M016 · Newtonian contact mechanics

Accelerating Wedge & Sliding Block

A sliding wedge and frictionless block share a coupled motion. Change both masses, slope, release point and initial relative speed; apply a signed horizontal force to the wedge. Compare the fixed-base reference, inspect reaction forces and account for horizontal impulse and internal normal-force work.

Recorded time $0\,\mathrm{s}$

Distance down the face $1.4\,\mathrm{m}$

Signed velocity down the face $0\,\mathrm{m/s}$

Acceleration along the face $0\,\mathrm{m/s^2}$

Normal force on block $0\,\mathrm{N}$

Wedge displacement $0\,\mathrm{m}$

Wedge velocity $0\,\mathrm{m/s}$

Wedge acceleration $0\,\mathrm{m/s^2}$

Block kinetic energy $0\,\mathrm{J}$

Wedge kinetic energy $0\,\mathrm{J}$

Normal-force work on block $0\,\mathrm{J}$

Horizontal external work $0\,\mathrm{J}$

Total horizontal momentum $0\,\mathrm{\frac{kg\,m}{s}}$

Horizontal impulse residual $0\,\mathrm{\frac{kg\,m}{s}}$

Record end time $0\,\mathrm{s}$

Energy-ledger residual $0\,\mathrm{J}$

PREDICT / RELEASE / ACCOUNT

Can a frictionless normal force transfer energy?

Physical forcesOne fixed force scale per trialRelease slotInitial coordinate on the moving faceContact reaction on baseOpposite normal force at the contactSlope angle

\theta

30.00\,{}^{\circ}

Block mass

m

1.00\,\mathrm{kg}

Release fraction along the face

s_0/\ell

0.35\,

Initial velocity down the face

\dot s_0

0.00\,\mathrm{m/s}

Horizontal force on wedge

H

0.00\,\mathrm{N}

Wedge mass

M

1.00\,\mathrm{kg}

Inspect recorded time

t/t_{\mathrm{end}}

0.00\,\%

Drag the release and slope handles below, or use keyboard sliders. Editing creates a fresh trial. Rightward wedge force is positive; the block coordinate increases downhill. Normal force can exchange energy between the bodies but its total internal work cancels. Gray traces are separate ideal references. The zero-speed action is enabled only for a future relative turning point within the record.

Rigid bodies, frictionless horizontal support and frictionless face; the wedge remains level. The initial wedge speed is zero. The finite track ends the record before an impact or detachment. Supports and block dimensions are schematic.

Start with the essentials

Focus question: Can a frictionless normal force transfer energy between two bodies?
One-sentence intuition: The normal does no work on relative motion along the face, yet the translating contact transfers laboratory-frame energy from one body to the other.

Core mathematical model

Define the two coordinates

x_b=X+(s-\ell/2)\cos\theta,\qquad y_b=(\ell-s)\sin\theta

The wedge moves right with the ground-frame coordinate; the block moves down its face with the relative coordinate. A constant schematic normal offset of the drawn block does not change accelerations or energy differences.

Coupled Newton equations

(M+m)\ddot X+m\cos\theta\,\ddot s=H,\qquad\ddot s+\cos\theta\,\ddot X=g\sin\theta

The first equation is horizontal momentum balance for both bodies. The second projects the block equation onto the frictionless face. External horizontal force acts on the wedge only.

Recoil and contact

\ddot X=\frac{H-mg\sin\theta\cos\theta}{M+m\sin^2\theta},\qquad N=m(g\cos\theta+\ddot X\sin\theta)

With no external force the wedge recoils left. Contact force remains positive throughout the allowed control domain. The block and wedge do not detach in this model.

Momentum and total energy

p_x=(M+m)\dot X+m\dot s\cos\theta=p_x(0)+Ht,\qquad\Delta(K_b+K_w+U)=H(X-X_0)

Both kinetic energies are calculated in the laboratory frame, including the velocity cross term for the block. At zero external force total energy and horizontal momentum remain constant.

A frictionless contact can exchange work

W_{N,b}=N\sin\theta\,(X-X_0),\qquad W_{N,w}=-W_{N,b}

The normal is perpendicular to relative displacement along the face, but not to the laboratory-frame displacement of the block. It can transfer energy between bodies while its net internal work cancels.

Common difficulties

A fixed-base equation has a boundary

Typical misconceptionThe block always has the same downhill acceleration on a given slope.

Better mental modelA moving wedge changes the constraint acceleration. Compare the free and heavy-base presets with the gray fixed-base reference.

Normal force can do ground-frame work

Typical misconceptionA normal force never does work.

Better mental modelIt does no work along relative motion, but can transfer energy through the translating support. The two normal-work contributions are opposite.

A frame force is not another contact

Typical misconceptionThe animation adds a centrifugal or pseudo-force to the inertial balance.

Better mental modelThe 3D arrows show actual physical forces only. Relative acceleration follows from the laboratory-frame constraint equations.

Run the experiment

01
Release the base
Use the free-wedge preset and complete the record.
What to observe: Leftward wedge momentum balances the block momentum; the relative slide is faster than the fixed-base reference.
02
Make the base heavy
Increase wedge mass while keeping the block and slope fixed.
What to observe: Recoil shrinks and the block approaches the fixed-base limit.
03
Drive the wedge
Select driven-uphill and inspect relative velocity.
What to observe: An accelerating support can send a frictionless block uphill; external work and impulse account for the motion.
04
Inspect the transfer
Show the contact reaction and inspect normal-force work.
What to observe: The block and wedge exchange kinetic energy through their contact, without frictional dissipation.