DPST1021 Chap.3 Newton's Laws and Uniform Circular Motion
Newton's Laws and Uniform Circular Motion
Week 2 of Physics 1A moves from describing motion to explaining it. Lecture 4 opens with uniform circular motion: a body travelling at constant speed on a circle still accelerates, because its velocity keeps turning, and that acceleration, of size v squared over r, always points toward the centre of the circle.
The same lecture introduces free body diagrams and Newton's first law, together with the idea of an inertial frame in which that law holds.
Lecture 5 adds the third law and the distinction between internal and external forces. Choosing a system carefully lets internal pairs cancel, so the second law can be written using external forces only.
Lecture 6 then puts the three laws to work on problems that need the net force on a body, brings in Hooke's law for springs, and applies the second law to circular paths such as the conical pendulum and the banked road.
The central skill is a dependable routine: isolate one body, draw every force acting on it, choose axes that suit the acceleration, and write one component equation per axis.
Circular motion fits the same routine once the acceleration is recognised as centripetal. These lectures sit within the mechanics material assessed in Test 1, and the second weekly homework quiz covers Lectures 4 to 7, so fluency here supports the friction, energy and rotation topics that follow later in the term.
What this chapter covers
- 01
Circular motion quantities: link period, angular speed, speed and radius through v = rω and v = 2πr/T, then find the centripetal acceleration.
- 02
Average and instantaneous acceleration: separate the change in velocity over an interval from the centre-pointing value at a single point on the circle.
- 03
Free body diagrams: isolate one body and draw its weight, every tension and every normal force acting on it.
- 04
First law and inertial frames: recognise constant velocity as the zero-net-force state and identify accelerating frames where the law appears to fail.
- 05
Third law and internal forces: pair forces acting on different bodies, and choose systems so internal pairs cancel.
- 06
Hooke's law: model a spring force as proportional and opposite to the extension, then combine it with the second law.
- 07
Circular dynamics: apply the second law with a centripetal acceleration to conical pendulums, banked turns and other horizontal circles.
Worked example · free
A puck circling on a spring
- 1Free body diagram: weight and normal force balance vertically; the spring force, pointing to the pin, is the only horizontal force.
- 1Hooke's law: x = 0.500 − 0.400 = 0.100 m, so F = 150 × 0.100 = 15.0 N inward.
- 1Second law toward the centre, with r = 0.500 m: 15.0 = 0.300v²/0.500.
- 1Solve: v² = 15.0 × 0.500 / 0.300 = 25.0, so v = 5.00 m/s.
- 1Period: T = 2πr/v = 2π(0.500)/5.00 = 0.628 s.
Key terms
- Centripetal acceleration
- The centre-directed acceleration of a body moving on a circle, of size v squared over r or rω squared.
- Angular speed
- The rate of sweeping out angle, in radians per second, equal to 2π over the period.
- Free body diagram
- A sketch of one isolated body showing every external force on it as an arrow.
- Inertial frame
- A reference frame in which a body with zero net force keeps a constant velocity.
- Normal force
- The contact force a surface exerts perpendicular to itself, sized to stop the body sinking in.
- Internal force
- A force between two parts of the same chosen system; such forces cancel in pairs.
- Spring constant
- The stiffness k in Hooke's law, the force needed per metre of extension, measured in newtons per metre.
Newton's Laws and Uniform Circular Motion FAQ
Is centripetal force a separate force to draw on a diagram?
No. Centripetal describes a direction, not a source. On a free body diagram you draw only real forces such as tension, weight, normal force or friction; the centripetal force is the net component of those forces that points to the centre. An extra arrow labelled centripetal counts that push twice.
Why are weight and the normal force not a third-law pair?
Both act on the same body, while a third-law pair always acts on two different bodies. The weight of a book pairs with the book's gravitational force on the Earth; the table's push on the book pairs with the book's push on the table. The two can also differ in size, as in an accelerating lift.
How do I decide which bodies to include in a system?
Include bodies that share an acceleration when you want that acceleration quickly, because the forces between them then cancel. Isolate a single body when you need one of those connecting forces, such as a rope tension or a spring force. Using both choices in turn gives a built-in check.
Why does a conical pendulum's period not depend on the mass?
Mass appears in both the vertical and horizontal equations, so it divides out when you form tanθ = v squared over rg. The period then depends only on the string length, the angle and g. Heavy and light bobs at the same angle go round together.
What makes a frame non-inertial?
Any frame accelerating or rotating relative to the ground. Inside a braking train, objects appear to move with no force pushing them. Problems are normally solved from the ground frame instead, where every acceleration has a real force behind it.
Exam move
Make the free body diagram a habit: for each practice problem, draw the diagram, list the forces with what exerts each one, and write component equations before substituting. Then practise choosing systems by solving connected-body questions twice, once treating the bodies together and once separately, and confirm that both routes give the same acceleration.
For circular motion, start with the kinematics of period, angular speed and centripetal acceleration in radians, then treat each dynamics problem as an ordinary second-law problem with the acceleration pointing inward. Work the conical pendulum and the frictionless banked road side by side so the shared geometry becomes obvious.
Finally, attempt the second homework quiz on Lectures 4 to 7 and track which force you most often forget to draw.
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