Mechanics questions from Oxford Physics entrance exam.

[Mr.A.Gibson]

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]

 

[pasmith]

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
$$ma = F(x)$$
or
$$m\ddot x = F(x)$$
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.

 

[Mr.A.Gibson]

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.

 

[pasmith]

Based on the graph,
$$F(x) = \begin{cases}<br /> -10, & x \geq 0 \\<br /> -10 - x, & x < 0<br /> \end{cases}$$

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

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 $x < 0$. Basically when $x > 0$ the particle behaves as it would under constant gravity, but in $x < 0$ 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 $x > 0$ with sinusoidal motion in $x < 0$ 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.

 

[Mr.A.Gibson]

so long as you recognise this as ballistic motion in $x \geq 0$ and sinusoidal oscillation about $x = -10$ in $x < 0$.

Thanks, that's the part I missed.