Figluma mechanics worksheet

Crate pulled across a rough floor

A 12 kg crate is pulled to the right across a rough horizontal floor by a 60 N rope at 30 degrees above the horizontal. The coefficient of kinetic friction is 0.20. Find the acceleration of the crate.

MechanicsForcesHorizontal motionFriction

Diagram State

Horizontal floor with an angled pulling force resolved into horizontal and vertical components.

crate
floor
applied pull
pull components
normal force
weight
kinetic friction

Givens

Symbol	Quantity	Value	Unit
$m$	Mass	$12$	kg
$P$	Applied pull	$60$	N
$θ$	Pull angle above horizontal	$30$	deg
$μ_{k}$	Kinetic friction coefficient	$0.20$	unitless
$g$	Gravitational field strength	$9.8$	m s^-2

Unknowns

Symbol	Quantity	Value	Unit
$a$	Acceleration to the right	$?$	m s^-2

Coordinate System / Sign Convention

+x is to the right, in the direction of motion. +y is upward.

Assumptions / Constraints Checklist

The crate is already sliding right, so kinetic friction applies.
The rope force is constant and acts 30 degrees above the horizontal.
The crate stays in contact with the floor, so vertical acceleration is zero.
Air resistance is negligible.
Vertical forces balance because the crate does not lift off or sink through the floor.
The upward component of the pull reduces the normal force.
Kinetic friction points left because the crate slides right.
The contact model requires P sin(theta) < mg.

Student Solve Checklist

Mark each row only after your setup matches the diagram state; worked equations stay in the teacher key.

SetupAxes, sign convention, model constraints, and linked-motion/origin choices are stated.
Choose horizontal and vertical axes
ComponentsResolved components, force directions, normal/friction setup, or velocity split are correct.
Balance vertical forcesWrite kinetic friction from the reduced normal force
Net-force / governing equationThe main Newton's law or motion equation uses the right model, signs, and shared variables.
Apply Newton's second law horizontally
ResultThe final rearrangement, numeric value, units, and direction/speed interpretation are correct.
Divide by mass and calculate acceleration

Student Working Area

Solution / Answer Key

1. Choose horizontal and vertical axes
$+x = right, +y = upward$
The crate accelerates horizontally, so standard horizontal and vertical axes keep the motion equation direct.
Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Choose horizontal and vertical axes" the right next equation?Listen for: The crate accelerates horizontally, so standard horizontal and vertical axes keep the motion equation direct.Flag if: Student can only quote "+x = right, +y = upward" without connecting it to the diagram state or givens.
2. Balance vertical forces
$N = m g - P sin (θ)$
The crate has no vertical acceleration. The upward rope component helps support the crate, so the floor supplies less normal force than mg.
Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Balance vertical forces" the right next equation?Listen for: The crate has no vertical acceleration. The upward rope component helps support the crate, so the floor supplies less normal force than mg.Flag if: Using N = mg and ignoring that the angled pull reduces the normal force.; Swapping the pull components by using P sin(theta) horizontally and P cos(theta) vertically.
3. Write kinetic friction from the reduced normal force
$f_{k} = μ_{k} N = μ_{k} (m g - P sin (θ))$
Kinetic friction is proportional to the normal force, and here the normal force is reduced by the upward component of the pull.
Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Write kinetic friction from the reduced normal force" the right next equation?Listen for: Kinetic friction is proportional to the normal force, and here the normal force is reduced by the upward component of the pull.Flag if: Using static-friction threshold logic after the problem states the crate is sliding.
4. Apply Newton's second law horizontally
$P cos (θ) - μ_{k} (m g - P sin (θ)) = ma$
The horizontal component of the pull drives the crate right. Kinetic friction acts left and must be subtracted.
Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Apply Newton's second law horizontally" the right next equation?Listen for: The horizontal component of the pull drives the crate right. Kinetic friction acts left and must be subtracted.Flag if: Adding friction to the pulling force instead of subtracting it.
5. Divide by mass and calculate acceleration
$a = [P cos (θ) - μ_{k} (m g - P sin (θ))] / m = 2.9 m s^{- 2}$
Substituting P = 60 N, theta = 30 degrees, mu_k = 0.20, and m = 12 kg gives about 2.9 m s^-2 to the right.
Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Divide by mass and calculate acceleration" the right next equation?Listen for: Substituting P = 60 N, theta = 30 degrees, mu_k = 0.20, and m = 12 kg gives about 2.9 m s^-2 to the right.Flag if: Putting ma in the vertical equation even though the acceleration is horizontal.; Reporting the net force without dividing by mass.

Diagnostic Checklist

Key checkpoint equations

1. Choose horizontal and vertical axes
$+x = right, +y = upward$
2. Balance vertical forces
$N = m g - P sin (θ)$
3. Write kinetic friction from the reduced normal force
$f_{k} = μ_{k} N = μ_{k} (m g - P sin (θ))$
4. Apply Newton's second law horizontally
$P cos (θ) - μ_{k} (m g - P sin (θ)) = ma$
5. Divide by mass and calculate acceleration
$a = [P cos (θ) - μ_{k} (m g - P sin (θ))] / m = 2.9 m s^{- 2}$

Common Wrong Paths

Using N = mg and ignoring that the angled pull reduces the normal force.
Swapping the pull components by using P sin(theta) horizontally and P cos(theta) vertically.
Adding friction to the pulling force instead of subtracting it.
Putting ma in the vertical equation even though the acceleration is horizontal.
Reporting the net force without dividing by mass.
Using static-friction threshold logic after the problem states the crate is sliding.

Wrong Answer Signals

Using N = mg and ignoring that the angled pull reduces the normal force.Usually indicates the "Balance vertical forces" checkpoint needs review.
$N = m g - P sin (θ)$
Swapping the pull components by using P sin(theta) horizontally and P cos(theta) vertically.Usually indicates the "Balance vertical forces" checkpoint needs review.
$N = m g - P sin (θ)$
Adding friction to the pulling force instead of subtracting it.Usually indicates the "Apply Newton's second law horizontally" checkpoint needs review.
$P cos (θ) - μ_{k} (m g - P sin (θ)) = ma$
Putting ma in the vertical equation even though the acceleration is horizontal.Usually indicates the "Divide by mass and calculate acceleration" checkpoint needs review.
$a = [P cos (θ) - μ_{k} (m g - P sin (θ))] / m = 2.9 m s^{- 2}$
Reporting the net force without dividing by mass.Usually indicates the "Divide by mass and calculate acceleration" checkpoint needs review.
$a = [P cos (θ) - μ_{k} (m g - P sin (θ))] / m = 2.9 m s^{- 2}$
Using static-friction threshold logic after the problem states the crate is sliding.Usually indicates the "Write kinetic friction from the reduced normal force" checkpoint needs review.
$f_{k} = μ_{k} N = μ_{k} (m g - P sin (θ))$

Tutor Marking Rubric

Tutor score rows use curated Figluma checkpoints as marking cues. They are not automated grading or a symbolic mark scheme.

Tutor Mark Sheet

Manual tutor mark sheet only. Use observed work and leave rows blank when evidence is copied from a reveal.

Total: ___ / 8

Setup

Choose horizontal and vertical axes

___ / 2

Axes, sign convention, model constraints, and linked-motion/origin choices are stated.If weak, reteach from "Choose horizontal and vertical axes" before assigning another variant.

Score descriptions

0No usable evidence for this row, or the work contradicts "Choose horizontal and vertical axes".
1Partly correct, but review this row's checkpoint signal: evidence reaches "Choose horizontal and vertical axes" but is not yet consistent across the row
2Complete row: Axes, sign convention, model constraints, and linked-motion/origin choices are stated.

Watch signals

Use the checkpoint titles when no curated wrong-path signal is listed for this row.

Components

Balance vertical forces; Write kinetic friction from the reduced normal force

___ / 2

Resolved components, force directions, normal/friction setup, or velocity split are correct.If weak, reteach the row from this signal: Using N = mg and ignoring that the angled pull reduces the normal force.

Score descriptions

0No usable evidence for this row, or the work contradicts "Balance vertical forces".
1Partly correct, but review this row's checkpoint signal: Using N = mg and ignoring that the angled pull reduces the normal force.
2Complete row: Resolved components, force directions, normal/friction setup, or velocity split are correct.

Watch signals

Using N = mg and ignoring that the angled pull reduces the normal force.
Swapping the pull components by using P sin(theta) horizontally and P cos(theta) vertically.

Net-force / governing equation

Apply Newton's second law horizontally

___ / 2

The main Newton's law or motion equation uses the right model, signs, and shared variables.If weak, reteach the row from this signal: Adding friction to the pulling force instead of subtracting it.

Score descriptions

0No usable evidence for this row, or the work contradicts "Apply Newton's second law horizontally".
1Partly correct, but review this row's checkpoint signal: Adding friction to the pulling force instead of subtracting it.
2Complete row: The main Newton's law or motion equation uses the right model, signs, and shared variables.

Watch signals

Adding friction to the pulling force instead of subtracting it.

Result

Divide by mass and calculate acceleration

___ / 2

The final rearrangement, numeric value, units, and direction/speed interpretation are correct.If weak, reteach the row from this signal: Putting ma in the vertical equation even though the acceleration is horizontal.

Score descriptions

0No usable evidence for this row, or the work contradicts "Divide by mass and calculate acceleration".
1Partly correct, but review this row's checkpoint signal: Putting ma in the vertical equation even though the acceleration is horizontal.
2Complete row: The final rearrangement, numeric value, units, and direction/speed interpretation are correct.

Watch signals

Putting ma in the vertical equation even though the acceleration is horizontal.
Reporting the net force without dividing by mass.

Total: ___ / 8Add the four row scores after checking actual student evidence. Leave it blank when the work is incomplete or only copied from reveals.Reteach from the first 0 or 1 row before assigning another variant or adding a new problem.

Setup

Axes, sign convention, model constraints, and linked-motion/origin choices are stated.

012

Score guide

0No usable evidence for this row, or the work contradicts "Choose horizontal and vertical axes".
1Partly correct, but review this row's checkpoint signal: evidence reaches "Choose horizontal and vertical axes" but is not yet consistent across the row
2Complete row: Axes, sign convention, model constraints, and linked-motion/origin choices are stated.

Checkpoints

Choose horizontal and vertical axes
$+x = right, +y = upward$

Watch for

Use the checkpoint equations for this row.

Components

Resolved components, force directions, normal/friction setup, or velocity split are correct.

012

Score guide

0No usable evidence for this row, or the work contradicts "Balance vertical forces".
1Partly correct, but review this row's checkpoint signal: Using N = mg and ignoring that the angled pull reduces the normal force.
2Complete row: Resolved components, force directions, normal/friction setup, or velocity split are correct.

Checkpoints

Balance vertical forces
$N = m g - P sin (θ)$
Write kinetic friction from the reduced normal force
$f_{k} = μ_{k} N = μ_{k} (m g - P sin (θ))$

Watch for

Using N = mg and ignoring that the angled pull reduces the normal force.
Swapping the pull components by using P sin(theta) horizontally and P cos(theta) vertically.
Using static-friction threshold logic after the problem states the crate is sliding.

Net-force / governing equation

The main Newton's law or motion equation uses the right model, signs, and shared variables.

012

Score guide

0No usable evidence for this row, or the work contradicts "Apply Newton's second law horizontally".
1Partly correct, but review this row's checkpoint signal: Adding friction to the pulling force instead of subtracting it.
2Complete row: The main Newton's law or motion equation uses the right model, signs, and shared variables.

Checkpoints

Apply Newton's second law horizontally
$P cos (θ) - μ_{k} (m g - P sin (θ)) = ma$

Watch for

Adding friction to the pulling force instead of subtracting it.

Result

The final rearrangement, numeric value, units, and direction/speed interpretation are correct.

012

Score guide

0No usable evidence for this row, or the work contradicts "Divide by mass and calculate acceleration".
1Partly correct, but review this row's checkpoint signal: Putting ma in the vertical equation even though the acceleration is horizontal.
2Complete row: The final rearrangement, numeric value, units, and direction/speed interpretation are correct.

Checkpoints

Divide by mass and calculate acceleration
$a = [P cos (θ) - μ_{k} (m g - P sin (θ))] / m = 2.9 m s^{- 2}$

Watch for

Putting ma in the vertical equation even though the acceleration is horizontal.
Reporting the net force without dividing by mass.

Diagram State

Givens

Unknowns

Coordinate System / Sign Convention

Assumptions / Constraints Checklist

Student Solve Checklist

Student Working Area

1. Choose horizontal and vertical axes

2. Balance vertical forces

3. Write kinetic friction from the reduced normal force

4. Apply Newton's second law horizontally

5. Divide by mass and calculate acceleration

Diagnostic Checklist

Key checkpoint equations

Common Wrong Paths

Wrong Answer Signals

Tutor Marking Rubric

Tutor Mark Sheet

Setup

Components

Net-force / governing equation

Result

Setup

Score guide

Checkpoints

Watch for

Components

Score guide

Checkpoints

Watch for

Net-force / governing equation

Score guide

Checkpoints

Watch for

Result

Score guide

Checkpoints

Watch for