# Mechanical vibrations: maximum velocity

• vxr
In summary, the conversation discusses solving a problem and calculating the maximum velocity for a given position and time. The formula for position and velocity at a specific time are given, and the maximum velocity is determined to be equal to Aω. The correctness of the calculation is confirmed.
vxr
Homework Statement
An air-track glider is attached to a spring pulled at distance ##d = 0.2m## to the right. Starting from released stage at ##t = 0## it subsequently makes ##n = 15## complete oscillations in time ##t = 10s##. Determine the period of oscillation ##T## and the object's maximum speed (velocity) ##v##, as well as its position and velocity ##v## at time ##t = 0.8s##.
Relevant Equations
##x(t) = Acos(\omega t + \theta)##
So I am almost sure I know how to solve this, just curious about the maximum velocity. Anyway, if you could double check my calculations, here it is.

##T = \frac{t}{n} = \frac{10s}{15} = \frac{2}{3}s##

##\omega = \frac{2\pi}{T} = 2\pi \frac{3}{2} = 3\pi##

a). position at ##t = 0.8s##:

##x(t) = Acos{(\omega t + \theta)} = Acos\Big(\frac{2\pi t}{T} + \theta\Big) = Acos\Big( 3\pi t \Big)##

##x(0.8s) = 0.2 cos (7,54) = 0.061 m##

b). velocity at ##t = 0.8s##:

##v = \frac{d}{dt}x = \frac{d}{dt}\Big( Acos(\omega t + \theta) \Big) = -A\omega sin(\omega t + \theta)##

##v(0.8s) = -0.2 * 3\pi sin (3\pi * 0.8) = -\frac{3}{5} sin (\frac{12}{5}\pi) =~ -1.79 \frac{m}{s}##

c). maximum velocity. During my classes something like that was done:

##v_{max} = A\omega \cos (0) = A \omega = 0.2 * 9.42 = 1.884 \frac{m}{s}##

I understand that ##cos(0) = 1## but is this correct?

Yes. You are correct.
You have ##v= -A\omega sin(f(t))##
The largest this value can be is when the sin is -1.
Hence: ##A\omega##

vxr
Thanks.

## 1. What is mechanical vibration?

Mechanical vibration refers to the rapid back-and-forth motion of an object or system around an equilibrium point. This motion can be caused by a force or disturbance, and can result in various forms of energy, such as sound or heat.

## 2. How is maximum velocity defined in mechanical vibrations?

Maximum velocity in mechanical vibrations is the highest point reached by an object or system during its back-and-forth motion. It is typically measured in meters per second (m/s) or feet per second (ft/s).

## 3. What factors affect the maximum velocity of a vibrating object?

The maximum velocity of a vibrating object can be affected by a variety of factors, including the frequency and amplitude of the vibration, the mass and stiffness of the object, and any external forces or damping present in the system.

## 4. How is maximum velocity calculated in mechanical vibrations?

The maximum velocity of a vibrating object can be calculated by taking the square root of the product of the frequency and the amplitude of the vibration. This can be represented by the equation Vmax = √(f x A), where Vmax is the maximum velocity, f is the frequency, and A is the amplitude.

## 5. Why is maximum velocity important in mechanical vibrations?

Maximum velocity is an important aspect of mechanical vibrations because it can determine the energy and intensity of the vibration, as well as its potential impact on the object or system. It can also be used to calculate other parameters, such as maximum acceleration and displacement, which are important in understanding and controlling vibrations.

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