Object movement under the infuence of Forces on the x,y axes

• Michael_0039
In summary: So, in summary, the first step to find the V(t) would be to find the acceleration using Newton's second law and then solve the two differential equations for the component functions ##v_x(t)## and ##v_y(t)## to obtain the components of the vector velocity ##\vec V(t)##. Then, to find the route's length, you would integrate the velocity to get the position as a function of time. However, it is important to be careful with the bounds of integration and the use of the modulus of velocity to ensure accurate results.

Michael_0039

Homework Statement
Mass object m is moving on plane x-y under the influence of constant force F. The projection Force on x'x is F*cos(ωt) and on y'y F*sin(ωt), ω=constant. For t=0 the Velocity=0. Find the V(t) and the route's length of the object until it stops for the first time.
Relevant Equations
F*cos(ωt)
F*sin(ωt)
Which could be the first step to find the V(t) ?

$$\vec F = m\vec a = m \frac {d\vec V}{dt}$$
$$(F_x, F_y) = m \frac {d} {dt} (v_x, v_y)$$

So just solve the two differential equations for the component functions ##v_x(t)## and ##v_y(t)##. Those are the components of the vector velocity ##\vec V(t)##

Michael_0039
The first step would be to find the acceleration. You could use Newton's second law.

Michael_0039
This is my try:

tnich
Now, for the route's length, I have to integrate one more time [ ds/dt=v ]? And try to find the s. Is that process correct ?

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Michael_0039 said:
Now, for the route's length, I have to integrate one more time [ ds/dt=v ]? And try to find the s. Is that process correct ?
Yeah, you have the formula ##\mathrm{ds}=|\mathbf{v}|\mathrm{dt}##.

Michael_0039
I try this, but the solution is negative, how this is explained ? And how will I work for "...until it stops for the first time "

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Try just integrating the velocity to get the position as a function of time. From this, the shape of the object's trajectory should be obvious, and it will allow you to easily calculated the distance traveled.

archaic
tnich said:
Try just integrating the velocity to get the position as a function of time. From this, the shape of the object's trajectory should be obvious, and it will allow you to easily calculated the distance traveled.

but I do not understand what kind of motion describe this one?
The obvious might be a helicoidal movement, because of the Fx=F*cos(ωt) & Fy=F*sin(ωt).

Michael_0039 said:
View attachment 250192
but I do not understand what kind of motion describe this one?
The obvious might be a helicoidal movement, because of the Fx=F*cos(ωt) & Fy=F*sin(ωt).
you know what's weird? at ##t=0s## your object has moved a negative distance.

jbriggs444
Michael_0039 said:
The obvious might be a helicoidal movement, because of the Fx=F*cos(ωt) & Fy=F*sin(ωt).
You are on the right track here, but it's a 2-dimensional path. What would your helix look like if z(t) = 0?

I think your expression for ##v_y## is wrong, shouldn't it be ##v_y(t)=-\frac{F\cos{\omega t}}{m\omega}##? You have integrated between ##0## and ##t##, it should've been ##\int_{t_0}^{t'}dt##, and since velocity is zero at ##t=t_0##, you'll have ##v_y(t')=v(t')-v(t_0)=v(t')##. Same thing for ##v_x## but you've fortunate enough that ##\sin{0}=0##.

EDIT : No, I'm wrong.

As @archaic points out in post #10, your expression for S does not give the right value at t=0. This is because you did not apply the bounds correctly. When you fix that you will find the constant of integration will ensure your result is positive, at least initially. But you may notice it still goes down after a while, which is also infeasible.

The next problem is that having taken the modulus of velocity to get speed, you must be careful what happens to your integral when the integrand reaches zero. Blindly integrating through that point might result in your inadvertently using negative values for speed. This is probably why the question specifies "until it comes to rest for the first time".

The whole thing becomes simpler if you get rid of the surd, which you can do by considering the half angle ##(\cos(\frac 12\omega t))##.

This is what you've done so far :
$$|\vec v|=\sqrt{v_x^2+v_y^2}=\frac{F\sqrt{2}}{\omega m}\sqrt{1-\cos{\omega t}}=\frac{2F}{\omega m}\sin{\frac{\omega}{2}t}$$
My calculation for ##S(t)## gives this, following haruspex's advice. You can see that it is zero in the beginning.
$$S(t)=\int_0^{t}|\vec v(t')|dt'=\frac{4F}{\omega^2m}(1-\cos{\frac{\omega}{2}t})$$
You probably have a stray minus sign somewhere.

Michael_0039
archaic said:
$$|\vec v|=\sqrt{v_x^2+v_y^2}=\frac{F\sqrt{2}}{\omega m}\sqrt{1-\cos{\omega t}}=\frac{2F}{\omega m}\sin{\frac{\omega}{2}t}$$
##\frac{2F}{\omega m}|\sin(\frac{\omega}{2}t)|##

archaic and Michael_0039
Hello all !

Now I see, using this trigonometric transformation formula

, it is much better.

haruspex said:
##\frac{2F}{\omega m}|\sin(\frac{\omega}{2}t)|##
I often miss that after rooting a square, thank you!

Michael_0039 said:
Hello all !

Now I see, using this trigonometric transformation formula
View attachment 250429

, it is much better.
I have something to say about your work, and I think I am right this time.
When you are looking to get the anti-derivative but without caring about the accumulation of change (to mathematically vulgarize it : ##\int_a^b df = f(a+dx)-f(a)+f(a+2dx)-f(a+dx)+...+f(b)-f(b-dx)##), you should do an indefinite integration and find the constant using given conditions.
Consider some 1D motion with a constant acceleration ##a## and an initial velocity ##v_0##.
$$\int_{t_0}^t a\,dt'=a(t-t_0)=v(t)-v(t_0)\neq v(t)=\int a\,dt=at+c\text{, }v(0)=v_0=c$$
Or ##v(t)=\int_{t_0}^t a\,dt+v(t_0)##
Someone correct me if I'm wrong!

archaic
Michael_0039 said:

Are you in agreement with this?Thanks
Looks good - well done.

1. What is the difference between force and motion?

Force is a push or pull on an object, while motion is the change in an object's position over time. Forces can cause objects to start moving, stop moving, or change direction.

2. How do forces affect the movement of an object on the x and y axes?

Forces can cause an object to accelerate or decelerate in a certain direction on the x and y axes. This movement is determined by the direction and magnitude of the force applied.

3. What is the relationship between mass and acceleration in object movement?

The greater the mass of an object, the more force is needed to accelerate it. This is expressed by Newton's second law of motion: F=ma, where F is force, m is mass, and a is acceleration.

4. How do different types of forces impact object movement on the x and y axes?

Different types of forces, such as friction, gravity, and applied forces, can impact object movement in different ways. Friction can slow down an object's movement, gravity can pull an object towards the ground, and applied forces can cause an object to move in a certain direction.

5. How can we calculate the net force on an object moving on the x and y axes?

To calculate the net force on an object, we must take into account all the forces acting on the object. We can use vector addition to determine the net force on an object, taking into consideration both magnitude and direction of each force.