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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 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.

State

Givens

$m$ Mass: $12 kg$
$P$ Applied pull: $60 N$
$θ$ Pull angle above horizontal: $3 0^{\circ}$
$μ_{k}$ Kinetic friction coefficient: $0.20$
$g$ Gravitational field strength: $9.8 m s^{- 2}$

Unknowns

$a$ Acceleration to the right: $? m s^{- 2}$

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

Assumptions

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.

Constraints

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.

Crate pulled across a rough floor

Active checkpoint: Choose horizontal and vertical axes

LookSetupChoose horizontal and vertical axes

ChooseChoose horizontal and vertical axesThe crate accelerates horizontally, so standard horizontal and vertical axes keep the motion equation direct.

AvoidWrong turnStudent can only quote "+x = right, +y = upward" without connecting it to the diagram state or givens.

Equation lens

+ x

right, +y = upward

Choose horizontal and vertical axesBefore revealing the worked equation, decide what "Choose horizontal and vertical axes" must establish from the diagram and givens.

Hint before equation

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

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

Step 1 of 5SetupChoose horizontal and vertical axes

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

Checkpoints

Choose horizontal and vertical axes
Not yet
Set the axes, origin, sign convention, or linked-motion choice before writing algebra.
Balance vertical forces
Not yet
Resolve only onto the chosen axes and check which part of each vector belongs in this step.
Watch: Using N = mg and ignoring that the angled pull reduces the normal force.
Write kinetic friction from the reduced normal force
Not yet
Separate driving terms from opposing terms, then assign signs from the positive direction.
Watch: Using static-friction threshold logic after the problem states the crate is sliding.
Apply Newton's second law horizontally
Not yet
Separate driving terms from opposing terms, then assign signs from the positive direction.
Watch: Adding friction to the pulling force instead of subtracting it.
Divide by mass and calculate acceleration
Not yet
Substitute after the model equation is settled, then check units and the requested interpretation.
Watch: Putting ma in the vertical equation even though the acceleration is horizontal.

Common traps

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.

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 horizontal and vertical axes
Use the checkpoint cues before moving on.
ComponentsResolved components, force directions, normal/friction setup, or velocity split are correct.
Balance vertical forcesWrite kinetic friction from the reduced normal force
3 common traps 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 horizontally
1 common trap tied to this row.
ResultThe final rearrangement, numeric value, units, and direction/speed interpretation are correct.
Divide by mass and calculate acceleration
2 common traps tied to this row.

Why this equation?

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

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