Calculating time by the force which depends on x

In summary, the conversation discusses a problem involving an object with mass m starting at point (-l,0) on the x-axis with a velocity v and experiencing a force that depends on its distance from the origin. After T seconds, the object reaches the origin and the question is to find T. The solution involves using equations for position, velocity, acceleration, and force, and integrating to find the relation between x and v. The final step involves using this relation to solve for T.
  • #1
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Homework Statement



An object with mass m start its journey on x-axis at point (-l,0) with velocity v.
The object feels force which depends on the object's distance to origin.
After T second it reaches the origin. What is T?

Homework Equations



x(t=0)=-l
v(t=0)=v

f(x)=-kx

x(t=T)=0

T=?

The Attempt at a Solution



1: dx/dt = v(t)

2: dt = dx/v(t) [from 1]

3: dv/dt = a(t) = (-k/m) * x(t)

4: dt = m * dv/(-k * x(t)) [from 3]

5: v(t) * dv = (-k/m) * x(t) * dx [put 2 to 3]

6: integrating both sides of 5 and putting the values x=0 and the velocity at the origin (which can be calculated by the energy change with integrating f(x) dx from -l to zero) gives the relation between x and v.

7: Putting x in terms of v (found in 6) to 4 and integrating right side from initial velocity to velocity at the origin gives a time (dt at the left side).

But i am not sure about the answer. Any ideas will be great.
Thank you.
 
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  • #2
Steps 1 to 5 are correct, and the final step you've written (step 5) is saying that the change in KE of the object is equal to the change in PE of the object (since PE depends on its distance from the origin). This is all correct.

And you're right, this gives the relation between x and v. And then yes, you can calculate what the final speed should be. And then yes, you could put x in terms of v to equation 4, and integrate the right hand side from initial to final speed, and that gives the time on the left hand side.

So yes, it is all correct. But the integral that you must do to integrate the speed is a bit difficult. Have you tried it already? I reckon you'll need to do a sinusoidal substitution (or just look it up in an integral table if you're allowed to do that).

There are probably loads of other ways to do the problem, but I think your way is correct (although its a little difficult).
 

1. How do you calculate time using force that depends on x?

To calculate time using force that depends on x, you would need to use the formula: t = F(x) / m, where t is time, F(x) is the force that depends on x, and m is the mass. This formula is derived from Newton's Second Law of Motion, which states that force is equal to mass times acceleration.

2. What is the relationship between time and force that depends on x?

The relationship between time and force that depends on x is inverse. This means that as the force that depends on x increases, the time it takes for an object to move decreases. Similarly, as the force decreases, the time increases.

3. Can you give an example of a situation where time is calculated using force that depends on x?

One example is calculating the time it takes for a ball to roll down a ramp with varying inclines. The force that depends on x in this situation would be the gravitational force, which increases as the angle of the ramp increases. By using the formula t = F(x) / m, you can calculate the time it takes for the ball to reach the bottom of the ramp for each incline.

4. Are there any limitations to using this method of calculating time?

Yes, there are limitations to using this method. The force that depends on x must be a constant force, meaning it does not change with time. Additionally, this method assumes that there is no friction or other external forces acting on the object, which may not always be the case in real-world scenarios.

5. How is this method of calculating time useful in scientific research?

This method is useful in scientific research because it allows for the prediction and analysis of the motion of objects under the influence of a force that depends on x. It can also be used to compare and analyze the effects of different forces on the time it takes for an object to move. This can be applied to various fields such as physics, engineering, and biomechanics.

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