riaology
Unit 2 · Lesson 7 of 11

A constraint determines the allowed slope.

A particle constrained to a circular guide cannot change its horizontal position independently of its height.

Change the input. Follow the mathematics.

What you'll be able to do
  • Set up the model and calculate its dy/dx.
  • Connect the equation to the changing visual.
  • Explain the assumptions and identify a misleading interpretation.
Predict before exploring

What will change if you increase horizontal coordinate x? Predict the direction, then explain the response.

A model you can changeMove one control at a time.
Read the modelx2+y2=9,2x+2ydydx=0,dydx=−xyx^2+y^2=9,\quad2x+2y\frac{dy}{dx}=0,\quad\frac{dy}{dx}=-\frac xy

Upper semicircle of radius 3 m. Endpoints with vertical tangents are excluded from the control range.

x (m) versus y (m). Upper circle, Tangent. Selected dy/dx: -0.57735 . A constrained particle moving horizontally must change its vertical coordinate to stay on the circle.0-31.5033y (m)x (m)
Upper circleTangent · dashed

dy/dxy=9−1.52=2.59808,dydx=−1.52.59808=−0.57735y=\sqrt{9-1.5^2}=2.59808,\quad\frac{dy}{dx}=-\frac{1.5}{2.59808}=-0.57735

A constrained particle moving horizontally must change its vertical coordinate to stay on the circle.

Assumptions, units, and limits

Upper semicircle of radius 3 m. Endpoints with vertical tangents are excluded from the control range.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

01

From the situation to an equation

Differentiate the constraint with respect to x while treating y as a function of x. Every derivative of y requires dy/dx by the chain rule. Solve the resulting algebraic equation for the slope; denominators identify potential vertical tangents.

x2+y2=9,2x+2ydydx=0,dydx=−xyx^2+y^2=9,\quad2x+2y\frac{dy}{dx}=0,\quad\frac{dy}{dx}=-\frac xy
02

Keep the reasoning visible

ddx(y2)=2ydydx\frac d{dx}(y^2)=2y\frac{dy}{dx}

A constrained particle moving horizontally must change its vertical coordinate to stay on the circle.

03

What makes this model useful?

Upper semicircle of radius 3 m. Endpoints with vertical tangents are excluded from the control range.

The circle has upper and lower branches. A single y(x) expression or finite slope may fail at its left and right endpoints.

A worked example

Calculate the starting scenario

  1. Use horizontal coordinate x = 1.5 m.
  2. y=9−1.52=2.59808,dydx=−1.52.59808=−0.57735y=\sqrt{9-1.5^2}=2.59808,\quad\frac{dy}{dx}=-\frac{1.5}{2.59808}=-0.57735
  3. A constrained particle moving horizontally must change its vertical coordinate to stay on the circle. The checked quantity is dy/dx.
Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the dy/dx using horizontal coordinate x 1.5 m. State the meaning of its sign and units.

Checked quantity: dy/dx · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Differentiate the constraint with respect to x while treating y as a function of x. Every derivative of y requires dy/dx by the chain rule. Solve the resulting algebraic equation for the slope; denominators identify potential vertical tangents.

Explore the visual solution
Read the modelx2+y2=9,2x+2ydydx=0,dydx=−xyx^2+y^2=9,\quad2x+2y\frac{dy}{dx}=0,\quad\frac{dy}{dx}=-\frac xy

Upper semicircle of radius 3 m. Endpoints with vertical tangents are excluded from the control range.

x (m) versus y (m). Upper circle, Tangent. Selected dy/dx: -0.57735 . A constrained particle moving horizontally must change its vertical coordinate to stay on the circle.0-31.5033y (m)x (m)
Upper circleTangent · dashed

dy/dxy=9−1.52=2.59808,dydx=−1.52.59808=−0.57735y=\sqrt{9-1.5^2}=2.59808,\quad\frac{dy}{dx}=-\frac{1.5}{2.59808}=-0.57735

A constrained particle moving horizontally must change its vertical coordinate to stay on the circle.

Assumptions, units, and limits

Upper semicircle of radius 3 m. Endpoints with vertical tangents are excluded from the control range.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

Differentiate the constraint with respect to x while treating y as a function of x. Every derivative of y requires dy/dx by the chain rule. Solve the resulting algebraic equation for the slope; denominators identify potential vertical tangents.

x2+y2=9,2x+2ydydx=0,dydx=−xyx^2+y^2=9,\quad2x+2y\frac{dy}{dx}=0,\quad\frac{dy}{dx}=-\frac xy
Calculate and explain

Apply the idea to a new scenario

At x=2x=2 m on the upper radius-3 m circle, find dy/dx.

Checked quantity: dy/dx · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Differentiate the constraint with respect to x while treating y as a function of x. Every derivative of y requires dy/dx by the chain rule. Solve the resulting algebraic equation for the slope; denominators identify potential vertical tangents.

Explore the visual solution
Read the modelx2+y2=9,2x+2ydydx=0,dydx=−xyx^2+y^2=9,\quad2x+2y\frac{dy}{dx}=0,\quad\frac{dy}{dx}=-\frac xy

Upper semicircle of radius 3 m. Endpoints with vertical tangents are excluded from the control range.

x (m) versus y (m). Upper circle, Tangent. Selected dy/dx: -0.89443 . A constrained particle moving horizontally must change its vertical coordinate to stay on the circle.0-31.5033y (m)x (m)
Upper circleTangent · dashed

dy/dxy=9−22=2.23607,dydx=−22.23607=−0.894427y=\sqrt{9-2^2}=2.23607,\quad\frac{dy}{dx}=-\frac{2}{2.23607}=-0.894427

A constrained particle moving horizontally must change its vertical coordinate to stay on the circle.

Assumptions, units, and limits

Upper semicircle of radius 3 m. Endpoints with vertical tangents are excluded from the control range.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

Differentiate the constraint with respect to x while treating y as a function of x. Every derivative of y requires dy/dx by the chain rule. Solve the resulting algebraic equation for the slope; denominators identify potential vertical tangents.

x2+y2=9,2x+2ydydx=0,dydx=−xyx^2+y^2=9,\quad2x+2y\frac{dy}{dx}=0,\quad\frac{dy}{dx}=-\frac xy
Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual
Read the modelx2+y2=9,2x+2ydydx=0,dydx=−xyx^2+y^2=9,\quad2x+2y\frac{dy}{dx}=0,\quad\frac{dy}{dx}=-\frac xy

Upper semicircle of radius 3 m. Endpoints with vertical tangents are excluded from the control range.

x (m) versus y (m). Upper circle, Tangent. Selected dy/dx: -0.57735 . A constrained particle moving horizontally must change its vertical coordinate to stay on the circle.0-31.5033y (m)x (m)
Upper circleTangent · dashed

dy/dxy=9−1.52=2.59808,dydx=−1.52.59808=−0.57735y=\sqrt{9-1.5^2}=2.59808,\quad\frac{dy}{dx}=-\frac{1.5}{2.59808}=-0.57735

A constrained particle moving horizontally must change its vertical coordinate to stay on the circle.

Assumptions, units, and limits

Upper semicircle of radius 3 m. Endpoints with vertical tangents are excluded from the control range.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. Differentiate the constraint with respect to x while treating y as a function of x. Every derivative of y requires dy/dx by the chain rule. Solve the resulting algebraic equation for the slope; denominators identify potential vertical tangents.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

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