During simple harmonic motion (SHM), acceleration (α) changes. "ω^2 = (ωo)^2 + 2αθ" is for constant angular accelelation, so can't be used in this question.hidemi said:Homework Statement:: At the instant its angular displacement is 0.32 rad, the angular acceleration of a physical pendulum is -630 rad/s2. What is its angular frequency of oscillation?
A) 6.6 rad/s
B) 14 rad/s
C) 20 rad/s
D) 44 rad/s
E) 200 rad/s
The answer is D
Relevant Equations:: ω^2 = (ωo)^2 + 2αθ
ω^2 - (ωo)^2
= 2 (-630) (0.32)
= -403.2
This is what I have now and I stuck here.
Thank you. I think I got it but just want to double check:Steve4Physics said:During simple harmonic motion (SHM), acceleration (α) changes. "ω^2 = (ωo)^2 + 2αθ" is for constant angular accelelation, so can't be used in this question.
For linear SHM in the x-direction say, the essential relationships (the things that make it simple harmonic motion) are that:
- the magnitude of acceleration (a) is proportional to the magnitude of displacement (x) and
- acceleration acts in the opposite direction to the displacement.
This can be expressed as: a = -kx where k is a positive constant.. Note that a and x are functions of time, a(t) and x(t). With some calculus you can show k = ω² where ω is the angular frequency. So we can write
a = -ω²x
This is a key formula you need to understand/learn.
For rotational SHM (e.g. thinking about a pendulum in terms of angular changes) the equivalent formula is
α = -ω²θ
(It's basically the same formula as "a = -ω²x" but α(t) is angular acceleration and θ(t) is angular displacement.)
If you use this formula you can find ω.
Yes, use the last equation. It's a straightforward substitution.hidemi said:Thank you. I think I got it but just want to double check:
θ = A * sin(ωt)
ω = Aω * cos(ωt)
α = -Aω^2 * sin(ωt)
the last formula is what you were referring, correct?
Thank you so much.kuruman said:Yes, use the last equation. It's a straightforward substitution.
Hi @hidemi. I'm not sure if there is some confusion here, so I'm adding this. The three formulae you list are:hidemi said:Thank you. I think I got it but just want to double check:
θ = A * sin(ωt)
ω = Aω * cos(ωt)
α = -Aω^2 * sin(ωt)
the last formula is what you were referring, correct?
Thanks for the clarification!Steve4Physics said:Hi @hidemi. I'm not sure if there is some confusion here, so I'm adding this. The three formulae you list are:
θ = A sin(ωt)
ω = Aω cos(ωt)
α = -Aω² sin(ωt)
You cannot use the last formula directly, because you do not know A or t.
However note that if you substitute "θ = A sin(ωt)" (the first formula) into the last formula, you can easily show that α = -ω²θ. And "α = -ω²θ" is the formula you need to solve this problem.
The angular frequency of oscillation is a measure of how quickly a system oscillates or rotates around a central point. It is typically denoted by the Greek letter omega (ω) and is measured in radians per second.
The angular frequency of oscillation is directly related to the frequency of oscillation by the equation ω = 2πf, where f is the frequency in hertz (Hz). This means that as the frequency increases, the angular frequency also increases.
The angular frequency of oscillation is affected by the mass, stiffness, and damping of the system. A higher mass or stiffness will result in a lower angular frequency, while a higher damping will result in a higher angular frequency.
The angular frequency of oscillation is used in a variety of real-world applications, including in the design of mechanical systems, electrical circuits, and musical instruments. It is also used in the study of waves and vibrations in physics and engineering.
Yes, there is a difference between angular frequency and angular velocity. Angular frequency is a measure of how quickly a system oscillates or rotates, while angular velocity is a measure of how quickly an object rotates around an axis. Angular frequency is measured in radians per second, while angular velocity is measured in radians per unit time (e.g. radians per second or radians per minute).