Equations of Rotational Motion Video Lessons

Concept

Problem: Rotational motion with a constant nonzero acceleration is not uncommon in the world around us. For instance, many machines have spinning parts. When the machine is turned on or off, the spinning parts tend to change the rate of their rotation with virtually constant angular acceleration. Many introductory problems in rotational kinematics involve motion of a particle with constant, nonzero angular acceleration. The kinematic equations for such motion can be written as:θ(t) = θ0 + ω0t + 1/2 αt2andω(t) = ω0 + αtIn answering the following questions, assume that the angular acceleration is constant and nonzero:C) True or false: The quantity represented by ω0 is a function of time (i.e., is not constant).D) True or false: The quantity represented by ω is a function of time (i.e., is not constant).E) Which of the following equations is not an explicit function of time t? Keep in mind that an equation that is an explicit function of time involves t as a variable.A. θ = θ0 + ω0t + 1/2 αt2B. ω = ω0 + αtC. ω2 = ω02 + 2α(θ - θ0)F) In the equation ω = ω0 + αt, what does the time variable t represent? Choose the answer that is always true. Several of the statements may be true in a particular problem, but only one is always true.A. the moment in time at which the angular velocity equals ω0B. the moment in time at which the angular velocity equals ωC. the time elapsed from when the angular velocity equals ω0 until the angular velocity equals ωG) Consider two particles A and B. The angular position of particle A, with constant angular acceleration, depends on time according to θA(t) = θ0 + ω0t + 1/2 αt2. At time t = t1, particle B, which also undergoes constant angular acceleration, has twice the angular acceleration, half the angular velocity, and the same angular position that particle A had at time t = 0.Which of the following equations describes the angular position of particle B?A. θB(t) = θ0 + 2 ω0t + 1/4 αt2B. θB(t) = θ0 + 1/2 ω0t + αt2C. θB(t) = θ0 + 2 ω0(t - t1) + 1/4 α(t - t1)2D. θB(t) = θ0 + 1/2 ω0(t - t1) + α(t - t1)2E. θB(t) = θ0 + 2 ω0(t + t1) + 1/4 α(t + t1)2F. θB(t) = θ0 + 1/2 ω0(t + t1) + α(t + t1)2

FREE Expert Solution

C)

The quantity ω0 is the initial angular velocity of the particle.

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Problem Details

Rotational motion with a constant nonzero acceleration is not uncommon in the world around us. For instance, many machines have spinning parts. When the machine is turned on or off, the spinning parts tend to change the rate of their rotation with virtually constant angular acceleration. Many introductory problems in rotational kinematics involve motion of a particle with constant, nonzero angular acceleration. The kinematic equations for such motion can be written as:

θ(t) = θ0 + ω0t + 1/2 αt2

and

ω(t) = ω0 + αt

In answering the following questions, assume that the angular acceleration is constant and nonzero:

C) True or false: The quantity represented by ω0 is a function of time (i.e., is not constant).

D) True or false: The quantity represented by ω is a function of time (i.e., is not constant).

E) Which of the following equations is not an explicit function of time t? Keep in mind that an equation that is an explicit function of time involves t as a variable.

A. θ = θ0 + ω0t + 1/2 αt2

B. ω = ω0 + αt

C. ω2 = ω02 + 2α(θ - θ0)

F) In the equation ω = ω0 + αt, what does the time variable t represent? Choose the answer that is always true. Several of the statements may be true in a particular problem, but only one is always true.

A. the moment in time at which the angular velocity equals ω0

B. the moment in time at which the angular velocity equals ω

C. the time elapsed from when the angular velocity equals ω0 until the angular velocity equals ω

G) Consider two particles A and B. The angular position of particle A, with constant angular acceleration, depends on time according to θA(t) = θ0 + ω0t + 1/2 αt2. At time t = t1, particle B, which also undergoes constant angular acceleration, has twice the angular acceleration, half the angular velocity, and the same angular position that particle A had at time t = 0.

Which of the following equations describes the angular position of particle B?

A. θB(t) = θ0 + 2 ω0t + 1/4 αt2

B. θB(t) = θ0 + 1/2 ω0t + αt2

C. θB(t) = θ0 + 2 ω0(t - t1) + 1/4 α(t - t1)2

D. θB(t) = θ0 + 1/2 ω0(t - t1) + α(t - t1)2

E. θB(t) = θ0 + 2 ω0(t + t1) + 1/4 α(t + t1)2

F. θB(t) = θ0 + 1/2 ω0(t + t1) + α(t + t1)2