08Classical Mechanics
Kinematics describes how bodies move; dynamics explains why. A force is an interaction between a body and its surroundings, and Newton’s second law, \(\sum\vec F = m\vec a\), ties the forces to the acceleration they produce. Choose a physical system, isolate each body in its free-body diagram, and the laboratory writes the second law along each axis, solves the system for whatever the statement asks, checks the result body by body and animates the motion.
Once ΣF = ma gives the acceleration, the motion is a uniformly accelerated one.
For every body \(\sum F_x = m a_x\) and \(\sum F_y = m a_y\), with the axes along the motion and perpendicular to it. These component equations are the invariant of dynamics; the condition of the statement (at rest, moving at constant velocity, accelerating) decides whether \(a = 0\).
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.
Substituting the results back into every equation is the last step of the invariant: the units must agree and every sum must match m a.
A body keeps its state of rest or of uniform motion in a straight line unless external forces oblige it to change it. With no net force the velocity is constant and the acceleration zero; this is the condition of equilibrium, \(\sum\vec F = 0\). The law also defines the inertial frames, those in which it holds; an accelerating elevator is not one of them.
The acceleration of a body is directly proportional to the net external force and inversely proportional to its mass:
\[\sum\vec F_{\text{ext}} = m\,\vec a.\]It is a vector equation, so it is applied component by component, \(\sum F_x = m a_x\), \(\sum F_y = m a_y\). The mass measures the inertia, the resistance to changes of velocity; the newton is the force that gives 1 kg an acceleration of 1 m/s².
To every action exerted by a body corresponds a reaction of equal magnitude and opposite direction exerted by the other body. The two forces act on different bodies, so they never cancel in the free-body diagram of a single one; that is why a horse can pull a cart. When both forces act on the same body, as in a tug of war viewed as one system, there is no apparent motion.
The weight is the gravitational force on a body, \(\vec W = m\vec g\), measured in newtons; the mass, in kilograms, does not change from place to place. The old technical unit is the kilogram-force: \(1\ \text{kgf} = (1\ \text{kg})(9.8\ \text{m/s}^2) = 9.8\ \text{N}\). A scale does not measure the weight but the normal force it exerts, which differs from \(mg\) when the scale accelerates.
The course notes summarise the method as an invariant:
For connected bodies that move together, the course proposes to take the acceleration positive in the direction of motion and write for the whole system
\[F_{\text{mayor}} - F_{\text{menor}} = (\textstyle\sum m)\,a.\]The larger force decides the direction of motion and the internal tensions cancel in pairs. For the Atwood machine it gives at once \(m_2 g - m_1 g = (m_1 + m_2)\,a\); the tension then comes from the equation of either body.
A rope pulls along itself with a tension \(T\); if it is massless, the tension is the same all along it, and an ideal pulley (massless and frictionless) only changes its direction. If it does not stretch, the bodies it joins share the magnitude of their acceleration. A surface pushes perpendicular to itself with the normal force \(N\), whose value is whatever prevents the body from sinking into it: it is not always \(mg\).
Resnick insists on examining the results before trusting them. On an incline, \(a = g\sin\theta\) gives \(0\) for \(\theta = 0\) and \(g\) for \(\theta = 90°\), as it should. For the Atwood machine, \(m_1 = m_2\) gives \(a = 0\), and \(m_1 = 0\) gives free fall. The tension must lie between \(m_1 g\) and \(m_2 g\). If a limit gives something absurd, there is an error.
A body in uniform circular motion has a centripetal acceleration \(v^2/r\), so the net force on it points to the centre and has magnitude
\[\sum F_r = m\,\frac{v^2}{r}.\]This centripetal force is not a new kind of force: it is provided by a tension, a normal, friction or gravity. Taking a radial axis (towards the centre) and a perpendicular one, the invariant applies as always: at the top and bottom of a vertical circle, in a banked curve or in a conical pendulum.
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 sled of mass \(m = 7.5\ \text{kg}\) is pulled along a frictionless horizontal surface by a rope with a constant force \(P = 21.0\ \text{N}\). Analyse the motion if the rope makes an angle \(\theta = 15°\) with the horizontal.
The same sled, pulled with the rope horizontal: find the acceleration and the normal force.
With the rope at \(15°\), how large must \(P\) be for the normal force to vanish, so that the sled is about to leave the surface? What would the acceleration be?
A box of mass m is placed on a frictionless incline that makes an angle φ with the horizontal. Determine the acceleration of the box after it is released. (Take m = 5 kg and φ = 30°, then change them: the acceleration does not depend on m.)
A block of mass \(m = 18.0\ \text{kg}\) is held in place by a rope on a frictionless incline at \(27°\). Find the tension in the rope and the normal force exerted on the block by the plane.
The rope that held the \(18.0\ \text{kg}\) block on the \(27°\) incline is cut. Analyse the subsequent motion.
Two blocks of masses \(m_1 = 2.00\ \text{kg}\) and \(m_2 = 1.50\ \text{kg}\) are in contact on a frictionless horizontal surface. A horizontal force \(F = 7.00\ \text{N}\) is applied to \(m_1\). Find the acceleration of the blocks and the contact force between them.
Two blocks joined by a rope of negligible mass are dragged by a horizontal force \(F = 68.0\ \text{N}\); \(m_1 = 12.0\ \text{kg}\), \(m_2 = 18.0\ \text{kg}\). Ignoring friction here, find the tension and the acceleration. (With friction, see the friction page.)
A passenger of mass \(72.2\ \text{kg}\) rides in an elevator standing on a scale. What does the scale read when the cab descends at constant velocity?
The same passenger, when the cab ascends with an acceleration of \(3.20\ \text{m/s}^2\).
A person weighs a fish of mass \(3.00\ \text{kg}\) on a spring scale attached to the ceiling of an elevator. Show that if the elevator accelerates the scale reads a different weight: here the elevator accelerates downwards at \(2.00\ \text{m/s}^2\).
If the cable breaks and the elevator falls freely (a = −g), what does the scale read?
Solve the Atwood machine: what are the accelerations and the tensions of the rope on both sides of the pulley? Since the rotational inertia of the pulley is ignored, the tension is the same on both sides. (Take \(m_1 = 2.00\ \text{kg}\) and \(m_2 = 3.00\ \text{kg}\).)
In an Atwood machine \(m_1 = 2.00\ \text{kg}\) rises with an acceleration of \(1.96\ \text{m/s}^2\). What is \(m_2\), and what is the tension?
A block of mass \(m_1 = 4.00\ \text{kg}\) on a frictionless horizontal surface is pulled by a light rope that passes over a pulley and is attached to a hanging block \(m_2 = 2.00\ \text{kg}\). Find the tension and the acceleration of each block.
A block \(m_1 = 9.5\ \text{kg}\) on a frictionless incline at \(34°\) is joined over a pulley to a hanging block \(m_2 = 2.6\ \text{kg}\). The system starts from rest. Describe the motion. (A negative a means that the motion goes the other way: m₁ slides down.)
A traffic light weighing \(122\ \text{N}\) hangs from a cable tied to two other cables fastened to a support; the upper cables make angles of \(37.0°\) and \(53.0°\) with the horizontal. Find the tension in the three cables.
A block of mass \(m = 15.0\ \text{kg}\) hangs from a knot held by two ropes at \(30°\) and \(45°\) with the horizontal. What are the tensions in the ropes?
A conical pendulum has a string of length \(L = 1.2\ \text{m}\) that makes an angle \(\theta = 25°\) with the vertical. What is the period of the motion? (It does not depend on the mass.)
Consider a conical pendulum with an \(80.0\ \text{kg}\) bob on a \(10.0\ \text{m}\) wire making an angle of \(5.00°\) with the vertical. Determine the tension in the wire and the radial acceleration of the bob.
A civil engineer wants to design a curved exit ramp of radius \(35.0\ \text{m}\) so that a car will not have to rely on friction to round it at \(13.4\ \text{m/s}\). At what angle should the curve be banked?
A highway curve of radius \(200\ \text{m}\) is designed for \(90\ \text{km/h}\). What banking angle does it need, and what normal force does the road exert on a \(1200\ \text{kg}\) car?
A \(0.400\ \text{kg}\) object swings in a vertical circular path on a string \(0.500\ \text{m}\) long. If its speed is \(4.00\ \text{m/s}\) at the top of the circle, what is the tension in the string there?
A bucket of water is whirled in a vertical circle of radius \(1.00\ \text{m}\). What is its minimum speed at the top if no water is to spill? (At the limit the tension vanishes.)
Tarzan (\(m = 85.0\ \text{kg}\)) tries to cross a river swinging on a vine \(10.0\ \text{m}\) long; his speed at the bottom will be \(8.00\ \text{m/s}\). The vine has a breaking strength of \(1000\ \text{N}\). Does he make it across?
A \(40.0\ \text{kg}\) child swings on a swing supported by two chains \(3.00\ \text{m}\) long. The tension in each chain at the lowest point is \(350\ \text{N}\). Find the child’s speed at the lowest point. (The two chains together pull with 700 N.)
A car passes over a bump that follows the arc of a circle of radius \(20.0\ \text{m}\). What is the greatest speed it can have at the top without leaving the road? (At the limit N = 0.)
A light string can support a stationary hanging load of \(25.0\ \text{kg}\) before breaking (\(245\ \text{N}\)). A \(3.00\ \text{kg}\) object attached to it rotates on a frictionless table in a circle of radius \(0.800\ \text{m}\). What range of speeds can it have before the string breaks?
Theory after R. Resnick, D. Halliday and K. Krane, Física, vol. 1, chapter 5 and section 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 Newton’s laws.