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6.2.4Energy and Momentum of Rotating Systems

Angular Impulse Graphs — AP PHYSICS 1 Revision Notes & Topical Questions

5 min read 9 parts Lesson 7 of 13 1 quiz

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Lesson summary

6.2.4 Angular Impulse Graphs — AP PHYSICS 1 notes: Interpret torque–time and angular momentum–time graphs, finding angular impulse from areas and torque from gradients.Area — the area under a net torque–time graph equals the angular impulse, which equals ΔL.Sign — areas above the…

  • Rectangle (constant torque)
  • Straight line
  • Horizontal line
  • Curve with increasing gradient
  • Constant I
  • Gradient of ω–t

Section 1 of 9Torque–time graphs

1Area
  • Area the area under a net torque–time graph equals the angular impulse, which equals ΔL.
2Sign
  • Sign areas above the axis are positive and below are negative.
3Rectangle (constant torque)
  • τΔt.
4Triangle
  • ½ × base × height.
5Trapezium
  • ½(a + b) × width.
6Total
  • Total add areas of simple shapes and subtract those below the axis.

Section 2 of 9Angular momentum–time graphs

Angular momentum–time graphs
Gradient
  • Gradient equals the net torque, since τ = ΔL/Δt.
Straight line
  • Straight line constant torque.
Horizontal line
  • Horizontal line zero net torque, so L is conserved.
Curve with increasing gradient
  • Curve with increasing gradient growing torque.
Relation
  • Relation the τ–t graph is the gradient of the L–t graph and the L–t graph is built from the area under the τ–t graph.

Section 3 of 9Angular velocity from graphs

Constant I
  • ω = L/I, so an L–t graph has the shape of an ω–t graph scaled by 1/I.
Gradient of ω–t
  • α = τ/I.
Final ω
  • Final ω find the area (ΔL), divide by I, and add the initial ω.
Example
  • Example a triangle of 15 N·m·s and a rectangle of −10 N·m·s give ΔL = 5 N·m·s;
    • with I = 5.0 kg·m2 and ω0 = −2.0 rad/s, ω = −1.0 rad/s.

Section 4 of 9Reading graphs carefully

Axes and units
  • Axes and units check them before computing areas.
Force graph
  • Force graph multiply by the lever arm to get torque.
Irregular shapes
  • Irregular shapes count squares and convert using the value of one square.
Below the axis
  • Below the axis the torque reverses and ΔL reduces.
Torque returning to zero
  • Torque returning to zero the object then keeps ω constant.

Section 5 of 9Impulse versus work graphs

GraphHorizontal axisArea gives
τ against tTimeAngular impulse ΔL
τ against θAngleWork
Check
  • Check read the axis label to tell which is shown.
Analogy
  • Analogy area under F–t gives Δp; area under τ–t gives ΔL.

Section 6 of 9Sketching from a description

1Steady torque for 3.0 s then coasting
  • τ–t is a rectangle ending at 3.0 s; L–t rises linearly then becomes horizontal.
2With constant opposing friction
  • L rises more gently during the push, then falls with a small negative gradient.
3Check
  • Check positive area minus negative area equals the final ΔL; equal areas return L to zero.
4Labels
  • Labels label axes with quantities and units, and mark key times.

Section 7 of 9More from the specification

More from the specification
Rotation
  • Motion in which an object turns about an axis.
Collision
  • Short interaction with large internal forces.
Newton
  • Unit of force, kg m s-2.
Rotational inertia
  • Resistance to angular acceleration, I = Σmr2.
Average torque
  • Constant torque giving the same angular impulse in the same time.
External torque
  • Torque produced by a force from outside the system.
Rigid system
  • System that holds its shape while rotating.
Velocity
  • Rate of change of displacement.
Inertia
  • Resistance of an object to changes in its motion.
Distance
  • Total length of path travelled, a scalar.

Section 8 of 9Key terms

TermMeaning
Torque–time graphGraph whose area gives the angular impulse.
GradientSlope of a graph; on an L–t graph it is the net torque.
Area under a curveProduct of the two axis quantities.
Angular impulseτΔt, equal to the change in angular momentum.
Constant torqueHorizontal line on a τ–t graph, sloped line on an L–t graph.

Section 9 of 9Exam technique

1
Taking the gradient of a torque–time graph instead of the area to find ΔL.
2
Mixing the meaning of the area under τ–t (impulse) and τ–θ (work).
3
Forgetting to subtract areas below the axis.
4
Using the maximum torque instead of the average for a varying torque.
5
Not dividing by I to convert ΔL to Δω.

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