M016 · Newtonian contact mechanics

Accelerating Wedge & Sliding Block

Can a frictionless normal force transfer energy? Yes. In the ground frame, a freely recoiling wedge gains the energy that the normal force removes from the block. A horizontal drive can reverse this transfer. In the wedge frame, the normal force does no work on the relative slide; the energy balance includes inertial-force work.

Release or horizontally drive a frictionless wedge. Compare ground and wedge observers at the same instant, with consistent force arrows, inertial forces, kinetic energies and work.

Recorded time $0.228\,\mathrm{s}$

Distance down the face $1.6\,\mathrm{m}$

Velocity relative to the face $1.79\,\mathrm{m\,s^{-1}}$

Acceleration along the face $7.85\,\mathrm{m\,s^{-2}}$

Normal force on block $6.8\,\mathrm{N}$

Block speed in selected frame $1.18\,\mathrm{m\,s^{-1}}$

Block horizontal acceleration in selected frame $3.4\,\mathrm{m\,s^{-2}}$

Horizontal inertial force on block $0\,\mathrm{N}$

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

Wedge displacement $-0.0883\,\mathrm{m}$

Wedge velocity $-0.775\,\mathrm{m\,s^{-1}}$

Wedge acceleration in ground frame $-3.4\,\mathrm{m\,s^{-2}}$

Block kinetic energy $0.7\,\mathrm{J}$

Wedge kinetic energy $0.3\,\mathrm{J}$

Normal work in selected frame $-0.3\,\mathrm{J}$

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

Total horizontal momentum $0\,\mathrm{kg\,m\,s^{-1}}$

Horizontal impulse residual $0\,\mathrm{kg\,m\,s^{-1}}$

Record end time $0.814\,\mathrm{s}$

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

Predict the motion, then track the energy

Can a frictionless normal force transfer energy?

Physical forcesAll forces share one fixed scale in each trial.Release slotMarks where the block starts on the moving face.Contact reaction on baseShows the opposite normal force acting on the base at the contact.Slope 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\,\%

The face and horizontal guide are frictionless, with no rotational degree of freedom. Force on the wedge is positive rightward; block displacement is positive downhill. Switching observers preserves the trial and time. Violet dashed inertial forces appear only in the accelerating wedge frame; velocity, energy and work readouts use that selected frame, except the explicitly labelled ground-frame wedge acceleration. The run ends at the first face endpoint, six model seconds or two metres of wedge travel. Charts are analytic predictions; subsequent motion and impacts are outside this observation window.

Model assumptions

Both bodies are rigid. The floor and the wedge face are frictionless, and the wedge stays level. The wedge starts at rest. The run ends at a face endpoint, six model seconds or two metres of wedge travel, whichever comes first; subsequent motion, impact or detachment is outside the displayed window. The supports and block sizes are schematic.

Physics tutorial

The same contact transfers energy differently in different frames

BackgroundA freely moving wedge recoils as the block slides down. On the ground, the normal force can transfer energy between them; riding with the wedge changes the displacement used to calculate that work.

Why it mattersWatch one motion from the ground and from the moving base. OpenStax 5.7 and 9.3 provide the force and momentum principles; MIT Engineering Dynamics lecture 8 explains inertial forces in accelerating observers.

Start with the essentials

Focus question: Why does normal-force work change when the observer rides with the wedge?
One-sentence intuition: The normal is perpendicular to the slide relative to the face. Its ground-frame work can be nonzero because the face moves too; the accelerating frame needs inertial-force work in its energy balance.

Core mathematical model

Two coordinates describe the motion

\begin{aligned}x_b&=X+(s-\ell/2)\cos\theta\\y_b&=(\ell-s)\sin\theta\end{aligned}

The wedge moves horizontally; the block coordinate increases downhill along its face. The drawn body has a fixed schematic normal offset, which changes no acceleration or energy difference.

Coupled equations of motion

\begin{aligned}(M+m)\ddot X+m\cos\theta\,\ddot s&=H\\\ddot s+\cos\theta\,\ddot X&=g\sin\theta\end{aligned}

The horizontal drive acts on the wedge. The floor and face are frictionless, and the assembly has no rotational degree of freedom.

Recoil and contact force

\begin{aligned}\ddot X&=\frac{H-mg\sin\theta\cos\theta}{M+m\sin^2\theta}\\N&=m(g\cos\theta+\ddot X\sin\theta)\end{aligned}

A free wedge accelerates left. A suitable rightward drive can reverse the recoil or make the block slide uphill relative to the wedge. Contact remains compressive over the available ranges.

Momentum and energy in the ground frame

\begin{aligned}p_x&=(M+m)\dot X+m\dot s\cos\theta\\\dot p_x&=H\\\Delta(K_b+K_w+U)&=H\Delta X\end{aligned}

With no horizontal drive, total horizontal momentum and mechanical energy stay constant. Each kinetic energy uses the full ground-frame velocity.

Contact work on the two bodies

\begin{aligned}W_{N,b}&=N\sin\theta\,\Delta X\\W_{N,w}&=-W_{N,b}\end{aligned}

During free recoil, the block loses energy through contact and the wedge gains it. A drive can reverse the transfer; the two contact-work contributions still cancel.

Riding with the accelerating wedge

\begin{aligned}\mathbf a_b^{\prime}&=\mathbf a_b-\ddot X\hat{\mathbf x}\\\mathbf F_{\mathrm{in},b}&=-m\ddot X\hat{\mathbf x}\\\Delta(K_b^{\prime}+U)&=W_{\mathrm{in},b}\end{aligned}

This observer translates without rotating. Normal-force work on the relative slide is zero; the inertial force accounts for the relative energy balance. Switching observers preserves the same physical trial and current time.

Common difficulties

The base motion changes the slide

Typical misconceptionThe fixed-incline acceleration applies to every wedge.

Better mental modelA moving base accelerates the face. Compare Free wedge with Heavy base; the recoil and relative slide both change.

Work depends on the observer

Typical misconceptionA normal force can never transfer energy.

Better mental modelLook at normal work in the ground frame, then switch to the wedge frame. The relative slide is perpendicular to the normal, while the ground displacement includes base motion.

Inertial forces belong to an accelerating description

Typical misconceptionThe violet arrow is another physical contact.

Better mental modelViolet dashed arrows appear in the wedge frame. They account for the observer acceleration; the teal contact forces are the same physical interaction in either view.

Run the experiment

01
Release both bodies
Watch the Free wedge preset.
What to observe: The base recoils left as the block slides down. Ground-frame normal work on the block is negative.
02
Ride with the wedge
Pause, then choose Wedge frame.
What to observe: The base stays still and the ground datum moves. Violet inertial forces appear, and normal work on the block becomes zero.
03
Keep the same instant
Switch between the two observers.
What to observe: The trial and time are preserved. The selected-frame speed, work and kinetic energies change together.
04
Drive the block uphill
Choose Driven uphill and inspect the relative velocity.
What to observe: The rightward acceleration of the support drives relative uphill motion. External work and impulse account for the ground-frame balance.
05
Keep the block at one place on the face
Choose Ride with the wedge.
What to observe: Both bodies accelerate horizontally while their relative position stays fixed.
06
Catch a relative turning point
Choose Uphill release. Open Work, constraints and diagnostics and press Exact zero speed, then Step 1 ms.
What to observe: The relative velocity reaches zero and changes sign. The block starts sliding down the face again.

Core mathematical model

Two coordinates describe the motion

Coupled equations of motion

Recoil and contact force

Momentum and energy in the ground frame

Contact work on the two bodies

Riding with the accelerating wedge

Common difficulties

The base motion changes the slide

Work depends on the observer

Inertial forces belong to an accelerating description

Run the experiment

Release both bodies

Ride with the wedge

Keep the same instant

Drive the block uphill

Keep the block at one place on the face

Catch a relative turning point