# Solving Oscillatory Motion Homework: Frequency, Equation & Max Velocity

• JoanF
In summary, the system consists of two masses, A and B, accelerating to the right with an acceleration of 7.5 m/s^2, while mass C accelerates downwards. Once mass A is stopped, mass B begins to oscillate in simple harmonic motion, connected to mass A by a wire with a length of 0.85 m. The frequency of the pendulum motion can be calculated using the period equation for a simple pendulum. The equation for the elongation of the motion of body B can be written as x(t) = A sin(ωt + Q). The maximum velocity of the motion can be found using the equation for velocity and the first instant in which it is reached can be calculated by setting the acceleration
JoanF

## Homework Statement

mass of A=5,0kg
mass of B=2,0kg
friction coefficient between A and the plan=0,30

The system is moving with an acceleration of 7,5 m /s^2 and the angle theta is constant and equals to 37º.

Then, a mechanism makes the body A and the body B stops. B stop when it is in the position of the figure and it begins to oscillate in an simple harmonic motion.

lenght of the wire that connects A to B = 0,85 m

1) Calc the value of the frequency of pendulum motion.

2)Write the equation of the elongation of the motion of body B.

3) Calc the value of the maximum velocity of the motion and the first instant in which that velocity is reached.

## The Attempt at a Solution

1)it gaves me 1,83 but it's wrong

2) x(t)=Asin(wt+Q)
A=0,51
f=1,83
Q= -pi/2 (this is wrong but I don't know why =S )
x(t)=0,51sin(11,5t-pi/2)

3) v(t)=[x(t)] '
a(t)= [v(t)] '

v_max => a(t) = 0 <=> v_max=5,9 m/s

I haven't done the instant because this was already wrong

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JoanF said:

## Homework Statement

mass of A=5,0kg
mass of B=2,0kg
friction coefficient between A and the plan=0,30

The system is moving with an acceleration of 7,5 m /s^2 and the angle theta is constant and equals to 37º.

Then, a mechanism makes the body A and the body B stops. B stop when it is in the position of the figure and it begins to oscillate in an simple harmonic motion.

lenght of the wire that connects A to B = 0,85 m

1) Calc the value of the frequency of pendulum motion.

2)Write the equation of the elongation of the motion of body B.

3) Calc the value of the maximum velocity of the motion and the first instant in which that velocity is reached.

## The Attempt at a Solution

1)it gaves me 1,83 but it's wrong

2) x(t)=Asin(wt+Q)
A=0,51
f=1,83
Q= -pi/2 (this is wrong but I don't know why =S )
x(t)=0,51sin(11,5t-pi/2)

3) v(t)=[x(t)] '
a(t)= [v(t)] '

v_max => a(t) = 0 <=> v_max=5,9 m/s

I haven't done the instant because this was already wrong

For a simple pendulum, the period is given by $2 \pi \sqrt{ \frac{L}{g} }$ where L is the length of the pendulum. The period and the frequency are related by T = 1/f .

The purpose of this complicated arrangement is essentially to give the pendulum bob an initial velocity when it starts its oscillation. You will have to calculate this initial velocity in order to calculate Q. The way to do that will be to use the equation for the position of the pendulum bob and the equation for its velocity at the start of the oscillation (t = 0).

That should get you started.

AEM said:
For a simple pendulum, the period is given by $2 \pi \sqrt{ \frac{L}{g} }$ where L is the length of the pendulum. The period and the frequency are related by T = 1/f .

The purpose of this complicated arrangement is essentially to give the pendulum bob an initial velocity when it starts its oscillation. You will have to calculate this initial velocity in order to calculate Q. The way to do that will be to use the equation for the position of the pendulum bob and the equation for its velocity at the start of the oscillation (t = 0).

That should get you started.

But why Q= -pi/2 it's wrong? I don't understand :S

Last edited:
JoanF said:
But why Q= -pi/2 it's wrong? I don't understand :S

I'm sorry I couldn't respond earlier. I was away.

How did you calculate Q?

AEM said:
I'm sorry I couldn't respond earlier. I was away.

How did you calculate Q?

x(t)=Asin(wt+Q)

x(0)= -A
-A=Asin(w.0+Q)
-1=sin Q
sin (-pi/2)=sinQ
Q= -pi/2

AEM said:
For a simple pendulum, the period is given by $2 \pi \sqrt{ \frac{L}{g} }$ where L is the length of the pendulum. The period and the frequency are related by T = 1/f .

The purpose of this complicated arrangement is essentially to give the pendulum bob an initial velocity when it starts its oscillation. You will have to calculate this initial velocity in order to calculate Q. The way to do that will be to use the equation for the position of the pendulum bob and the equation for its velocity at the start of the oscillation (t = 0).

That should get you started.

I just started to solve your problem, but I find that there is some information missing. As I read the problem statement and look at the diagram you included, here's what I think is happening: The masses A, and B, are accelerating to the right while C is accelerating downwards. This means that when block A is stopped, the mass B will have a velocity in the X direction. This velocity should be taken into account when you try to solve the equations for oscillatory motion. Therefore, you will need to calculate the motion of the system as a whole before you worry about the pendulum. You will need to know either how long the system accelerated, OR for how far it moved while it was accelerated. Then you can calculate the velocity of B in the X direction.

Once you have the velocity of B horizontally, you will need to convert it into an initial angular velocity, call it $\theta_0$

The frequency of oscillation is easy to calculate, as I mentioned above. However, you will have to use two equations simultaneously to find the amplitude and the phase angle (your Q).

These equations are

$$\theta(t) = A cos(\omega t + \phi)$$

and

$$\frac{d \theta}{dt} = - \omega A sin(\omega t + \phi)$$

In your work, you have chosen to use $x(t) = A sin( \omega t + Q)$. I think it would be better to work with angles than linear variables. Also, one can use either sine or cosine for the function representing the motion, but you should be aware that you will get different values for the angle $\phi$ ( your Q ). This angle reflects both which function you choose to express your answer in and how the motion begins. For example, if you start a pendulum from rest by pulling it back, holding it for an instant and then letting go, AND use the cosine expression above, the angle $\phi$ will be zero.

You solve the two equations I wrote above by putting in the values that you have for the initial value of theta and initial angular velocity, and t = 0. It is also better for you to work with radian measure, rather than expressing your angles in degrees.

If you can tell me how long the acceleration lasted, or for how far block A moved, I'll solve the problem and see if there are any other hints I can give you.

i'm heading a problem in mathematical equation about the initial condition, what methode should we used if there is an addition force, such as damping force. This equation will be applied to performed oscillatory system in fortran language programme .

Could you give us the differential equation that you've come up with? It should be a second order linear DE (unless you haven't made the small angle approximation) with constant coefficients. Such DEs can be solve exactly without much difficulty and the solution method can be found in any "Introduction to Differential Equations" textbook.

## 1. What is oscillatory motion and why is it important to study?

Oscillatory motion is a type of periodic motion in which a system repeatedly moves back and forth around a central equilibrium point. It is important to study because it is a fundamental concept in physics and is applicable to many real-world systems such as pendulums, springs, and waves. Understanding oscillatory motion can also help us predict and control the behavior of these systems.

## 2. What is frequency and how is it related to oscillatory motion?

Frequency is the number of cycles or oscillations that occur in a given time period. In oscillatory motion, frequency is directly related to the speed at which the system oscillates. This means that as the frequency increases, the system oscillates faster and completes more cycles in a given time period.

## 3. What is the equation for calculating frequency in oscillatory motion?

The equation for frequency in oscillatory motion is f = 1/T, where f is the frequency and T is the time period of one complete oscillation. This means that if we know the time period, we can easily calculate the frequency of the oscillations.

## 4. How does maximum velocity relate to oscillatory motion?

Maximum velocity is the highest speed that a system reaches during its oscillations. In oscillatory motion, maximum velocity is directly related to the amplitude, or maximum displacement, of the system. This means that as the amplitude increases, the maximum velocity also increases.

## 5. Can oscillatory motion occur in different dimensions?

Yes, oscillatory motion can occur in one, two, or three dimensions. For example, a pendulum oscillates in one dimension, while a water wave oscillates in two dimensions. The principles of oscillatory motion remain the same in different dimensions, but the equations and calculations may differ.

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