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Mechanics

Block on a rough inclineForces / Inclined planes / Friction Crate pulled across a rough floorForces / Horizontal motion / Friction Table block and hanging massConnected particles / Friction / Newton's laws Passenger in an accelerating elevatorNewton's laws / Apparent weight / Vertical motion Projectile launched from a cliffProjectile motion / Kinematics / Vectors

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.

State

Givens

$m$ Mass: $2.0 kg$
$θ$ Incline angle: $3 0^{\circ}$
$μ_{k}$ Kinetic friction coefficient: $0.20$
$g$ Gravitational field strength: $9.8 m s^{- 2}$

Unknowns

$a$ Acceleration down the slope: $? m s^{- 2}$

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

Assumptions

The block is already sliding, so kinetic friction applies.
Air resistance is negligible.
The incline angle and friction coefficient are constant.

Constraints

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.

Block on a rough incline

Active checkpoint: Choose the slope-aligned axes

LookSetupChoose the slope-aligned axes

ChooseChoose the slope-aligned axesThe slope-aligned axes make the acceleration one-dimensional. The block accelerates along x, while the y forces balance.

AvoidWrong turnReporting the acceleration with the opposite sign or with force/speed units.

Equation lens

+ x

down slope, +y = normal outward

Choose the slope-aligned axesBefore revealing the worked equation, decide what "Choose the slope-aligned axes" must establish from the diagram and givens.

Hint before equation

Set the axes, origin, sign convention, or linked-motion choice before writing algebra.

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

Equation-choice checkWhat feature of the diagram, sign convention, or givens makes "Choose the slope-aligned axes" the right next equation?

Step 1 of 5SetupChoose the slope-aligned axes

Use the diagram focus, problem state, and scratch workspace first.

Checkpoints

Choose the slope-aligned axes
Not yet
Set the axes, origin, sign convention, or linked-motion choice before writing algebra.
Watch: Reporting the acceleration with the opposite sign or with force/speed units.
Resolve weight perpendicular to the plane
Not yet
Resolve only onto the chosen axes and check which part of each vector belongs in this step.
Watch: Swapping sin(theta) and cos(theta) for the slope and normal components.
Write friction from the normal force
Not yet
Separate driving terms from opposing terms, then assign signs from the positive direction.
Watch: Swapping sin(theta) and cos(theta) for the slope and normal components.
Apply Newton's second law along the slope
Not yet
Separate driving terms from opposing terms, then assign signs from the positive direction.
Watch: Using mg instead of mg sin(theta) along the slope.
Cancel mass and calculate acceleration
Not yet
Substitute after the model equation is settled, then check units and the requested interpretation.
Watch: Keeping mass in the final acceleration even though it cancels.

Common traps

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.

Solve Workspace

Solve checklist0 / 4 marked

Checkpoint reveal0 / 5 revealed

Checkpoint progress0 / 5 worked; 0 checked

Equation reasons0 / 5 noted

SetupAxes, sign convention, model constraints, and linked-motion/origin choices are stated.
Choose the slope-aligned axes
1 common trap tied to this row.
ComponentsResolved components, force directions, normal/friction setup, or velocity split are correct.
Resolve weight perpendicular to the planeWrite friction from the normal force
1 common trap tied to this row.
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
3 common traps tied to this row.
ResultThe final rearrangement, numeric value, units, and direction/speed interpretation are correct.
Cancel mass and calculate acceleration
1 common trap tied to this row.

Why this equation?

Local note for the active checkpoint. It is copied into the attempt snapshot for validation.

Scratch working