AP Physics C: Mechanics – Practice Set

The position of a particle moving along the $x$-axis is given by $x(t)=2t^3-3t^2+4$ (in metres, $t$ in seconds).(a) Derive expressions for the velocity $v(t)$ and acceleration $a(t)$.(b) Evaluate $v$ and $a$ at $t=2\,\text{s}$.

A particle moves with velocity $v(t)=3t^2-12t+9\,\text{m s}^{-1}$.(a) Find the times, for $t\geq0$, at which the particle is momentarily at rest.(b) Find the net displacement of the particle between $t=0\,\text{s}$ and $t=4\,\text{s}$.

A variable force acting on an object along the $x$-axis is $F(x)=3x^2\,\text{N}$ (with $x$ in metres). Calculate the work done by this force as the object moves from $x=0$ to $x=4\,\text{m}$.

A block of mass 4.0 kg, initially at rest, is acted on by a time-varying net force $F(t)=6t^2\,\text{N}$ for $0\leq t\leq3\,\text{s}$.(a) Calculate the impulse delivered to the block over this interval.(b) Calculate the block’s velocity at $t=3\,\text{s}$.

A uniform disk of mass 2.0 kg and radius 0.30 m, free to rotate about its central axis, experiences a constant net torque of $5.0\,\text{N m}$.(a) Calculate the moment of inertia of the disk about its axis (use $I=\tfrac12MR^2$).(b) Calculate the resulting angular acceleration.

A uniform rod of mass 3.0 kg and length 1.2 m rotates about a pivot at one end (use $I=\tfrac13ML^2$ for a rod about its end) at constant angular velocity $\omega=4.0\,\text{rad s}^{-1}$.(a) Calculate the moment of inertia of the rod about the pivot.(b) Calculate its angular momentum and rotational kinetic energy.

A uniform solid disk (mass $M$, radius $R$, $I=\tfrac12MR^2$) is released from rest and rolls without slipping down a ramp, descending a vertical height $h=2.0\,\text{m}$.(a) Using energy conservation, show that $v=\sqrt{\dfrac{4gh}{3}}$ for the speed of the centre of mass at the bottom.(b) Calculate this speed numerically.

Explain why the disk in the previous question reaches the bottom of the ramp with a smaller speed than an identical-mass block sliding down a frictionless ramp of the same height, even though both start from rest at the same height.

A particle undergoes simple harmonic motion with position $x(t)=0.10\cos(5t)$ (SI units).(a) Derive $v(t)$ and $a(t)$ by differentiation.(b) State the maximum speed and maximum magnitude of acceleration of the particle.

For the oscillator in the previous question, verify by substitution that $x(t)=0.10\cos(5t)$ satisfies the SHM differential equation $\dfrac{d^2x}{dt^2}=-\omega^2x$, and state the value of $\omega$ and the period $T$.

A block–spring system undergoes SHM. At a certain instant the block is at displacement $x=0.060\,\text{m}$ from equilibrium and has speed $0.80\,\text{m s}^{-1}$; the amplitude of the motion is $0.10\,\text{m}$. Using energy conservation ($\tfrac12kA^2=\tfrac12kx^2+\tfrac12mv^2$), and given the mass is $0.50\,\text{kg}$, calculate the spring constant $k$.

A physical pendulum consists of a uniform rod of length $L=0.80\,\text{m}$ pivoted at one end, oscillating with small amplitude. Its period is given by $T=2\pi\sqrt{\dfrac{I}{Mgd}}$, where $I=\tfrac13ML^2$ is the moment of inertia about the pivot and $d=L/2$ is the distance from the pivot to the centre of mass. Show that $T=2\pi\sqrt{\dfrac{2L}{3g}}$ and calculate its numerical value.