Mechanics questions from Oxford Physics entrance exam.

AI Thread Summary
The discussion centers on a mechanics problem from the Oxford Physics entrance exam involving a mass moving under a force dependent on its position. Participants express confusion about deriving velocity as a function of time, suggesting that the question might be misprinted and should instead focus on position. The force acting on the mass is defined piecewise, indicating different behaviors in different regions of motion. It is clarified that the motion can be understood as ballistic in one region and sinusoidal in another, requiring continuity in velocity and acceleration at the transition point. The key takeaway is the recognition of the motion's periodic nature and the need to connect different motion types at the origin.
Mr.A.Gibson
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I'm ok with this question up until a3, I have no idea how to get velocity as a function of time from the information available. This question is taken from the Oxford Physics entrance exam. I'm not sure if it's a miss-print, perhaps the questions should be as a function of x, because that seems a lot easier and a similar level to questions from other years. Or perhaps I'm missing something.

22. A point like object with mass m = 1 kg starts from rest at point x0 = 10 m and moves without any friction under a force F which depends on the coordinate x as illustrated in figure below. The motion is confined to one dimension along x.

http://theonlinephysicstutor.com/Blog/Entries/2012/7/12_Entry_1_files/shapeimage_2.png

a1 What is its speed at x=0? [2]
a2 Sketch its kinetic energy as a function of x. [4]
a3 Sketch its velocity as well as its acceleration as a function of time t. [6]

Now consider a case when, in addition, a friction force of a magnitude of 1 N is present for x ≥ 0.
b1 Sketch how the velocity depends on x in that case. [6]
b2 How many meters this point like object traveled during the time when its position coordinate x was ≥ 0? [2]
 
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Mr.A.Gibson said:
I'm ok with this question up until a3, I have no idea how to get velocity as a function of time from the information available. This question is taken from the Oxford Physics entrance exam. I'm not sure if it's a miss-print, perhaps the questions should be as a function of x, because that seems a lot easier and a similar level to questions from other years. Or perhaps I'm missing something.

22. A point like object with mass m = 1 kg starts from rest at point x0 = 10 m and moves without any friction under a force F which depends on the coordinate x as illustrated in figure below. The motion is confined to one dimension along x.

http://theonlinephysicstutor.com/Blog/Entries/2012/7/12_Entry_1_files/shapeimage_2.png

a1 What is its speed at x=0? [2]
a2 Sketch its kinetic energy as a function of x. [4]
a3 Sketch its velocity as well as its acceleration as a function of time t. [6]

You are given the force on the object and told its mass. Thus you have
<br /> ma = F(x)<br />
or
<br /> m\ddot x = F(x)<br />
You can determine an expression for F(x) from the graph. From there you can hopefully solve the resulting ODE for x, and then determine \dot x and \ddot x by differentiation.
 
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Surely that will give you acceleration and velocity as a function of displacement not time? Since F and x vary with time i can't see how to make the differential work, in fact I cannot get any equation as a function of time.
 
Based on the graph,
<br /> F(x) = \begin{cases}<br /> -10, &amp; x \geq 0 \\<br /> -10 - x, &amp; x &lt; 0<br /> \end{cases}<br />

Conveniently m = 1\,\mathrm{kg} so we have
<br /> \frac{d^2x}{dt^2} = \begin{cases}<br /> -10, &amp; x \geq 0 \\<br /> -10 - x, &amp; x &lt; 0<br /> \end{cases}<br />

It's not necessary to solve this ODE so long as you recognise this as ballistic motion in x \geq 0 and sinusoidal oscillation about x = -10 in x &lt; 0. Basically when x &gt; 0 the particle behaves as it would under constant gravity, but in x &lt; 0 it's suddenly attached to a Hookean spring. Both of these should be covered in either A-level physics or maths/further maths, so should be familiar to someone sitting an Oxford physics entrance paper.

The difficulty is to patch together ballistic motion in x &gt; 0 with sinusoidal motion in x &lt; 0 in such a manner that both velocity and acceleration are continuous when the particle is at the origin. This requires finding the times at which x(t) = 0. In fact the motion is periodic in time; this follows from consideration of the KE graph.
 
pasmith said:
so long as you recognise this as ballistic motion in x \geq 0 and sinusoidal oscillation about x = -10 in x &lt; 0.

Thanks, that's the part I missed.
 
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