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

Connected Objects II: Ramps with Pulleys, Stacked Blocks and Accelerating Vehicles

B2.4 B2.5 A1.1 A1.2 A1.4 A1.10About 75 minutes
Learning goals
  • I can decide which way a ramp-and-pulley system will move before I include friction, and then solve for its acceleration and tension.
  • I can find the greatest acceleration a vehicle can have before a load on it slides, and what happens if that limit is exceeded.
  • I can analyse stacked objects in which static friction between them provides the acceleration.
  • I can plan an inquiry to test a dynamics prediction: a testable question, a hypothesis, variables, equipment, safety and an analysis plan.

Words to know

hypothesis
A testable prediction with a reason, often in the form "If ..., then ..., because ...".
independent variable
The variable the experimenter deliberately changes.
dependent variable
The variable that is measured to see how it responds.
controlled variables
Variables kept the same so that they do not affect the result.
limiting static friction
The largest static friction force possible, μs FN. It sets the largest acceleration that friction alone can give an object.

Will the toolbox slide?

A pickup truck stops at a red light with a heavy toolbox sitting loose in its bed. The light turns green and the driver accelerates. Nothing pushes the toolbox forward except static friction from the truck bed. If the truck accelerates gently, friction is enough, and the toolbox speeds up with the truck. If the truck accelerates hard, friction cannot keep up, and the toolbox slides toward the tailgate.

This lesson tackles three classic Grade 12 systems: a block on a ramp tied to a hanging mass, objects carried by friction on an accelerating vehicle, and stacked blocks. It ends with how to plan an inquiry that tests a prediction about one of them.

Ramp and pulley: decide the direction first

A block on a ramp is tied by a string, over a pulley at the top, to a hanging mass. Which way does it go? Friction cannot decide that, because friction only ever opposes motion. So:

  1. Ignore friction and compare the two forces that drive the system along the string: the hanging weight m₂g and the down-slope component m₁g sin θ.
  2. The larger one wins. Now you know the direction of motion, and friction points the opposite way.
  3. If the system starts from rest, check that the difference is larger than the maximum static friction; if not, nothing moves.
  4. Write ΣF = ma for each object (positive along the string) and add the equations.
Worked example

Example 1: Block on a ramp, mass hanging

A 4.0 kg block sits on a 30° ramp with μk = 0.15. It is tied by a string over an ideal pulley at the top of the ramp to a 3.0 kg hanging mass. The system is moving. Find its acceleration and the tension.

  1. Driving forces: hanging weight = 3.0 × 9.8 = 29.4 N; down-slope component = 4.0 × 9.8 × sin 30° = 19.6 N. The hanging mass wins: it moves down, and the block moves up the ramp.
  2. Friction on the block points down the slope: Ff = 0.15 × 4.0 × 9.8 × cos 30° = 5.09 N.
  3. Block (+ up the slope): FT − 19.6 − 5.09 = 4.0a.
  4. Hanging mass (+ down): 29.4 − FT = 3.0a.
  5. Add: 29.4 − 19.6 − 5.09 = 7.0a, so a = 4.71 ÷ 7.0 = 0.673 m/s².
  6. FT = 3.0(9.8 − 0.673) = 27.4 N. Check: 4.0(0.673) + 19.6 + 5.09 = 27.4 ✓.

a ≈ 0.67 m/s² (block up the ramp) and FT ≈ 27 N.

30°4.0 kg3.0 kgFT29.4 NFT (up the slope)mg sin θ = 19.6 NFf = 5.1 N (down the slope)
The ramp-and-pulley system of Example 1. The hanging weight (29.4 N) beats the down-slope component of the block's weight (19.6 N), so the block moves up the slope, and kinetic friction (5.1 N) points down the slope.
Worked example

Example 2: The toolbox in the truck

A 50 kg crate rests in a truck bed. Between the crate and the bed, μs = 0.35 and μk = 0.25. (a) What is the greatest acceleration the truck can have without the crate sliding? (b) The truck accelerates at 3.0 m/s². What friction force acts on the crate? (c) The truck accelerates at 4.0 m/s² instead. Find the crate's acceleration relative to the ground and relative to the truck.

  1. Only static friction can push the crate forward: μs mg ≥ ma, so a(max) = μs g. The mass cancels.
  2. (a) a(max) = 0.35 × 9.8 = 3.43 m/s².
  3. (b) 3.0 < 3.43, so the crate moves with the truck. Ff = ma = 50 × 3.0 = 150 N (less than the maximum, 0.35 × 490 = 171.5 N).
  4. (c) 4.0 > 3.43, so the crate slides. Kinetic friction still pushes it forward: a(crate) = μk g = 0.25 × 9.8 = 2.45 m/s² forward relative to the ground.
  5. Relative to the truck: 2.45 − 4.0 = −1.55 m/s², so the crate accelerates 1.55 m/s² backward relative to the truck bed.

(a) 3.4 m/s² (b) 150 N of static friction (c) 2.5 m/s² forward relative to the ground, but 1.6 m/s² backward relative to the truck. The crate does not get "thrown back": it just cannot keep up. You will revisit this in Inertial and Non-inertial Frames of Reference.

cratetruck: a = 3.0 m/s²FBD of the crateFNFgstatic friction is the onlyforward force on the crateFs
While the crate rides along without slipping, the only horizontal force on it is static friction from the truck bed, pointing forward, in the direction of the acceleration.
Worked example

Example 3: Stacked blocks

A 2.0 kg book sits on a 5.0 kg box on a frictionless floor. μs between the book and the box is 0.40. A horizontal force F pulls the box. What is the largest F for which the book does not slip?

  1. The book is accelerated only by static friction from the box, so its largest acceleration is μs g = 0.40 × 9.8 = 3.92 m/s².
  2. If the book does not slip, both move together: F = (2.0 + 5.0) × 3.92 = 27.4 N.

About 27 N. Pull any harder and the box slides out from under the book. This is the physics of the famous tablecloth trick.

Planning an inquiry

B2.4 asks you to predict forces in a system and then test your prediction. A complete plan has these parts:

  • Testable question: How does the hanging mass affect the acceleration of a cart pulled up a ramp?
  • Hypothesis: If the hanging mass increases, then the acceleration increases but levels off, because the driving force grows while the total mass being accelerated grows too: a = (m₂g − m₁g sin θ − Ff) ÷ (m₁ + m₂).
  • Variables: independent (hanging mass), dependent (acceleration), controlled (cart mass, ramp angle, surface, release point, distance timed).
  • Equipment and safety: list everything, including padding where the mass lands and keeping fingers clear of the pulley.
  • Procedure: numbered steps detailed enough for someone else to repeat, with several trials per setting.
  • Analysis plan: a = 2Δd ÷ Δt², a graph of measured and predicted a against m₂, and percent differences.
  • Conclusion: state whether the evidence supports the hypothesis, with the sources of error and suggested improvements.
Just for fun

The tablecloth trick works because of Example 3. Yanked fast enough, the cloth gives the dishes only a small, brief kick of kinetic friction, so they barely move. Please practise it with a sheet of paper and a plastic cup, not your family's good dishes.

Hands-on activity

Paper, coin and cup: the limits of static friction

You need: a strip of paper about 5 cm × 25 cm, a plastic cup (or a stack of two or three), a coin, and a smooth table.

  1. Put the paper strip at the edge of the table with the cup on it, near the edge.
  2. Pull the paper slowly. The cup moves with the paper: static friction gives it the same acceleration.
  3. Pull the paper quickly (a sharp tug, slightly downward over the table edge, works best). The cup stays nearly in place: the paper's acceleration is larger than μs g, so the cup slips.
  4. Put a coin on top of the cup and repeat. Does the coin ride along with the cup?
  5. Plan an inquiry on paper (you do not have to carry it out): "How does the mass in the cup affect the fastest pull that still carries the cup along?" Write the hypothesis, the independent, dependent and controlled variables, a procedure, and how you would measure the pull. Use Example 3 to predict the answer. (Hint: μs g does not depend on mass.)

Your turn

1

A 3.0 kg block on a frictionless 30° ramp is tied over a pulley at the top to a 2.0 kg hanging mass. Which way does the system move? Find the acceleration and tension.

2

A box sits on a flatbed with μs = 0.30. What is the greatest acceleration the truck can have without the box sliding?

3

A 10 kg block on a frictionless 30° ramp is tied to a 4.0 kg hanging mass over a pulley at the top. Which way does it move, and with what acceleration?

4

A 3.0 kg book rests on a 6.0 kg box on a frictionless floor; μs = 0.20 between them. Find the largest horizontal pull on the box for which the book does not slip.

5

For the question "How does the ramp angle affect the acceleration of a ramp-and-pulley system?", name the independent, dependent and two controlled variables.

6

Write a hypothesis for the question above.

More practice: warm-up

1

Can friction decide which way a system moves?

2

What force accelerates a box riding without slipping on an accelerating truck?

3

Does a(max) = μs g depend on the box's mass?

4

Which variable does the experimenter deliberately change?

5

Why should you do several trials for each setting?

More practice: on your own

1

A 5.0 kg block on a 40° ramp (μk = 0.20) is tied over a pulley at the top to a 2.0 kg hanging mass. The block slides down the ramp. Find the acceleration and tension.

2

A truck carrying an unsecured crate (μs = 0.50) brakes at 5.0 m/s². Does the crate slide? Which way?

3

What hanging mass will pull a 3.0 kg block up a frictionless 25° ramp at constant velocity?

4

A 20 kg toolbox (μs = 0.30) rides in a truck accelerating at 2.0 m/s². Find the friction force on the toolbox. Does it slide?

5

A student times a ramp-and-pulley system over 1.00 m from rest: 1.42, 1.38, 1.45, 1.40 and 1.35 s. The predicted acceleration is 1.10 m/s². Find the measured acceleration and the percent difference.

Challenge

1

In Example 3, the floor now has μk = 0.10 under the box (the book still has μs = 0.40 on the box). Find the largest pull for which the book does not slip.

2

A 4.0 kg block rests on a 30° ramp with μs = 0.40, tied over a pulley at the top to a hanging mass m. For what range of m does the system stay at rest?

3

In Example 2(c), the crate starts 1.5 m from the open tailgate. How long until it slides off the back of the truck?

Check your understanding

Quiz

1.

A block on a ramp is tied over a pulley to a hanging mass. To decide which way the system moves, you first compare:

2.

A crate in a truck bed has μs = 0.40. The greatest acceleration of the truck without the crate sliding is about:

3.

A truck accelerates too quickly and an unsecured box slides toward the tailgate. Relative to the ground, the box:

4.

In a test of how hanging mass affects acceleration, the ramp angle should be:

5.

A 2.0 kg block sits on a 3.0 kg block on a frictionless floor (μs = 0.50 between blocks). The largest pull on the bottom block without slipping is about:

6.

Which statement is a testable hypothesis?

For parents and teachers

Big idea: These are the curriculum's own sample systems for B2.4 and B2.5: an inclined-plane pulley system, a load on an accelerating vehicle, and connected or stacked objects. The lesson also teaches how to plan an inquiry (A1.1 question and hypothesis, A1.2 variables and methods, A1.4 safety, A1.10 conclusions).

How to help: Before your teen calculates, ask: "Which way will it go, and how do you know?" For the truck problem, ask: "What force pushes the box forward?" Many students answer "inertia" or "the truck", but it is static friction.

Common difficulties:

  • Choosing the friction direction before deciding the direction of motion.
  • Thinking a sliding load is "pushed backward". It accelerates forward, just less than the vehicle.
  • Confusing independent and controlled variables when planning.

Safety: the paper-and-cup activity uses plastic cups only.