 N

*T* _{2} = 8.3 N

__10__ .*a* = 6.0 m/s^{2}

*T* = 8.0 N

__11__ .*a* = 8.0 m/s^{2}

*T* _{1} = 10 N

*T* _{2} = 4.0 N

__12__ .*a* = 5.0 m/s^{2}

*T* = 15 N

__13__ .*a* = 3.3 m/s^{2}

*T* _{1} = 13 N

*T* _{2} = 20 N

__14__ .*a* = 0.22 m/s^{2}

*T* _{1} = 20 N

*T* _{2} = 29 N

**Step-by-Step Solution to Problem 2**

__Step 1__* :* Free-body diagrams:

No components are necessary, so on to the next step.

__Step 2__* :* Write Newton’s second law for each block, calling the rightward direction positive:

*T* − 0 = *ma* *F* − *T* = (2*m* )*a*

__Step 3__* :* Solve algebraically. It’s easiest to add these equations together, because the tensions cancel:

*F* = (3*m* )*a* , so *a* = *F* /3*m* = (10 N)/3(1 kg) = 3.3 m/s^{2} .

To get the tension, just plug back into *T* − 0 = *ma* to find *T* = *F* /3 = 3.3 N.

**Step-by-Step Solution to Problem 5**

__Step 1__* :* Free-body diagrams:

No components are necessary, so on to the next step.

__Step 2__* :* Write Newton’s second law for each block, calling clockwise rotation of the pulley positive:

(2*m* )*g* − *T* = (2*m* )*a* *T* − *mg* = *ma*

__Step 3__* :* Solve algebraically. It’s easiest to add these equations together, because the tensions cancel:

*mg* = (3*m* )*a* , so *a* = *g* /3 = 3.3 m/s^{2}

To get the tension, just plug back into *T* − *mg* = *ma* : *T* = *m* (*a* + *g* ) = (4/3)*mg* = 13 N.

**Inclined Planes**

**How to Do It**

Use the following steps to solve these kinds of problems: (1) Draw a free-body diagram for the object (the normal force is perpendicular to the plane; the friction force acts along the plane, opposite the velocity); (2) break vectors into components, where the parallel component of weight is *mg* (sin *θ* ); (3) write Newton’s second law for parallel and perpendicular components; and (4) solve the equations for whatever the problem asks for.

Don’t forget, the normal force is *not* equal to *mg* when a block is on an incline!

**The Drill**

Directions: For each of the following situations, determine:

(a) the acceleration of the block down the plane

(b) the time for the block to slide to the bottom of the plane

In each case, assume a frictionless plane unless otherwise stated; assume the block is released from rest unless otherwise stated.

__1__ .

__2__ .

__3__ .

__4__ .

__5__ .

__6__ .

__7__ .

__8__ .

Careful—this one’s tricky.

**The Answers**

__1__ .*a* = 6.3 m/s^{2} , down the plane.

*t* = 2.5 s

See step-by-step solution below.

__2__ .*a* = 4.9 m/s^{2} , down the plane.

*t* = 2.9 s

__3__ .*a* = 5.2 m/s^{2} , down the plane.

*t* = 2.8 s

__4__ .*a* = 4.4 m/s^{2} , down the plane.

*t* = 3.0 s

** 5 .** Here the angle of the plane is 27° by trigonometry, and the distance along the plane is 22 m.

*a* = 4.4 m/s^{2} , down the plane.

*t* = 3.2 s

__6__ .*a* = 6.3 m/s^{2} , down the plane.

*t* = 1.8 s

__7__ .*a* = 6.3 m/s^{2} , down the plane.

*t* = 3.5 s

** 8 .** This one is complicated. Since the direction of the friction force changes depending on whether the block is sliding up or down the plane, the block’s acceleration is

*not*constant throughout the whole problem. So, unlike problem 7, this one can’t be solved in a single step. Instead, in order to use kinematics equations, you must break this problem up into two parts: up the plane and down the plane. During each of these individual parts, the acceleration is constant, so the kinematics equations are valid.

- up the plane:

*a* = 6.8 m/s^{2} , down the plane.

*t* = 0.4 s before the block turns around to come down the plane.

- down the plane:

*a* = 1.5 m/s^{2} , down the plane.

*t* = 5.2 s to reach bottom.

So, a total of *t* = 5.6 s for the block to go up and back down.

**Step-by-Step Solution to Problem 1**

__Step 1__* :* Free-body diagram:

__Step 2__* :* Break vectors into components. Because we have an incline, we use inclined axes, one parallel and one perpendicular to the incline:

__Step 3__* :* Write Newton’s second law for each axis. The acceleration is entirely directed parallel to the plane, so perpendicular acceleration can be written as zero:

*mg* sin *θ* − 0 = *ma* *F* _{N} − *mg* cos *θ* = 0

__Step 4__* :* Solve algebraically for *a* . This can be done without reference to the second equation. (In problems with friction, use *F* _{f} = *μF* _{N} to relate the two equations.)

*a* = *g* sin *θ* = 6.3 m/s^{2}

To find the time, plug into a kinematics chart:

Solve for *t* using the second star equation for kinematics (**): Δ*x* = *v* _{o}_{ }*t* + ½*at* ^{2} , where *v* _{o}_{ }is zero:

**Motion Graphs**

**How to Do It**

For a position–time graph, the slope is the velocity. For a velocity–time graph, the slope is the acceleration, and the area under the graph is the displacement.

**The Drill**

Use the graph to determine something about the object’s speed. Then play “Physics *Taboo* ”: suggest what object might reasonably perform this motion and explain in words how the object moves. Use everyday language. In your explanation, you may *not* use any words from the list below:

velocity

acceleration

positive

negative

increase

decrease

it

object

constant

**Answers with Explanations**

Note that our descriptions of the moving objects reflect our own imaginations. You might have come up with some very different descriptions, and that’s fine … provided that your answers are conceptually the same as ours.

- The average speed over the first 5 s is 10 m/s, or about 22 mph. So:

Someone rolls a bowling ball along a smooth road. When the graph starts, the bowling ball is moving along pretty fast, but the ball encounters a long hill. So, the ball slows down, coming to rest after 5 s. Then, the ball comes back down the hill, speeding up the whole way.

- This motion only lasts 1 s, and the maximum speed involved is about 5 mph. So:

A biker has been cruising up a hill. When the graph starts, the biker is barely moving at jogging speed. Within half a second, and after traveling only half a meter up the hill, the bike turns around, speeding up as it goes back down the hill.

- The maximum speed of this thing is 30 cm/s, or about a foot per second. So:

A toy racecar is moving slowly along its track. The track goes up a short hill that’s about a foot long. After 2 s, the car has just barely reached the top of the hill, and is perched there momentarily; then, the car crests the hill and speeds up as it goes down the other side.

- The steady speed over 200 s (a bit over 3 minutes) is 0.25 m/s, or 25 cm/s, or about a foot per second.

A cockroach crawls steadily along the school’s running track, searching for food. The cockroach starts near the 50 yard line of the football field; around three minutes later, the cockroach reaches the goal line and, having found nothing of interest, turns around and crawls at the same speed back toward his starting point.

- The maximum speed here is 50 m/s, or over a hundred mph, changing speed dramatically in only 5 or 10 s. So:

A small airplane is coming in for a landing. Upon touching the ground, the pilot puts the engines in reverse, slowing the plane. But wait! The engine throttle is stuck! So, although the plane comes to rest in 5 s, the engines are still on … the plane starts speeding up backwards! Oops …

- This thing covers 5 meters in 3 seconds, speeding up the whole time.

An 8-year-old gets on his dad’s bike. The boy is not really strong enough to work the pedals easily, so he starts off with difficulty. But, after a few seconds he’s managed to speed the bike up to a reasonable clip.

- Though this thing moves quickly—while moving, the speed is 1 m/s—the total distance covered is 1 mm forward, and 1 mm back; the whole process takes 5 ms, which is less than the minimum time interval indicated by a typical stopwatch. So we’ll have to be a bit creative:

In the Discworld novels by Terry Pratchett, wizards have developed a computer in which living ants in tubes, rather than electrons in wires and transistors, carry information. (Electricity has not been harnessed on the Discworld.) In performing a calculation, one of these ants moves forward a distance of 1 mm; stays in place for 3 ms; and returns to the original position. If this ant’s motion represents two typical “operations” performed by the computer, then this computer has an approximate processing speed of 400 Hz times the total number of ants inside.

- Though this graph
*looks*like problem 7, this one is a velocity–time graph, and so indicates completely different motion.

A small child pretends he is a bulldozer. Making a “brm-brm-brm” noise with his lips, he speeds up from rest to a slow walk. He walks for three more seconds, then slows back down to rest. He moved forward the entire time, traveling a total distance (found from the area under the graph) of 4 m.

- This stuff moves 300 million meters in 1 s at a constant speed. There’s only one possibility here: electromagnetic waves in a vacuum.

Light (or electromagnetic radiation of any frequency) is emitted from the surface of the moon. In 1 s, the light has covered about half the distance to Earth.

- Be careful about axis labels: this is an
*acceleration*–time graph. Something is accelerating at 1,000 cm/s^{2}for a few seconds. 1,000 cm/s^{2}= 10 m/s^{2}, about Earth’s gravitational acceleration. Using kinematics, we calculate that if we drop something from rest near Earth, after 4 s the thing has dropped 80 m.

One way to simulate the effects of zero gravity is to drop an experiment from the top of a high tower. Then, because everything that was dropped is speeding up at the same rate, the effect is just as if the experiment were done in the Space Shuttle—at least until everything hits the ground. In this case, an experiment is dropped from a 250-ft tower, hitting the ground with a speed close to 90 mph.

- 1 cm/s is ridiculously slow. Let’s use the world of slimy animals:

A snail wakes up from his nap and decides to find some food. He speeds himself up from rest to his top speed in 10 s. During this time, he’s covered 5 cm, or about the length of your pinkie finger. He continues to slide along at a steady 1 cm/s, which means that a minute later he’s gone no farther than a couple of feet. Let’s hope that food is close.

- This one looks a bit like those up-and-down-a-hill graphs, but with an important difference—this time the thing stops not just for an instant, but for five whole seconds, before continuing back toward the starting point.

A bicyclist coasts to the top of a shallow hill, slowing down from cruising speed (∼15 mph) to rest in 15 s. At the top, she pauses briefly to turn her bike around; then, she releases the brake and speeds up as she goes back down the hill.

**Simple Circuits**

**How to Do It**

Think “series” and “parallel.” The current through series resistors is the same, and the voltage across series resistors adds to the total voltage. The current through parallel resistors adds to the total current, and the voltage across parallel resistors is the same.

**The Drill**

For each circuit drawn below, find the current through and voltage across each resistor.

Note: Assume each resistance and voltage value is precise to two significant figures.

__1__ .

__2__ .

__3__ .

__4__ .

__5__ .

__6__ .

__7__ .

**The Answers**

__1__ .

__2__ .

__3__ .

__4__ .

__5__ .

__6__ .

__7__ .

**Step-by-Step Solution to Problem 2**

Start by simplifying the combinations of resistors. The 8 kΩ and 10 kΩ resistors are in parallel. Their equivalent resistance is given by

which gives *R* _{eq} = 4.4 kΩ.

Next, simplify these series resistors to their equivalent resistance of 6.4 kΩ.

6.4 kΩ (i.e., 6,400 Ω) is the total resistance of the entire circuit. Because we know the total voltage of the entire circuit to be 10 V, we can use Ohm’s law to get the total current

(more commonly written as 1.6 mA).

Now look at the previous diagram. The same current of 1.6 mA must go out of the battery, into the 2 kW resistor, and into the 4.4 kW resistor. The voltage across each resistor can thus be determined by V = (1.6 mA)R for each resistor, giving 3.2 V across the 2 kΩ resistor and 6.8 V across the 4.4 kΩ resistor.

The 2 kΩ resistor is on the chart. However, the 4.4 kΩ resistor is the equivalent of two parallel resistors. Because voltage is the same for resistors in parallel, there are 6.8 V across *each* of the two parallel resistors in the original diagram. Fill that in the chart, and use Ohm’s law to find the current through each:

*I* _{8k} = 6.8 V/8,000 Ω = 0.9 mA*I* _{10k} = 6.8 V/10,000 Ω = 0.7 mA

^{1}^{ }For numerical answers, it’s okay if you’re off by a significant figure or so.

^{2}^{ }And the curve must be steepest at the endpoints, where the net force *F* = *kx* is greatest; that’s where the speed must be changing most quickly.

****