Classical Mechanics — CheriMathLab

Key Concepts & Laws

Foundational principles that govern the motion of macroscopic objects.

Newton's Laws of Motion

1st: Inertia — an object stays at rest or uniform velocity unless acted on by a net force.
2nd: $$\vec{F} = m\vec{a}$$ 3rd: Every action has an equal and opposite reaction.

Projectile Motion

Horizontal: $$x = v_0 \cos\theta \cdot t$$ Vertical: $$y = v_0 \sin\theta \cdot t - \tfrac{1}{2}gt^2$$ Range: $$R = \frac{v_0^2 \sin 2\theta}{g}$$

Work-Energy Theorem

Net work equals change in kinetic energy: $$W_{\text{net}} = \Delta KE = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mv_0^2$$

Conservation Laws

Momentum: $\sum \vec{p} = \text{const}$ (no external forces)
Energy: $KE + PE = \text{const}$ (conservative forces)
Angular momentum: $L = I\omega = \text{const}$

Circular Motion

Centripetal acceleration: $$a_c = \frac{v^2}{r}$$ Centripetal force: $$F_c = \frac{mv^2}{r}$$ Period: $$T = \frac{2\pi r}{v}$$

Rotational Dynamics

Torque: $$\vec{\tau} = \vec{r} \times \vec{F} = I\alpha$$ Rotational KE: $$KE = \tfrac{1}{2}I\omega^2$$ Parallel axis theorem: $$I = I_{cm} + Md^2$$

Simple Harmonic Motion

Equation: $$x(t) = A\cos(\omega t + \phi)$$ Period (spring): $$T = 2\pi\sqrt{\frac{m}{k}}$$ Period (pendulum): $$T = 2\pi\sqrt{\frac{L}{g}}$$

Gravitation

Newton's law: $$F = \frac{GMm}{r^2}$$ Gravitational PE: $$U = -\frac{GMm}{r}$$ Escape velocity: $$v_e = \sqrt{\frac{2GM}{R}}$$

Problems

12 challenging problems spanning real-world and conceptual scenarios.

Easy

A 1200 kg car accelerates from rest to 27 m/s in 8 seconds on a level road. Assuming constant acceleration, find (a) the net force on the car, (b) the distance traveled during this interval, and (c) the work done by the engine if friction provides a constant 400 N opposing force.

Show Hint

Use $a = \Delta v / \Delta t$ to find acceleration, then $F_{\text{net}} = ma$. Distance via $s = \tfrac{1}{2}at^2$. The engine force = net force + friction, and work $W = F_{\text{engine}} \times s$.

Medium

A soccer ball is kicked at 25 m/s at an angle of 35° above the horizontal from ground level. (a) Find the maximum height reached. (b) Find the total time of flight. (c) A 3 m tall wall stands 45 m away. Does the ball clear the wall? By how much?

Show Hint

Decompose into horizontal and vertical components. At the wall, find the time to travel 45 m horizontally, then compute the height at that time. Compare with 3 m.

Medium

A 0.5 kg ball on a string of length 1.2 m is swung in a vertical circle. What is the minimum speed at the top of the circle so the string remains taut? What is the tension in the string at the bottom if the ball moves at 7 m/s there?

Show Hint

At the top, minimum speed occurs when tension = 0, so $mg = \frac{mv^2}{r}$. At the bottom, apply Newton's 2nd law radially: $T - mg = \frac{mv^2}{r}$.

Hard

A 60 kg skier starts from rest at the top of a 30 m high frictionless slope. At the bottom, the terrain becomes horizontal with a coefficient of kinetic friction $\mu_k = 0.15$. (a) What is the skier's speed at the bottom of the slope? (b) How far does the skier travel on the horizontal surface before stopping? (c) If the slope itself had $\mu_k = 0.08$, how would the answers change?

Show Hint

Use energy conservation: $mgh = \tfrac{1}{2}mv^2$ for part (a). On the flat, friction does work $W = -\mu_k mg\, d$. For part (c), include friction work on the slope using the slope length (you'll need the angle or use $d = h/\sin\theta$).

Medium

Two ice skaters, one of mass 50 kg and the other 80 kg, stand facing each other at rest on frictionless ice. They push apart. If the lighter skater moves at 3 m/s, find the speed of the heavier skater and the total kinetic energy generated. Where did this energy come from?

Show Hint

Conservation of momentum: total initial momentum = 0, so $m_1 v_1 + m_2 v_2 = 0$. The energy comes from internal (chemical/muscular) energy of the skaters.

Hard

A bullet of mass 10 g traveling at 400 m/s embeds itself in a 2 kg wooden block suspended from a string (ballistic pendulum). (a) Find the velocity of the block+bullet immediately after collision. (b) Find the maximum height the block swings to. (c) What fraction of kinetic energy is lost in the collision?

Show Hint

The collision is perfectly inelastic — use conservation of momentum to find the combined velocity. Then use energy conservation for the swing height. Compare initial and final KE.

Hard

A solid cylinder of mass 4 kg and radius 0.1 m rolls without slipping down a 25° incline. (a) Find the acceleration of the center of mass. (b) Find the friction force. (c) What minimum coefficient of static friction is needed to prevent slipping?

Show Hint

For rolling without slipping: $a = \frac{g\sin\theta}{1 + I/(mr^2)}$. For a solid cylinder, $I = \tfrac{1}{2}mr^2$. Friction provides the torque: $f = \frac{I\alpha}{r}$.

Advanced

A figure skater spinning with arms extended has moment of inertia $4.5\;\text{kg}\!\cdot\!\text{m}^2$ and angular velocity 2 rev/s. She pulls her arms in, reducing her moment of inertia to $1.5\;\text{kg}\!\cdot\!\text{m}^2$. (a) Find her new angular velocity. (b) Find the ratio of final to initial rotational kinetic energy. (c) Explain the source of the extra kinetic energy.

Show Hint

Angular momentum is conserved: $I_1\omega_1 = I_2\omega_2$. The extra energy comes from the work done by the skater's muscles as she pulls her arms inward against centripetal acceleration.

Medium

A mass of 0.3 kg is attached to a spring with spring constant $k = 120\;\text{N/m}$ on a frictionless table. It is displaced 8 cm from equilibrium and released. (a) Find the angular frequency and period of oscillation. (b) Find the maximum speed. (c) At what displacement is the kinetic energy equal to the potential energy?

Show Hint

$\omega = \sqrt{k/m}$. Max speed at equilibrium: $v_{\max} = A\omega$. When $KE = PE$, total energy is split equally: $\tfrac{1}{2}kx^2 = \tfrac{1}{2}\!\left(\tfrac{1}{2}kA^2\right)$, so $x = A/\sqrt{2}$.

Hard

Calculate the orbital speed and period of the International Space Station, which orbits at an altitude of approximately 408 km above Earth's surface. Earth's mass is $5.972 \times 10^{24}\;\text{kg}$ and radius is 6371 km. How does the gravitational acceleration at ISS altitude compare to $g$ at the surface?

Show Hint

Orbital radius $r = R_E + h$. Set gravitational force equal to centripetal force: $\frac{GMm}{r^2} = \frac{mv^2}{r}$. The ratio of $g$ values is $(R_E/r)^2$.

Advanced

A particle of mass $m$ slides without friction inside a hemispherical bowl of radius $R$. It is released from rest at the rim. Using energy conservation and Newton's laws, find (a) the speed at the bottom, (b) the normal force at the bottom, and (c) show that the normal force at an angle $\theta$ from the bottom is $N = mg(3\cos\theta - 2)$.

Show Hint

At angle $\theta$ from the bottom, height $= R(1 - \cos\theta)$. Use energy conservation to find $v(\theta)$, then apply Newton's 2nd law radially: $N - mg\cos\theta = \frac{mv^2}{R}$.

Advanced

Thought Experiment: Imagine a tunnel bored straight through Earth along a diameter. Assuming uniform density, a ball dropped into the tunnel undergoes simple harmonic motion. (a) Show the restoring force is proportional to displacement from the center. (b) Find the period of oscillation. (c) Compare this period with the orbital period of a surface-skimming satellite. What do you notice?

Show Hint

Inside Earth at distance $r$ from center, only mass within radius $r$ contributes: $M(r) = M(r/R)^3$. The force becomes $F = -\frac{GMm}{R^3}r$, which is Hooke's law. The period equals the surface satellite period: $T = 2\pi\sqrt{R/g} \approx 84\;\text{min}$.