doPhysics · Dynamics
Accelerated Objects
Newton's second law put to work on bodies that speed up. The acceleration first and then the forces, one axis at a time. Pushes and pulls at an angle, the smooth incline, strings and pulleys, the lift, the pseudo force, and impulse.
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Why this chapter
Why this chapter exists
In the last chapter every force balanced, and nothing accelerated. Most of the interesting world is not like that. Trolleys are pushed, blocks slide down ramps, lifts start and stop, and a fielder stops a cricket ball in a tenth of a second. This chapter puts \(\sum \vec F = m\vec a\) to work on such bodies. The method is always the same: decide where the body accelerates, put an axis along it, and write the law one axis at a time. With strings and pulleys a second rule joins it, the constraint that ties the motions of connected bodies together.
You will learn
By the end of this chapter you will
- Solve a single body pushed or pulled at an angle, or sliding on a smooth incline, by choosing axes along and across its acceleration.
- Explain why a light string has one tension, and solve bodies joined by strings over fixed and movable pulleys with the string constraint.
- Predict the reading of a weighing scale in a lift that speeds up, moves steadily, slows down, or falls freely.
- Use the pseudo force in an accelerating frame, and say honestly what it is and what it is not.
- Use impulse and the impulse-momentum theorem to explain catching, landing and the jerk of a string.
Before you begin
Before you begin
You should be able to draw a complete free-body diagram and write \(\sum \vec F = m\vec a\) along two axes, as in Forces and Newton's Laws. You should know the equations of motion for constant acceleration and relative acceleration from the Kinematics unit. You should be able to split a force into components with sine and cosine.
Lessons
1
The Second Law, One Axis at a Time
Newton's second law as two equations, one along each axis. Find the acceleration first, choose axes along it, then write the forces. Pushes and pulls at an angle, and the block on a smooth incline.
Second law along each axisPush and pull at an angleBlock on a smooth incline
2
Bodies Joined by a String
Why a light string pulls with the same tension at both ends, and how to solve two bodies that move together. Two blocks pulled across a table, and a block on a table pulled by a hanging mass.
One tension in a light stringTwo blocks pulled on a tableBlock and hanging mass
3
The Atwood Machine and String Constraints
Two masses over one pulley, and the rule that ties their accelerations together. A string that cannot stretch fixes how far each body moves; a movable pulley makes one body move twice as far as the other.
The atwood machineString constraint equal accelerationsMovable pulley halves the acceleration
4
Weight in a Lift
What a weighing scale really measures. The scale reads the normal force. It reads more while a lift accelerates upward, less while it accelerates downward, and zero in free fall.
Apparent weight in a liftFree fall and weightlessness
5
The Pseudo Force
Newton's laws seen from inside an accelerating vehicle. Why they fail there, the one correction that repairs them, and what that correction is not. The pendulum in a car and the block held still on a pushed wedge.
Pseudo force in an accelerating framePendulum in an accelerating carBlock on an accelerated wedge
6
Impulse and Momentum
The second law for forces that act for a very short time. Impulse as force times time, and as the area under a force-time graph. The impulse-momentum link, why a fielder draws the hands back, and the jerk of a string that goes taut.
Impulse and change in momentumImpulsive forces and catchingThe jerk of a string
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- apparent weight
- The reading of a weighing scale, which is the normal force on the body. It equals \(m(g + a)\) in a lift with upward acceleration \(a\).
- Atwood machine
- Two masses hung from the ends of one string over a fixed pulley. The heavier mass falls and the lighter rises with the same acceleration \((m_1 - m_2)g/(m_1 + m_2)\).
- component
- The part of a vector along one axis. A force at an angle \(\theta\) to the \(x\) axis has components \(F\cos\theta\) along \(x\) and \(F\sin\theta\) along \(y\).
- constraint
- A relation between the positions, and so the accelerations, of connected bodies. It comes from a string of constant length or a surface that a body must stay on.
- impulse
- The product of a force and the time for which it acts, or the area under a force-time graph. It equals the change in momentum and is measured in N s.
- impulsive force
- A very large force that acts for a very short time, such as a blow, a catch or the jerk of a string.
- incline
- A plane surface tilted at an angle to the horizontal, also called a ramp or an inclined plane.
- inertial frame
- A frame of reference in which a body with no net force on it keeps its velocity, so that Newton's laws hold. The ground is an inertial frame for everyday problems.
- inextensible string
- A string that does not stretch. Its length stays constant, which ties together the motions of the bodies tied to it.
- light string
- A string whose mass is negligible compared with the bodies it joins. The tension in a light string is the same at every point, even when it accelerates.
- momentum
- The product of the mass and the velocity of a body, \(\vec p = m\vec v\). A vector measured in kg m/s.
- movable pulley
- A pulley that hangs in a loop of the string instead of being fixed. It is held by two strands, so it feels twice the tension and moves half as far as the free end.
- net force
- The vector sum of all the forces acting on a body. By the second law it equals the mass times the acceleration.
- non-inertial frame
- A frame of reference that accelerates relative to an inertial frame. Newton's laws fail in it unless pseudo forces are added.
- normal force
- The push of a surface on a body pressed against it, perpendicular to the surface. It is whatever the second law across the surface requires, and it is not always equal to the weight.
- pseudo force
- The term \(-m\vec a_0\) added to the real forces on a body when its motion is described from a frame accelerating at \(\vec a_0\). It has no agent and no third-law partner.
- pulley
- A wheel that turns a string round a corner. A smooth, light pulley changes the direction of the tension but not its size.
- smooth surface
- A surface whose friction is small enough to neglect. It can push on a body only perpendicular to itself.
- tension
- The pull of a string, rope or cable on a body tied to it, directed along the string and away from the body.
- wedge constraint
- The rule that a body staying in contact with the face of a wedge has the same acceleration as the wedge perpendicular to that face.
- weightlessness
- The state of a body in free fall, in which nothing pushes on it and a scale under it reads zero. Its weight \(mg\) is still acting.