Chapter 3: Mechanics

The chapter problem: the roller coaster loop

A theme park has a roller coaster with a single vertical loop. A car is pulled to the top of a lift hill and released. The track falls away in a steep first drop, runs along the bottom, and rises into a circular loop. At the end of the ride, brakes bring the car to rest.

These are the facts to use all the way through the chapter.

• The car and two riders have a total mass of 500 kg.
• The car is released from rest at the top of the first drop.
• The first drop is a straight section of track at 60° to the horizontal, and the car descends 45 m vertically along it.
• The loop is a circle of radius 12 m.
• During the first drop, friction and air resistance are negligible.
• At the end of the ride the brakes bring the car to rest from 29.7 m s⁻¹ in 3.0 s.
• Take the acceleration due to gravity as 9.81 m s⁻².

By the end of the chapter you will work out the speed of the car at the bottom of the first drop and its acceleration down the slope. You will follow the energy through the ride, find the force that holds the car in the loop, and use momentum to find the braking force. At the end you will decide how much of the loop's height is paid for by the drop, and explain why a real loop is not a circle.

First contact: no working yet. Estimate. The car is released from rest 45 m above the bottom. At the top of the loop it is 24 m above the bottom, so it has given up some height but not all of it. Is its speed at the top of the loop closest to 10 m s⁻¹, 20 m s⁻¹ or 30 m s⁻¹? Write down your choice and one sentence of reasoning.

By the end of this chapter you will be able to

• use the equations of motion for constant acceleration, and say why each one applies;
• read a displacement–time graph and a velocity–time graph, using the gradient and the area;
• explain free fall and use $g$ in a calculation;
• state Newton's three laws of motion and apply $F = ma$ to a resultant force;
• draw a free-body diagram and find the resultant force on an object;
• analyse projectile motion as a constant horizontal velocity and a constant vertical acceleration;
• calculate work done by a force, including a force not along the line of motion;
• use conservation of energy, and the equations for kinetic and gravitational potential energy;
• calculate power from work and time, and from force and velocity;
• use the definition of momentum and the impulse–momentum relation;
• use conservation of momentum in one-dimensional collisions and explosions;
• convert between degrees and radians, and find angular velocity;
• use $v = \omega r$ and the equations for centripetal acceleration and force;
• describe simple harmonic motion, and use $a = -\omega^2 x$ and $x = A\cos\omega t$;
• calculate the period of a mass on a spring and of a simple pendulum.

Where it is used

Mechanics is the part of physics that describes how and why things move. A vehicle safety engineer uses momentum and impulse to design a crumple zone. A sports scientist uses projectiles to improve a throw. A roller-coaster designer uses energy and circular motion to keep riders safe at every point of the track. An engineer tuning a suspension bridge uses oscillations to stop the deck swinging in the wind.

The same ideas return throughout the book. Forces and energy carry into materials and electricity; circular motion and oscillations carry into waves and fields.

Your turn to connect it

Think of a journey in which your speed changes. It could be a bus pulling away from a stop, a lift, a bicycle going downhill, or a plane taking off. Write down two moments: one where the speed is rising, and one where it is falling. For one of them, say what force causes the change.

Roadmap

1. 3.1 Kinematics and motion graphs
2. 3.2 Dynamics and Newton's laws of motion
3. 3.3 Work, energy and power
4. 3.4 Momentum
5. 3.5 Circular motion
6. 3.6 Oscillations and simple harmonic motion
7. Review