Practice · Unit 6 · Aula Física

Angular kinematics

A wheel spins at 300 rpm. Calculate ω.

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State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Angular kinematics

ω₀=2.0 rad/s, α=3.0 rad/s², and t=4.0 s. Calculate ω_f.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Angular kinematics

With ω₀=2.0 rad/s, α=3.0 rad/s², and t=4.0 s, calculate Δθ.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Angular kinematics

At one instant ω>0 and α<0. Which statement is necessarily correct?

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.

Linear–angular relations

A point is 0.25 m from the axis of a wheel spinning at 12 rad/s. Calculate v.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Linear–angular relations

For r=0.40 m, ω=5.0 rad/s, and α=3.0 rad/s², calculate a_t and a_r.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Linear–angular relations

B is at twice A's radius on one disk. What is v_B/v_A?

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.

Linear–angular relations

A point has a_t=1.20 m/s² and a_r=10.0 m/s². Calculate |a|.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Moment of inertia

Two masses of 2.0 kg and 3.0 kg are 0.40 m and 0.20 m from the axis. Calculate I.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Moment of inertia

A solid disk with M=4.0 kg and R=0.50 m rotates about its central axis. Calculate I.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Moment of inertia

A rod with M=3.0 kg and L=1.20 m rotates about an axis 0.30 m from its center. Calculate I.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Moment of inertia

Two wheels have equal M and R; B has more mass near the rim. Which has greater I?

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.

Rotational energy

A flywheel has I=0.60 kg·m² and ω=10 rad/s. Calculate K_rot.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Rotational energy

A flywheel with I=0.80 kg·m² starts at rest and receives 48 J. Calculate ω_f.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Rotational energy

A and B rotate with the same ω, but I_B=2I_A. Then:

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.

Rotational energy

I=5.0 kg·m² and ω changes from 20 to 10 rad/s. Calculate ΔK_rot.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Torque

A 30 N tangential force acts 0.40 m from the axis. Calculate |τ|.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Torque

F=40 N, r=0.50 m, and φ=30°. Calculate |τ|.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Torque

Take counterclockwise positive. Torques +12 N·m and -8.0 N·m act. Calculate τ_net.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Torque

A nonzero force has a line of action through the axis. Its torque is:

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.

Rotational dynamics

A body with I=0.80 kg·m² receives τ_net=4.0 N·m. Calculate α.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Rotational dynamics

A solid disk with M=4.0 kg and R=0.50 m receives a 10 N tangential rim force. Calculate α.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Rotational dynamics

A pulley has R=0.20 m and I=0.080 kg·m². Opposing tensions T₁=18 N and T₂=10 N act. Calculate |α|.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Rotational dynamics

With the same net torque, doubling I makes |α|:

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.

Rolling

A wheel with R=0.25 m rolls without slipping at v_cm=5.0 m/s. Calculate |ω|.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Rolling

A solid disk with M=2.0 kg and R=0.30 m rolls without slipping at v_cm=4.0 m/s. Calculate total K.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Rolling

A solid cylinder rolls without slipping down a 30° slope. Use g=9.8 m/s². Calculate a_cm.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Rolling

During rolling without slipping, static friction:

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.

Work and power

A +6.0 N·m torque acts through a +5.0 rad rotation. Calculate W.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Work and power

A shaft receives τ=12 N·m and rotates at ω=20 rad/s in the same sense. Calculate P.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Work and power

τ(θ)=4θ N·m. Calculate W from θ=0 to 3.0 rad.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Work and power

A motor delivers 1500 W at 600 rpm. Calculate the corresponding average torque.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Angular momentum

A 0.50 kg particle travels at 8.0 m/s along a line 0.30 m from the origin. Calculate |L|.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Angular momentum

A body rotates about a principal axis with I=0.75 kg·m² and ω=8.0 rad/s. Calculate L_z.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Angular momentum

L_z changes from 2.0 to 8.0 kg·m²/s in 0.20 s. Calculate average external torque.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Angular momentum

A particle travels in a straight line. About an origin off that line, its angular momentum:

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.

Conservation and precession

With no external torque, I changes from 4.0 to 2.0 kg·m² while ω_i=3.0 rad/s. Calculate ω_f.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Conservation and precession

For the same process from I=4.0 to 2.0 kg·m² with ω_i=3.0 rad/s, calculate ΔK_rot.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Conservation and precession

I_p=3.0 kg·m² and a 2.0 kg person moves from r_i=1.0 m to r_f=0.50 m. If ω_i=2.0 rad/s, calculate ω_f.

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Conservation and precession

A gyroscope has M=0.80 kg, r=0.15 m, I=0.020 kg·m², and ω=120 rad/s. Calculate Ω≈Mgr/(Iω).

Request a hint

State the axis, sign convention, and validity conditions before substituting.

Review the solution

Principle
Select the governing rotational relation and state the axis and sign convention.
Representation
Represent every quantity about the same origin or axis.
Calculation
Substitute the data while preserving signs and units.

Angular kinematics

Linear–angular relations

Moment of inertia

Rotational energy

Torque

Rotational dynamics

Rolling

Work and power

Conservation and precession