09Classical Mechanics
Push a heavy box and at first nothing happens: static friction grows exactly as much as your push, up to a maximum \(\mu_s N\). Beyond it the box breaks loose and kinetic friction, \(\mu_k N\), usually smaller, opposes the sliding. That is why every friction problem starts with a classification: does the system stay at rest, is it about to move, or does it slide? Choose the system and the regime (or let the laboratory decide), and it writes the second law with the right friction force, solves for the unknowns and draws the free-body diagrams.
If the system slides, the acceleration is constant and the motion starts from rest.
The equations of each body are those of Newton’s laws plus the friction force along the surface, opposite to the (actual or impending) relative motion. The regime adds one more relation: \(a = 0\) at rest, \(f = \mu_s N\) and \(a = 0\) about to slide, \(f = \mu_k N\) while sliding.
Each body is isolated from the rest and drawn with every external force acting on it, including its weight. The axes are chosen along the motion and perpendicular to it, so that only one component of the acceleration survives.
The check of the results and, when the system is at rest, whether static friction can really hold it.
When the surface of one body slides, or tends to slide, over another, they exert friction forces on each other, along the surface and opposite to the relative motion. If we push a block gently it does not move: static friction balances the push. As the push grows, static friction grows with it up to a maximum; beyond it the block breaks loose, and a smaller force keeps it moving at constant velocity: kinetic friction.
For dry, unlubricated surfaces, both kinds of friction are approximately independent of the area of contact and proportional to the normal force (Leonardo da Vinci, Coulomb):
\[f_s \le \mu_s N,\qquad f_k = \mu_k N.\]The coefficients are dimensionless, depend on the pair of materials and usually satisfy \(\mu_s > \mu_k\). The equality \(f_s = \mu_s N\) holds only when the block is just about to slide.
The course notes ask to read the statement to obtain a first classification: does the motion happen at constant velocity or at rest, \(\sum\vec F = 0\), or with acceleration, \(\sum\vec F = m\vec a\)? With friction this is decisive: compare the force that tends to move the system with \(\mu_s N\). If it is smaller, nothing moves and \(f\) takes the value that balances it; if it is larger, the system slides and \(f = \mu_k N\).
Raise one end of a board with a body on it until it is about to slide (course homework). Then \(\sum F_x = mg\sin\theta - f_s = 0\) and \(N = mg\cos\theta\); with \(f_s = \mu_s N\):
\[\mu_s\, mg\cos\theta = mg\sin\theta \;\Rightarrow\; \mu_s = \tan\theta_s.\]In the same way, the angle at which it slides at constant velocity gives \(\mu_k = \tan\theta_k\), with \(\theta_k < \theta_s\). The mass cancels.
Friction opposes the relative sliding of the surfaces, not necessarily the motion of the body. A box on the floor of an accelerating truck is carried forward by static friction; a car moves because the road pushes its tyres forward; on a flat curve, friction points to the centre and provides the centripetal force.
Even a polished surface is rough at the atomic scale; the real area of contact is a tiny fraction of the apparent one and is proportional to the normal force, because the contact points deform under the load. Many of them become cold welds; friction is the force needed to break them as the surfaces slide (Resnick 6-2).
The shortest stopping distance is obtained with the tyres just about to skid, using the maximum static friction: \(a = \mu_s g\) and \(d = v_0^2/(2\mu_s g)\). If the wheels lock, kinetic friction, smaller, takes over and the distance grows; antilock brakes keep the wheels at the limit.
Problems adapted from Serway (chapters 2, 4, 5 and 6), Resnick and the course notes. Load any of them into the laboratory: the data fill the panel and the solver does the rest, so you can then change any value and see how the answer reacts.
A box is pushed with a force P directed at an angle below the horizontal over a horizontal floor; the static coefficient is \(\mu_s\). Find the minimum value of P that will move the box. (Take \(m = 20.0\ \text{kg}\), \(\mu_s = 0.40\) and the force \(30°\) below the horizontal, so θ = −30°.)
The \(7.5\ \text{kg}\) sled of Resnick is pulled with \(21.0\ \text{N}\) at \(15°\), now on snow with \(\mu_k = 0.10\). Find its acceleration and the normal force.
A \(20.0\ \text{kg}\) box rests on a floor with \(\mu_s = 0.30\) and \(\mu_k = 0.20\). It is pushed horizontally with \(50.0\ \text{N}\). Does it move? What is the friction force?
The same box is now pushed with \(80.0\ \text{N}\). Does it move? With what acceleration?
A block is at rest on an incline. When the angle is raised, sliding just begins at \(15°\). What is the coefficient of static friction? (The mass cancels; any value gives the same μ.)
A \(2.00\ \text{kg}\) block slides down an incline at \(30°\) with \(\mu_k = 0.20\). Find its acceleration and the friction force.
A \(3.00\ \text{kg}\) block rests on an incline at \(20°\) with \(\mu_s = 0.50\). Does it slide? How large is the friction force?
A \(2.00\ \text{kg}\) block is shoved up an incline at \(20°\) with \(\mu_k = 0.15\). What is its acceleration while it goes up?
A block slides down an incline at \(30°\) with an acceleration of \(2.50\ \text{m/s}^2\). What is the coefficient of kinetic friction?
A hanging block of \(8.5\ \text{kg}\) is connected by a rope over a pulley to a \(6.2\ \text{kg}\) block that slides on a flat table. If the coefficient of friction during the motion is \(0.2\), find the tension in the rope.
In the same tribometer the tension turns out to be 10 % larger than before, \(46.38\ \text{N}\). Find the coefficient of friction.
A \(10.0\ \text{kg}\) block on a table with \(\mu_s = 0.30\) is attached over a pulley to a hanging \(2.00\ \text{kg}\) block. Does the system move? What are the friction force and the tension?
Repeat the incline with a hanging block (\(m_1 = 9.5\ \text{kg}\) on \(34°\), \(m_2 = 2.6\ \text{kg}\)), now with friction between block 1 and the plane: \(\mu_s = 0.24\), \(\mu_k = 0.15\). Does it move, which way, and with what acceleration?
Two blocks joined by a rope of negligible mass are dragged by a horizontal force. \(F = 68.0\ \text{N}\), \(m_1 = 12.0\ \text{kg}\), \(m_2 = 18.0\ \text{kg}\) and the coefficient of kinetic friction between each block and the surface is \(0.100\). (a) Draw a free-body diagram for each block. (b) Determine the tension and the acceleration of the system.
A car moves at \(v_0 = 27\ \text{m/s}\) (60 mi/h) on a level road. If \(\mu_s = 0.60\) between tyres and road, what is the shortest distance in which it can stop, and how long does it take?
A car travelling at \(72\ \text{km/h}\) leaves skid marks \(30\ \text{m}\) long. What was the coefficient of friction?
A \(1500\ \text{kg}\) car moving on a flat horizontal road negotiates a curve of radius \(35.0\ \text{m}\). If \(\mu_s = 0.523\), find the maximum speed it can have and still make the turn.
A coin placed \(30.0\ \text{cm}\) from the centre of a rotating turntable slips when its speed is \(50.0\ \text{cm/s}\). What is the coefficient of static friction between coin and turntable?
In the rotor, a person stands against the wall of a cylinder of radius \(2.0\ \text{m}\); the floor drops away. If \(\mu_s = 0.40\) between clothes and wall, what minimum speed, and what rotation rate, keep the person from falling?
Theory after R. Resnick, D. Halliday and K. Krane, Física, vol. 1, sections 6-2 and 6-3; problems from R. Serway and J. Jewett, Física para ciencias e ingeniería, vol. 1, chapters 5 and 6, and from the course notes on friction.