Rotational Systems Analysis

Design a detailed experiment to investigate the conservation of angular momentum in a system where two ice skaters initially at rest push off from one another. Describe the variables you would measure, the data you would collect, and how you would analyze this data to confirm the conservation of angular momentum.

• Measure the initial and final velocities and directions of the skaters, calculate their momenta, and compare the total initial and final angular momenta.

• Measure the force exerted by each skater, calculate the change in kinetic energy, and verify energy conservation.

• Record the time it takes for each skater to stop, calculate the average acceleration, and use it to infer momentum.

• Observe the color changes in the skaters' outfits due to exertion and correlate this with energy expenditure.

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DOK Level 4: Extended Thinking

Propose a research project that explores the effects of varying torque on the angular acceleration of a rotating disk. Include a hypothesis, a detailed method for applying different torques, and a plan for measuring angular acceleration and analyzing the results.

• Hypothesize that increased torque leads to increased angular acceleration, use a motor to apply varying torques, measure the resulting angular accelerations with sensors, and analyze the data to confirm the relationship.

• Assume constant angular acceleration regardless of torque, use hand-spinning methods, and measure time to stop.

• Predict that torque has no effect on angular acceleration, apply constant torque, and measure disk vibrations instead of acceleration.

• Suggest that higher torque decreases angular acceleration, apply torque using weights, and measure the speed of sound in the disk material.

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DOK Level 4: Extended Thinking

Develop a comprehensive plan to analyze rotational kinetic energy in a system where multiple gears interact. Describe how you would model the system, the types of measurements you would need to take, and the calculations required to determine the total rotational kinetic energy of the system.

• Treat the system as a single object, measure total mass and average radius, and use translational kinetic energy formulas.

• Ignore the interactions between gears, measure the temperature increase of the system, and estimate energy based on heat produced.

• Focus on the material properties of the gears, calculate stress and strain instead of kinetic energy.

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DOK Level 4: Extended Thinking

Imagine a scenario where two rotating bodies collide and stick together. Outline a detailed investigation to analyze the angular momentum before and after the collision. Include how you would ensure the conservation of angular momentum and what factors might affect the outcome.

• Measure the angular velocities and moments of inertia before the collision, calculate initial angular momenta, simulate the collision, measure final angular velocity, and verify angular momentum conservation.

• Focus only on the final state of the bodies, assuming conservation of momentum without verification.

• Calculate the heat generated during the collision instead of focusing on angular momentum.

• Measure the sound emitted during the collision to infer changes in angular momentum.

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DOK Level 4: Extended Thinking

Formulate a project to study rotational equilibrium in a complex system with multiple forces acting at different points. Explain how you would model the system, identify all forces and torques, and describe the analytical approach to confirm that the system is in rotational equilibrium.

• Assume the system is always in equilibrium and simply measure the net force without considering torques.

• Focus on linear motion analysis, ignoring rotational effects.

• Measure electrical conductivity of the system to infer equilibrium state.

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DOK Level 4: Extended Thinking

Rotational inertia I represents an object’s resistance to angular acceleration.

Like mass, it is a ​ (a)   quantity

• scalar
• vector

Answer explanation

Rotational inertia I represents an object’s resistance to angular acceleration.

Like mass, it is a scalar quantity

Match the following

• F=ma

• x

• v

• Vf=Vo+at

• a

Answer explanation

Relationship between angular and linear motion

The linear displacement of a rotating object is given by rθ, where r is the distance from the rotational axis.

The linear speed of a rotating object is given by v = rω

The linear acceleration of a rotating object is given by a = rα.

Rotational kinematics -- Equations

The constant-acceleration kinematics equations for rotation are essentially identical to those for linear motion:

Match the following

• angular displacement

• moment of inertia

• torque

• angular acceleration

• angular velocity

​ ​ ​ (a)   is how fast an object is rotating at a specific moment in time.​

• Instantaneous angular velocity
• Average Angular Velocity
• Angular acceleration
• Angular Displacement

Answer explanation

Instantaneous angular velocity is how fast an object is rotating at a specific moment in time.

​ (a)   ω is angular displacement divided by the time interval over which that angular displacement occurred.  It is measured in rad/s.

• Average angular velocity
• Instantaneous Angular Velocity
• Angular acceleration
• Angular Displacement

Answer explanation

Average angular velocity ω is angular displacement divided by the time interval over which that angular displacement occurred.  It is measured in rad/s.