Sixth Term Examination Paper (STEP) Mechanics Syllabus

Mechanics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mechanics in Sixth Term Examination Paper (STEP), not a summary of it.

1. 1

   Kinematics

   5 topics

   • Displacement, velocity and acceleration in one dimension
   • Constant acceleration equations (suvat)
   • Motion graphs and interpretation of gradients/areas
   • Variable acceleration via calculus
     • Acceleration as a function of time, displacement or velocity
   • Projectile motion and the trajectory equation
     • Range, maximum height and time of flight
     • Projectiles on inclined planes
2. 2

   Forces and Newton's Laws

   5 topics

   • Resolving forces and equilibrium of a particle
   • Newton's three laws of motion
   • Friction and the model of limiting equilibrium
     • Coefficient of friction and motion on slopes
   • Connected particles, pulleys and tension
   • Motion in a vertical and horizontal circle
     • Conditions for completing a vertical circle
     • Conical pendulum
3. 3

   Momentum, Work and Energy

   5 topics

   • Impulse and conservation of linear momentum
   • Direct impact and Newton's experimental law (restitution)
     • Successive collisions
     • Oblique impact and impact with a surface
   • Work done by a force and the work-energy principle
   • Kinetic and gravitational potential energy
   • Conservation of mechanical energy and power
4. 4

   Rigid Bodies and Oscillations

   5 topics

   • Moments and equilibrium of rigid bodies
   • Centre of mass of systems and laminae
     • Centre of mass by integration
   • Toppling and sliding conditions
   • Elastic strings and springs (Hooke's law)
     • Elastic potential energy
   • Simple harmonic motion and its equation
     • Amplitude, period and energy in SHM
     • SHM from springs and pendulums

Mechanics flashcards for Sixth Term Examination Paper (STEP)

22 of 51 cards from the Mechanics deck — real questions with worked answers.

1. In one-dimensional motion, how are displacement, velocity and acceleration related by calculus?

   Velocity is the rate of change of displacement and acceleration is the rate of change of velocity: $v = \dfrac{dx}{dt}$ and $a = \dfrac{dv}{dt} = \dfrac{d^{2}x}{dt^{2}}$.

2. Define average velocity and average acceleration over a time interval.

   Average velocity is $\bar{v} = \dfrac{\Delta x}{\Delta t}$ (change in displacement over time); average acceleration is $\bar{a} = \dfrac{\Delta v}{\Delta t}$ (change in velocity over time).

3. What is the distinction between distance and displacement, and between speed and velocity?

   Distance and speed are scalars (magnitude only, always non-negative). Displacement and velocity are vectors carrying direction; displacement can be negative or zero even when distance travelled is large.

4. State the five constant-acceleration (suvat) equations.

   $v = u + at$; $\;s = ut + \tfrac{1}{2}at^{2}$; $\;s = vt - \tfrac{1}{2}at^{2}$; $\;v^{2} = u^{2} + 2as$; $\;s = \tfrac{1}{2}(u+v)t$.

5. Which suvat equation contains no time $t$, and which contains no final velocity $v$?

   No time: $v^{2} = u^{2} + 2as$. No final velocity: $s = ut + \tfrac{1}{2}at^{2}$.

6. Under what assumption are the suvat equations valid, and how is this commonly applied to vertical motion under gravity?

   They require constant (uniform) acceleration. For vertical motion near Earth's surface, $a = g \approx 9.8\,\text{m s}^{-2}$ directed downward, neglecting air resistance.

7. On a displacement–time graph, what does the gradient represent?

   The gradient of a displacement–time graph at a point gives the instantaneous velocity, $v = \dfrac{dx}{dt}$.

8. On a velocity–time graph, what do the gradient and the area under the curve represent?

   The gradient gives the acceleration $a = \dfrac{dv}{dt}$; the area between the graph and the time axis gives the displacement $s = \int v\,dt$.

9. On an acceleration–time graph, what does the area under the curve represent?

   The area under an acceleration–time graph gives the change in velocity, $\Delta v = \int a\,dt$.

10. For variable acceleration, how do you obtain velocity and displacement from acceleration, and vice versa?

    Differentiate to go down: $a = \dfrac{dv}{dt}$, $v = \dfrac{dx}{dt}$. Integrate to go up: $v = \int a\,dt$, $x = \int v\,dt$ (adding constants fixed by initial conditions).

11. Give the expression for acceleration as a function of displacement useful when $a$ depends on $x$.

    $a = v\dfrac{dv}{dx} = \dfrac{d}{dx}\!\left(\tfrac{1}{2}v^{2}\right)$, obtained from $a = \dfrac{dv}{dt} = \dfrac{dv}{dx}\dfrac{dx}{dt}$.

12. For a projectile launched at speed $u$ and angle $\theta$ to the horizontal (taking up as positive), write the horizontal and vertical components of position at time $t$.

    $x = u\cos\theta\,t$ and $y = u\sin\theta\,t - \tfrac{1}{2}g t^{2}$, with constant horizontal velocity $u\cos\theta$ and vertical acceleration $-g$.

13. Derive the equation of the trajectory $y$ in terms of $x$ for a projectile launched at angle $\theta$ and speed $u$.

    Eliminating $t$: $y = x\tan\theta - \dfrac{g x^{2}}{2u^{2}\cos^{2}\theta}$, a parabola.

14. State the formulae for the time of flight, maximum height and range of a projectile launched from level ground at speed $u$, angle $\theta$.

    Time of flight $T = \dfrac{2u\sin\theta}{g}$; maximum height $H = \dfrac{u^{2}\sin^{2}\theta}{2g}$; range $R = \dfrac{u^{2}\sin 2\theta}{g}$.

15. At what launch angle is the range of a projectile (on level ground) maximised, and what is that maximum range?

    At $\theta = 45^{\circ}$, since $\sin 2\theta = 1$; the maximum range is $R_{\max} = \dfrac{u^{2}}{g}$.

16. How do you resolve a force $F$ acting at angle $\theta$ into perpendicular components?

    Component along the chosen axis $= F\cos\theta$; component perpendicular $= F\sin\theta$ (angle measured from that axis).

17. State the condition for a particle to be in equilibrium under several forces.

    The vector sum of all forces is zero: $\sum \vec{F} = \vec{0}$, equivalently the resolved components in two perpendicular directions each sum to zero, $\sum F_x = 0$ and $\sum F_y = 0$.

18. State Newton's three laws of motion.

    1) A body remains at rest or in uniform motion unless acted on by a resultant force. 2) $\vec{F} = m\vec{a}$ (resultant force equals mass times acceleration). 3) For every action there is an equal and opposite reaction.

19. What is the relationship between weight and mass, and in what units?

    Weight is the gravitational force $W = mg$, measured in newtons (N); mass $m$ is in kilograms (kg) and $g \approx 9.8\,\text{m s}^{-2}$.

20. For a body on a surface, how is the normal reaction related to the friction force when sliding or on the point of sliding?

    Friction acts parallel to the surface opposing motion; at the point of sliding (limiting) $F = \mu R$, where $R$ is the normal reaction and $\mu$ is the coefficient of friction.

21. State the friction inequality in the model of limiting equilibrium.

    $F \leq \mu R$. Friction takes whatever value (up to $\mu R$) is needed to maintain equilibrium; sliding begins when $F$ would have to exceed $\mu R$.

22. Define the angle of friction $\lambda$ and relate it to $\mu$.

    The angle of friction is the angle between the total contact force and the normal at the point of slipping: $\tan\lambda = \mu$.

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