Figluma mechanics worksheet

Block on a rough incline

A 2.0 kg block is released from rest on a 30 degree rough incline. The coefficient of kinetic friction is 0.20. Find the acceleration down the slope.

MechanicsForcesInclined planesFriction

Diagram State

Rough incline with weight resolved into slope-parallel and normal components.

incline
block
normal force
kinetic friction
weight components
slope axes

Givens

Symbol	Quantity	Value	Unit
$m$	Mass	$2.0$	kg
$θ$	Incline angle	$30$	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 down the slope	$?$	m s^-2

Coordinate System / Sign Convention

+x is down the slope. +y is perpendicular outward from the slope.

Assumptions / Constraints Checklist

The block is already sliding, so kinetic friction applies.
Air resistance is negligible.
The incline angle and friction coefficient are constant.
Normal force is perpendicular to the plane.
Friction opposes motion, so it points up the plane.
Only the component of weight parallel to the plane accelerates the block.

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 the slope-aligned axes
ComponentsResolved components, force directions, normal/friction setup, or velocity split are correct.
Resolve weight perpendicular to the planeWrite friction from the 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 along the slope
ResultThe final rearrangement, numeric value, units, and direction/speed interpretation are correct.
Cancel mass and calculate acceleration

Student Working Area

Solution / Answer Key

1. Choose the slope-aligned axes
$+x = down slope, +y = normal outward$
The slope-aligned axes make the acceleration one-dimensional. The block accelerates along x, while the y forces balance.
Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Choose the slope-aligned axes" the right next equation?Listen for: The slope-aligned axes make the acceleration one-dimensional. The block accelerates along x, while the y forces balance.Flag if: Reporting the acceleration with the opposite sign or with force/speed units.
2. Resolve weight perpendicular to the plane
$N = m g cos (θ)$
There is no acceleration through the plane, so the normal force balances the perpendicular component of weight.
Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Resolve weight perpendicular to the plane" the right next equation?Listen for: There is no acceleration through the plane, so the normal force balances the perpendicular component of weight.Flag if: Swapping sin(theta) and cos(theta) for the slope and normal components.
3. Write friction from the normal force
$f_{k} = μ_{k} N = μ_{k} m g cos (θ)$
Kinetic friction is proportional to the normal force and points up the plane because motion is down the plane.
Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Write friction from the normal force" the right next equation?Listen for: Kinetic friction is proportional to the normal force and points up the plane because motion is down the plane.Flag if: Student can only quote "f_k = mu_k N = mu_k mg cos(theta)" without connecting it to the diagram state or givens.
4. Apply Newton's second law along the slope
$m g sin (θ) - μ_{k} m g cos (θ) = ma$
The down-slope component of weight is positive. Friction is negative because it acts up the slope.
Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Apply Newton's second law along the slope" the right next equation?Listen for: The down-slope component of weight is positive. Friction is negative because it acts up the slope.Flag if: Using mg instead of mg sin(theta) along the slope.; Adding friction in the direction of motion instead of subtracting it.
5. Cancel mass and calculate acceleration
$a = g (sin (θ) - μ_{k} cos (θ)) = 3.2 m s^{- 2}$
Mass cancels because both driving weight and friction scale with m. Substituting theta = 30 degrees and mu_k = 0.20 gives about 3.2 m s^-2 down the slope.
Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Cancel mass and calculate acceleration" the right next equation?Listen for: Mass cancels because both driving weight and friction scale with m. Substituting theta = 30 degrees and mu_k = 0.20 gives about 3.2 m s^-2 down the slope.Flag if: Keeping mass in the final acceleration even though it cancels.

Diagnostic Checklist

Key checkpoint equations

1. Choose the slope-aligned axes
$+x = down slope, +y = normal outward$
2. Resolve weight perpendicular to the plane
$N = m g cos (θ)$
3. Write friction from the normal force
$f_{k} = μ_{k} N = μ_{k} m g cos (θ)$
4. Apply Newton's second law along the slope
$m g sin (θ) - μ_{k} m g cos (θ) = ma$
5. Cancel mass and calculate acceleration
$a = g (sin (θ) - μ_{k} cos (θ)) = 3.2 m s^{- 2}$

Common Wrong Paths

Using mg instead of mg sin(theta) along the slope.
Adding friction in the direction of motion instead of subtracting it.
Dropping kinetic friction from the down-slope net-force equation.
Swapping sin(theta) and cos(theta) for the slope and normal components.
Reporting the acceleration with the opposite sign or with force/speed units.
Keeping mass in the final acceleration even though it cancels.

Wrong Answer Signals

Using mg instead of mg sin(theta) along the slope.Usually indicates the "Apply Newton's second law along the slope" checkpoint needs review.
$m g sin (θ) - μ_{k} m g cos (θ) = ma$
Adding friction in the direction of motion instead of subtracting it.Usually indicates the "Apply Newton's second law along the slope" checkpoint needs review.
$m g sin (θ) - μ_{k} m g cos (θ) = ma$
Dropping kinetic friction from the down-slope net-force equation.Usually indicates the "Apply Newton's second law along the slope" checkpoint needs review.
$m g sin (θ) - μ_{k} m g cos (θ) = ma$
Swapping sin(theta) and cos(theta) for the slope and normal components.Usually indicates the "Resolve weight perpendicular to the plane" checkpoint needs review.
$N = m g cos (θ)$
Reporting the acceleration with the opposite sign or with force/speed units.Usually indicates the "Choose the slope-aligned axes" checkpoint needs review.
$+x = down slope, +y = normal outward$
Keeping mass in the final acceleration even though it cancels.Usually indicates the "Cancel mass and calculate acceleration" checkpoint needs review.
$a = g (sin (θ) - μ_{k} cos (θ)) = 3.2 m s^{- 2}$

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 the slope-aligned axes

___ / 2

Axes, sign convention, model constraints, and linked-motion/origin choices are stated.If weak, reteach the row from this signal: Reporting the acceleration with the opposite sign or with force/speed units.

Score descriptions

0No usable evidence for this row, or the work contradicts "Choose the slope-aligned axes".
1Partly correct, but review this row's checkpoint signal: Reporting the acceleration with the opposite sign or with force/speed units.
2Complete row: Axes, sign convention, model constraints, and linked-motion/origin choices are stated.

Watch signals

Reporting the acceleration with the opposite sign or with force/speed units.

Components

Resolve weight perpendicular to the plane; Write friction from the normal force

___ / 2

Resolved components, force directions, normal/friction setup, or velocity split are correct.If weak, reteach the row from this signal: Swapping sin(theta) and cos(theta) for the slope and normal components.

Score descriptions

0No usable evidence for this row, or the work contradicts "Resolve weight perpendicular to the plane".
1Partly correct, but review this row's checkpoint signal: Swapping sin(theta) and cos(theta) for the slope and normal components.
2Complete row: Resolved components, force directions, normal/friction setup, or velocity split are correct.

Watch signals

Swapping sin(theta) and cos(theta) for the slope and normal components.

Net-force / governing equation

Apply Newton's second law along the slope

___ / 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: Using mg instead of mg sin(theta) along the slope.

Score descriptions

0No usable evidence for this row, or the work contradicts "Apply Newton's second law along the slope".
1Partly correct, but review this row's checkpoint signal: Using mg instead of mg sin(theta) along the slope.
2Complete row: The main Newton's law or motion equation uses the right model, signs, and shared variables.

Watch signals

Using mg instead of mg sin(theta) along the slope.
Adding friction in the direction of motion instead of subtracting it.

Result

Cancel 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: Keeping mass in the final acceleration even though it cancels.

Score descriptions

0No usable evidence for this row, or the work contradicts "Cancel mass and calculate acceleration".
1Partly correct, but review this row's checkpoint signal: Keeping mass in the final acceleration even though it cancels.
2Complete row: The final rearrangement, numeric value, units, and direction/speed interpretation are correct.

Watch signals

Keeping mass in the final acceleration even though it cancels.

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 the slope-aligned axes".
1Partly correct, but review this row's checkpoint signal: Reporting the acceleration with the opposite sign or with force/speed units.
2Complete row: Axes, sign convention, model constraints, and linked-motion/origin choices are stated.

Checkpoints

Choose the slope-aligned axes
$+x = down slope, +y = normal outward$

Watch for

Reporting the acceleration with the opposite sign or with force/speed units.

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 "Resolve weight perpendicular to the plane".
1Partly correct, but review this row's checkpoint signal: Swapping sin(theta) and cos(theta) for the slope and normal components.
2Complete row: Resolved components, force directions, normal/friction setup, or velocity split are correct.

Checkpoints

Resolve weight perpendicular to the plane
$N = m g cos (θ)$
Write friction from the normal force
$f_{k} = μ_{k} N = μ_{k} m g cos (θ)$

Watch for

Swapping sin(theta) and cos(theta) for the slope and normal components.

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 along the slope".
1Partly correct, but review this row's checkpoint signal: Using mg instead of mg sin(theta) along the slope.
2Complete row: The main Newton's law or motion equation uses the right model, signs, and shared variables.

Checkpoints

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

Watch for

Using mg instead of mg sin(theta) along the slope.
Adding friction in the direction of motion instead of subtracting it.
Dropping kinetic friction from the down-slope net-force equation.

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 "Cancel mass and calculate acceleration".
1Partly correct, but review this row's checkpoint signal: Keeping mass in the final acceleration even though it cancels.
2Complete row: The final rearrangement, numeric value, units, and direction/speed interpretation are correct.

Checkpoints

Cancel mass and calculate acceleration
$a = g (sin (θ) - μ_{k} cos (θ)) = 3.2 m s^{- 2}$

Watch for

Keeping mass in the final acceleration even though it cancels.

Diagram State

Givens

Unknowns

Coordinate System / Sign Convention

Assumptions / Constraints Checklist

Student Solve Checklist

Student Working Area

1. Choose the slope-aligned axes

2. Resolve weight perpendicular to the plane

3. Write friction from the normal force

4. Apply Newton's second law along the slope

5. Cancel 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