PHYS 141

Test 2 — Forces & Circular Motion

Chapters 4–5 · opens Jun 19, due Jun 20.

Opens Jun 19

Plan: master the Test 2 tab (essentials → worked examples → traps), then drill Practice. Your deadlines, grades, and tasks live in Profile.

How to read any formula

• Letters are stand-ins for numbers. "v" just means "whatever the speed is." You swap in the number from the problem.
• Units tell you what kind of thing it is. Meters = a distance, seconds = a time, newtons = a force. If your units do not match, something is off.
• Little letters below (subscripts) are labels, not multiplication. v_0 is "the starting speed," v_f is "the final speed." Same v, just tagged.
• Read it as a sentence. F = m·a literally says "force equals mass times acceleration." That is all a formula is — a sentence in shorthand.

Notation key (the squiggles)

Motion symbols (Test 1, and used everywhere)

Force symbols (Test 2 — Chapter 4)

Circular-motion symbols (Test 2 — Chapter 5)

Units, in plain words

Words that trip everyone up

• Mass vs weight: mass = how much stuff (kg, same everywhere). Weight = gravity pulling on it (N, less on the Moon).
• Speed vs velocity: speed = how fast. Velocity = how fast and which direction.
• Acceleration: any change in velocity — speeding up, slowing down, or turning.
• Net force: what is left after all the forces fight it out. It is the only thing that changes motion.
• Normal force: the surface pushing back, straight out of the surface. On flat ground it equals weight; on a ramp or in an elevator it does not.
• Friction: the force that fights sliding. More normal force → more friction (f = μ·N).
• Tension: the pull carried through a rope, string, or cable.
• Centripetal force: not a new force — it is the name for the net inward force that keeps something circling (a real force like friction or tension does the job).
• Period vs frequency: period = seconds per loop. Frequency = loops per second. They are flips of each other.
• Equilibrium: forces balanced, net force = 0 → no acceleration (sitting still, or moving at steady speed).
• Inertia: object's resistance to changing its motion. More mass = more inertia = harder to start or stop.

Phrase decoder — turn words into numbers

Test questions hide the numbers in phrases. Here is the translation:

• "starts from rest" / "dropped" → v_0 = 0
• "comes to rest" / "stops" → v_f = 0
• "at the highest point" → vertical velocity = 0 (but gravity still pulls — acceleration is NOT zero)
• "constant speed" / "constant velocity" → a = 0, and net force = 0
• "smooth" / "frictionless" → ignore friction (f = 0)
• "how far" → find distance d  ·  "how long" → find time t  ·  "how fast" → find velocity v

Same letter, different meaning

A few letters get reused. The units (and the topic) always tell you which one:

• T = tension (a force, in N) and period (a time, in s). Force problem or circle problem?
• a = plain acceleration; a_c = the special centripetal (inward) acceleration.
• F = a force; F_net = all forces combined; F_c = the inward force in a circle.
• v = speed now; v_0 = speed at the start; v_f = speed at the end.

Why moving in a circle is "accelerating" even at steady speed

Acceleration doesn't only mean "speeding up." It means velocity is changing — and velocity includes direction. Going around a circle, your direction is constantly turning, so your velocity is constantly changing, so you're accelerating, even if the speedometer never moves.

Analogy: swinging a ball on a string. Your hand keeps tugging it inward the whole time. That inward tug is what bends its path into a circle. Stop tugging (string snaps) and it shoots off in a straight line — it was never being pushed outward.

That inward acceleration is a_c = v²/r, and the inward force causing it is F_c = m v²/r.

What "centripetal force" really is

It's not a new kind of force you add to a diagram. It's a job title. Whatever real force happens to point toward the center — the string's tension, friction on tires, gravity on a satellite — that force is the centripetal force for that situation.

So on a test: first ask "what real thing pushes/pulls toward the center here?" Then set that force equal to m v²/r.

And there's no outward "centrifugal" force on the object. The feeling of being flung outward in a turning car is just your body's inertia wanting to go straight while the car turns under you.

Mass vs. weight (kilograms vs. newtons)

Mass is how much stuff you're made of — measured in kilograms. It's the same on Earth, the Moon, or in space.

Weight is how hard gravity pulls on that stuff — a force, measured in newtons. It changes with gravity: W = m·g.

Analogy: a brick has the same mass everywhere, but it "weighs" less on the Moon because the Moon pulls more gently. On a test, if you're given kg and need a force, multiply by g; if given newtons and need mass, divide by g.

"Net force" — the tug-of-war idea

Objects don't respond to each force separately; they respond to the one combined force left after all the pushes and pulls fight it out. Push right with 30 N, friction pulls back 12 N → the object only "feels" 18 N to the right.

Recipe: pick a positive direction, add forces with signs, get the net force, then use a = F_net/m. Only the leftover force accelerates anything.

➕➖ Signs — when is something negative (and why)?

A sign isn't "less than zero" — it's a direction. First pick which way is positive and stick with it the whole problem. Standard: up = +, down = − and right = +, left = −. The way something moves is usually +, which makes friction and "slowing down" negative.

The one trick that explains most of it: acceleration is negative whenever it points opposite your chosen + direction — braking, gravity, any backward push. Entering it: "how fast / how far / how much" wants the size → positive; a net force or component with a direction keeps the − when it points the negative way.

📐 sin vs cos — which one do I use?

The #1 mix-up, fixed by one rule: the side next to the angle uses cos; the side across from it uses sin (SOH-CAH-TOA: sin = opp/hyp, cos = adj/hyp).

For a force F at angle θ from the horizontal: F_x = F·cos θ (horizontal — next to the angle) and F_y = F·sin θ (vertical — across from it).

⚠️ If the angle is measured from the vertical instead, it flips (F_x = F·sin θ, F_y = F·cos θ). Always check which axis the angle opens from — the side touching the angle gets cos.

On a ramp tilted θ: gravity down the slope = mg·sin θ (steeper → slides more), into the ramp = mg·cos θ. Banked curve: tan θ = v²/(r·g). Worked: a 50 N pull at 30° → F_x = 50·cos30° = 43.3 N, F_y = 50·sin30° = 25 N.

➕ Adding vectors — components → resultant (your Wiley Ch 1)

To add A and B into R = A + B you never add the arrows directly — break each into x/y parts, add those, then rebuild:

1. components (watch signs): Vx = V·cos θ, Vy = V·sin θ — a part pointing left is −x, down is −y.  2. add: Rx = Ax + Bx, Ry = Ay + By (a left-pointing part subtracts → that's how you get a −4.97).  3. rebuild: R = √(Rx² + Ry²), θ = tan⁻¹(Ry/Rx).

The angle trap: tan⁻¹ only returns −90°…90°, so check the signs of Rx, Ry to place the arrow — if Rx is negative (points left), add 180° to the angle. Worked: Rx = 8, Ry = 6 → R = √(8²+6²) = 10 m, θ = tan⁻¹(6/8) = 36.9° above +x.

Positive and negative signs — what they actually mean

A minus sign is usually a direction label, not a mistake. First choose your positive direction, then every vector gets a sign compared with that choice.

Standard choice unless told otherwise: up = + and down = −; right = + and left = −; if the problem is one-dimensional, call the direction of motion + so friction/braking becomes −.

• Position / displacement: positive is one side of your origin; negative is the other side.
• Velocity: the sign tells which way the object is moving. If right is positive, moving right means v > 0; moving left means v < 0.
• Acceleration: the sign tells which way velocity is changing, or which way the net force points. Moving right but slowing down means v > 0 and a < 0.
• Forces: add pushes and pulls with signs. If right is positive, 30 N right + 12 N left becomes +30 + (-12) = +18 N.
• Circular motion: centripetal acceleration and force point inward. If inward is left or down on your diagram, the inward value can be negative in that axis.
• Mass, speed, distance, and time: these are sizes, so they are never negative. If the problem asks "how fast," "how far," or "how much force," it usually wants a positive size plus a direction word.

Best habit: write the sign key before the math: right/up = + or toward center = +. Then the answer's sign tells the direction.

sin, cos, x, and y signs

Cos and sin give sizes. The diagram gives the sign. The side next to the angle uses cos; the side across from the angle uses sin.

If the calculator gives a positive component size, attach the sign from the words or picture: left = negative x, down = negative y, backward = negative if forward was chosen positive.

Free-body diagrams — draw the forces, nothing else

Shrink the object to a dot. Draw an arrow for every real force touching it: gravity down (weight), the surface pushing back (normal), any pull/push, friction. Do not draw an arrow for "motion" or "centrifugal force" — those aren't forces.

Then split arrows into x and y, and write F_net,x = m·a_x and F_net,y = m·a_y. The diagram turns a word problem into equations.

Why velocity can be zero while acceleration isn't (review from Test 1)

Throw a ball straight up. At the very top it stops for an instant — velocity = 0. But gravity is still pulling it down the entire time (a = −9.8 m/s²); that's exactly why it doesn't hang there and instead falls back. Zero velocity is a single instant; the acceleration never paused.

Put it in DEGREE mode

Any problem with an angle (cos, sin, tan) is wrong if the calculator is in radians. Don't guess the mode — test it:

• If you get 0.5 → you're in degrees.
• If you get −0.952… → you're in radians. Fix it: press MODE, choose DEG (or use the DRG key), then re-test until cos(60) = 0.5.

Sanity values to remember: cos(60) = 0.5, sin(30) = 0.5, cos(0) = 1. If those come out right, your mode is right.

The 6 keys you actually need

• Square: the x² key. To square v, press v then x².
• Square root: the √ key (usually 2nd then x²). It opens √( — type what's inside, then close ).
• Negative sign: the (-) key (bottom row), NOT the big − minus. Use (-) for things like a = -9.8.
• Parentheses ( ): use them around any bottom of a fraction and around everything under a √.
• π: 2nd then the π key. (If unsure, just type 3.14159 — same answer.)
• Last answer: the Ans key reuses your previous result so you don't re-type it.

When to type a negative number

Use the calculator's (-) key when the value itself is negative. Use the big minus key only to subtract one value from another.

A negative final answer means "opposite my positive direction." If the problem asks for a magnitude, report the positive size plus the direction in words.

Button sequences for Test 2 formulas

Type these left to right, then press =. (Example numbers in parentheses.)

Go faster (without rushing)

• Write the equation first, numbers second. Don't re-derive on the test — pick the formula, then plug in. The Formulas tab is your map.
• List givens with units before touching the calculator: "m = 20 kg, μ = 0.30, F = 80 N, find a." Half the work is just sorting that out.
• Estimate first. Round to easy numbers in your head ("≈ 80/20 = 4"). If the calculator says 0.4 or 40, you fat-fingered something.
• One clean keystroke chain per number. Type it once, press =. If it looks off, re-key once slowly — don't stare and second-guess.
• Multiple choice = work backward. Eliminate answers with wrong units or impossible size; if two are close, the math decides. You don't always need the full solve.
• Stuck for 2 minutes? Flag it and move on. Come back after the easy points are banked. (Test 2 is 20 MC pts + 80 problem pts — don't lose easy ones to a hard one.)
• Show your work on the problem part. The equation + plugged-in numbers earns partial credit even if the final arithmetic slips.

Check & verify every answer (the 30-second pass)

• Units: did it come out in the right unit? Force → N, acceleration → m/s², speed → m/s. Wrong unit = wrong setup.
• Size: is it believable? A car's acceleration is a few m/s², not 5000. A centripetal force on a ball is a few N to tens of N.
• Sign: does the direction make sense? A braking force points backward; "up" forces are positive if you chose up as positive.
• Plug it back: put your answer into the original equation and see if both sides match.
• Did you square / root correctly? Most circular-motion errors are a missed x² or a missed √. Re-check those two keys.
• Angle mode: if a problem used cos/sin, re-confirm cos(60)=0.5 in your head — that's the radians trap.