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B. Dynamics · Dynamics: Forces and Motion in a Plane

Dynamics: Forces and Motion in a Plane: Unit Review

A1.1 A1.2 A1.3 A1.4 A1.5 A1.6 A1.7 A1.8 A1.9 A1.10 A1.11 A1.12 A1.13 A2.1 A2.2 B1.1 B1.2 B2.1 B2.2 B2.3 B2.4 B2.5 B2.6 B2.7 B3.1 B3.2 B3.3About 40 minutes
Learning goals
  • I can solve two-dimensional motion problems with vectors, projectiles and relative velocity.
  • I can draw free-body diagrams and apply Newton's laws to objects and systems in two dimensions, with friction and on inclines.
  • I can analyse uniform circular motion in horizontal and vertical planes, including banked curves, and explain fictitious forces in non-inertial frames.
  • I can plan and analyse dynamics investigations and assess technologies that use dynamics.

Words to know

free-body diagram
A diagram of one object with an arrow for every force acting on it.
centripetal acceleration
The acceleration toward the centre of a circle: v²/r = 4π²r/T².
fictitious force
A force added by an observer in an accelerating frame; it has no source and no third-law partner.
coefficient of static friction
μs: the ratio of the maximum static friction to the normal force.

Let's review!

In this unit you sharpened your vector tools and used them for projectiles launched from any height and for relative velocity. You applied Newton's laws in two dimensions with free-body diagrams, worked with static and kinetic friction, inclined planes and connected systems (pulleys, a dumbwaiter, stacked blocks and loads on accelerating trucks). You compared inertial and non-inertial frames, derived the equations of uniform circular motion, and applied them to flat and banked curves, vertical loops, an investigation with the whirling stopper, and technologies such as seat belts, centrifuges and runaway truck ramps.

The next unit, Work, Energy and Simple Harmonic Motion, links forces to energy. This review has questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do one or two sections at a time with a calculator and scrap paper. For every question: sketch the situation, draw a free-body diagram if forces are involved, choose a positive direction (along the acceleration), then write equations. Use g = 9.8 m/s², keep extra digits until the end, and round the final answer sensibly. When you miss a question, write down why (wrong sign? wrong component? forgot friction's direction?) and revisit that lesson. Finish with the 10-question quiz.

VectorsΔv, components (1)Projectilesx and y separate (2)Relative velocityv(AC) = v(AB) + v(BC) (3)Newton's laws in 2-DFBDs, ΣFx, ΣFy (4)Friction and rampsμs, μk, mg sin θ (5, 6)Systemspulleys, stacks, trucks (7, 8)Framesfictitious forces (9)Circular motionv²/r, 4π²r/T² (10, 11)Banks and loopstan θ = v²/rg; √(gr) (12, 13)Investigation (14) and technology, society and careers (15) run through all of it.
How the ideas in this unit connect. Numbers in brackets are the lessons.

Vectors in Two Dimensions

1

Add 4.0 km [N] and 3.0 km [W].

2

A velocity changes from 10 m/s [E] to 10 m/s [S] in 2.0 s. Find the average acceleration.

3

Write [S 30° E] as a standard-position angle.

Projectile Motion

1

A ball is thrown horizontally at 12 m/s from a 30 m cliff. How far out does it land?

2

Find the ball's speed and direction at impact (previous question).

3

Find the range of a ball launched at 18 m/s at 35° on level ground.

4

Why can't the range equation be used for the ball thrown from the cliff?

Relative Velocity

1

A boat moves at 5.0 m/s relative to the water. It must cross a 100 m river with a 2.0 m/s current and land directly opposite. Find the heading and the crossing time.

2

Car A moves 30 m/s [N]; car B moves 20 m/s [S]. Find v(AB).

3

Complete: v(PG) = v(PA) + ____.

Newton's Laws in Two Dimensions

1

A 20 kg sled is pulled with 100 N at 25° above the horizontal; μk = 0.20. Find the normal force and the acceleration.

2

A 3.0 kg lamp hangs from two cords, each at 45° above the horizontal. Find the tension in each.

3

Name the third-law partner of the normal force of a floor on a box.

Static and Kinetic Friction

1

A 50 kg crate has μs = 0.40. A 150 N horizontal push does not move it. What is the friction force?

2

A car skids to a stop from 15 m/s with μk = 0.70. Find the stopping distance.

3

Give one advantage and one disadvantage of friction for a car.

Inclined Planes

1

Find the acceleration of a block sliding down a 35° incline with μk = 0.25.

2

A coin starts to slip when a book is tilted to 31°. Find μs.

3

What is the normal force on a 10 kg box on a 20° ramp?

Pulleys and Connected Objects

1

An Atwood machine has 6.0 kg and 4.0 kg masses. Find the acceleration and tension.

2

A 2.0 kg block on a table (μk = 0.10) is pulled by a 1.0 kg hanging mass. Find the acceleration.

3

Why is a measured pulley-system acceleration usually less than the ideal prediction?

Ramps with Pulleys, Stacked Blocks and Accelerating Vehicles

1

What is the greatest acceleration a truck can have without a box (μs = 0.25) sliding in its bed?

2

A 3.0 kg block on a frictionless 30° ramp is tied over a pulley at the top to a 1.0 kg hanging mass. Which way does it move, and with what acceleration?

3

In an inquiry on how hanging mass affects acceleration, what is the dependent variable?

Inertial and Non-inertial Frames

1

A 55 kg person stands on a scale in an elevator accelerating upward at 2.0 m/s². What does the scale read?

2

A pendulum in a car hangs at 10° to the vertical. Find the car's acceleration.

3

Is centrifugal force a real force in the ground frame? Explain.

Uniform Circular Motion

1

A point 0.30 m from the centre of a wheel turns at 90 rpm. Find its speed and centripetal acceleration.

2

If the radius doubles at the same period, how does ac change?

3

State the key step in deriving a = v²/r.

Centripetal Force: Horizontal Circles

1

Find the maximum speed on a flat curve of radius 45 m with μs = 0.60.

2

What supplies the centripetal force for a rider in a rotor ride?

3

A 1.5 kg mass on a 0.80 m string moves in a horizontal circle on a frictionless table at 4.0 m/s. Find the tension.

Banked Curves

1

What bank angle gives a design speed of 20 m/s on a curve of radius 100 m?

2

Find the design speed of a curve of radius 150 m banked at 12°.

3

A car goes faster than the design speed. Which way does friction act?

Vertical Circles

1

What is the minimum speed at the top of a loop of radius 8.0 m?

2

A 70 kg rider passes the bottom of a loop of radius 8.0 m at 15 m/s. Find the seat's force.

3

Do you feel heavier or lighter at the crest of a hill? Why?

Investigating Circular Motion

1

A stopper makes 10 revolutions in 6.0 s. Find f.

2

For a 0.020 kg stopper at r = 0.50 m with f = 1.67 Hz, what tension is needed?

3

What graph gives a straight line for F = 4π²mrf², and what is its slope?

Dynamics at Work: Technologies, Their Impacts, and Careers

1

Explain how a seat belt's pendulum sensor locks the belt.

2

Give one social and one environmental impact of a technology from this unit.

3

Name two careers that use dynamics.

Ramp (sliding down)mgFNFftilt axes along the slopeTop of a loopFNmgboth toward the centreConical pendulummgFTnet force is horizontal
Three free-body diagrams worth knowing by heart. On a ramp, tilt the axes. At the top of a loop, both forces point to the centre. In a conical pendulum, tension and gravity add to a horizontal net force.
Just for fun

Newton's first law of studying: a textbook at rest tends to stay at rest, unless acted on by an outside force (usually an upcoming test).

Check your understanding

Quiz

1.

A projectile's horizontal velocity during its flight (no air resistance):

2.

A canoe heads straight across a river with a current. Compared with still water, the crossing time is:

3.

A rope pulls a box at an angle above the horizontal. The normal force is:

4.

A block just begins to slide on a ramp tilted at θ. Its μs is:

5.

A crate on a truck bed accelerates with the truck without sliding. The force that accelerates it is:

6.

In a car that is braking, a hanging air freshener swings forward. In the car's frame this is explained by:

7.

The centripetal acceleration of an object in uniform circular motion is:

8.

The design speed of a banked curve depends on:

9.

At the top of a vertical loop of radius r, the minimum speed is:

10.

In the whirling-stopper experiment, doubling the hanging mass (same r) multiplies the frequency by:

For parents and teachers

How to use the results: Match each kind of mistake to the lesson to revisit.

  • Component or angle errors, Δv mistakes: Vectors in Two Dimensions.
  • Projectile time, range or impact velocity: Projectile Motion.
  • Heading versus track, river and wind problems: Relative Velocity.
  • Normal force taken as mg when a force acts at an angle; missing forces on FBDs: Newton's Laws in Two Dimensions.
  • Using μs FN when the object is not slipping; mixing static and kinetic: Friction: Static and Kinetic Coefficients.
  • sin/cos swapped on slopes; wrong friction direction: Inclined Planes.
  • Pulley sign errors, tension equal to weight: Connected Objects I; ramp-pulley direction, truck loads, planning an inquiry: Connected Objects II.
  • Elevator readings, "centrifugal force" in the ground frame: Inertial and Non-inertial Frames.
  • rpm conversions, a = v²/r versus 4π²r/T²: Uniform Circular Motion.
  • Flat curves, conical pendulums, rotor rides: Centripetal Force; bank angles and design speeds: Banked Curves; tops and bottoms of loops, hill crests: Vertical Circles.
  • Graphs, slopes and error analysis: Investigating Circular Motion; technology analysis, impacts, careers: Dynamics at Work.

Tip: A quick daily habit of drawing one free-body diagram for something real (a backpack on a hook, a car on a ramp) keeps these skills sharp before the next unit.