Notes for Stability

•  Stability is a measure of an object’s ability to maintain its original position and can be explained by the position of its centre of gravity and moment of force due to its weight.

•  Centre of gravity (CG) of any object is defined as the point through which it whole weight appears to act.
• An external agent exerts a force on the object and the object is tilted through an angle away from the original position due to the force. The original position of the object is the position that the object takes before the action of the external agent. The part of the object which is still in contact with the ground/fixed point when the object is tilted is known as the contact point.
•  Most objects topple when they are tilted through a sufficiently large angle away from the original position.
•  An object can be in stable, unstable or neutral equilibrium.
•  An object is said to be in stable equilibrium if when slightly tilted (small angle away from the original position), it will return to its original position.
•  When slightly tilted, the CG of an object in stable equilibrium will rise. The line of action through its weight (i.e., the vertical line through the CG) still lies within its base. A moment due to its weight about the contact point (pivot) will cause the object to return to its original position.
•  An object is said to be in unstable equilibrium if when slightly tilted, it will topple over.
•  When slightly tilted, the CG of an object in unstable equilibrium will lower. The line of action through its weight will lie outside its base. A moment due to its weight about the contact point will cause the object to move away from the original position.
•  An object is said to be in neutral equilibrium if when slightly tilted, it remains in the new displaced position. Only objects with symmetrical forms (e.g. sphere) can be in neutral equilibrium.
•  When slightly tilted, the CG of an object in neutral equilibrium will remain at the same height. The line of action through its weight will lie within its base. No net moment is generated and it will remain in its new displaced position.
•  The more stable an object is, the larger the angle it can be tilted away from the original position before the line of action through its weight lie outside its base.
•  An object can be made more stable by lowering its CG or increasing its base area. Since the CG of an object is usually located near the region that has more mass, having more mass at the bottom can lower the CG of the object.

Manometer with two liquids

GCE O-level 5076 Science (Physics) 2015 November Paper 2 Question 4(c)

Question:

How would you find the density of liquid B? I believe most of you would find the pressure at point X and then equate it with the pressure at point Y. Have you ever wonder why this method works? At least I do.

Let us investigate by considering the pressure at points P and Q. Point P is at the junction between air and liquid A. Point Q is at the same horizontal level in liquid B.

Pressure at P = Atmospheric pressure

Pressure at Q = Atmospheric pressure + Pressure due to the column of liquid above Q

In other words, the pressure at A and at B are different. Hence, we cannot solve the question by equating the pressure at P and pressure at Q. This begs the following question: What's so special about points X and Y?

Explanation:

As the liquids are at rest (i.e. in equilibrium), the pressure at the bottom of the manometer due to the left column must be equal to the pressure due to the right column.

Pressure due to left liquid column = Pressure due to right liquid column.

For points X and Y,

Pressure at X + Pressure due to water column below X = Pressure at Y + Pressure due to water column below Y.

Since Pressure due to water column below X = Pressure due to water column below Y, Pressure at X = Pressure at Y.

For point A and B,
Pressure at P + Pressure due liquid column between P and X + Pressure due to water column below X = Pressure at Q + Pressure due liquid column between Q and Y + Pressure due to water column below Y.

Since Pressure due to water column below X = Pressure due to water column below Y,
Pressure at P + Pressure due liquid column between P and X = Pressure at Q + Pressure due liquid column between Q and Y.

Since liquid A and liquid B have different densities, pressure due liquid column between P and X is NOT equal to pressure due liquid column between Q and Y even though both liquid columns have the same vertical height (i.e. PX = QY). Hence, pressure at P is NOT equal to pressure at Q.

Isolated System for Principle of Conservation of Energy

According to Physics Matters GEO 'O' Level Physics (4th edition) by C. Chew, S.F. Chow and B.T. Ho:

The Principle of Conservation of Energy states that the energy cannot be created or destroyed, but can be converted from one form to another. The total energy in an isolated system is constant.

But what's an isolated system? According to Fundamentals of Physics (10th edition) by D. Halliday, R. Resnick and J. Jewett:

… the system is isolated from its environment; that is, no external force from an object outside the system causes energy changes inside the system.

Let us now consider a free-falling object. We often use it to illustrate the principle of conservation of energy (e.g. KE + GPE = constant). But are we "allowed" to do so? After all, we would not consider a free-falling object to be an isolated system because weight is an external force acting on the object. But why does the principle of conservation of energy holds for a free-falling object?

Hint: Redefine the system you consider to include …

Notes on Pressure

•  Pressure is an effect which occurs when a force is applied on a surface.
•  Pressure is defined as the force acting perpendicularly on per unit area.
•  Pressure is a scalar quantity.
•  The SI unit of pressure is newton per square metre (N/m^2), also known as the pascal (Pa).

Should velocity be defined as the rate of change of displacement?

What is displacement?

According to Fundamentals of Physics (10th edition) by D. Halliday, R. Resnick and J. Jewett:

The displacement of a particle is the change in its position.

Mathematically, displacement, x = final position - initial position and it is a vector quantity.

What is velocity?

According to Fundamentals of Physics (10th edition) by D. Halliday, R. Resnick and J. Jewett,

Velocity is the derivative of x (position) with respect to t (time).

In other words, velocity is the time rate of change of  position and it is also a vector quantity. Mathematically, velocity = dx/dt.

However, according to Physics Matters GEO 'O' Level Physics (4th edition) by C. Chew, S.F. Chow and B.T. Ho:

Velocity is the rate of change of displacement.

In other words, Physics Matters GEO 'O' Level Physics is saying that velocity is the rate of change of change of position. In my humble opinion, the definition might be technically incorrect.

Since I am on the topic of displacement and velocity, let me end off by posing a challenge to you: For motions in 1 dimension, are you able to determine the speed-time (distance-time) graph from its velocity-time graph (displacement-time) graph? How about velocity-time graph (displacement-time) from its speed-time (distance-time) graph?

O-level Science (Physics) N04/2/4bii

An athlete runs a 100 m race in 12.5 s. Explain why, for part of the race, the athlete must have been running faster than the speed calculated in (a).

GCE O-level 5052 Science (Physics) 2004 November Paper 2 Question 1(b)

Answer: The runner accelerates from rest.

Proof: Consider the area under an arbitrary speed-time graph that starts from rest and the area under another speed-time graph that is travelling at a constant speed, which is equal to the maximum speed of the first graph. Both graphs stop at time t. Thus, the area under the first speed-time graph  = average speed x t while the area under the second speed-time graph = maximum speed x t. Since first area < second area, hence average speed < maximum speed.

O-level Physics N15/2/4bii

Suggest why the hydraulic press does not work properly if the oil contains bubbles of air.

GCE O-level 5059 Physics 2015 November Paper 2 Question 4(b)(i)

1) Inefficient - Work done by piston with effort is more than work done by piston with load (because some work done to compress the gas).

2) Pascal principle - Pressure at piston with effort will not be equal to the pressure at piston with load.

3) Time lag - The piston with load will take some time to move after the effort is being exerted.

Note: Point 2 can be deduced from point 1. And I believe all these points might not be valid when we use the hydraulic press quasi-statically after it has reached dynamic equilibrium (pressure of gas = pressure of surrounding liquid).

Principle of Moments

According to Physics Matters GEO 'O' Level Physics (4th edition) by C. Chew, S.F. Chow and B.T. Ho:

The principle of moments states that when a body is in equilibrium, the sum of clockwise moments about a pivot is equal to the sum of anticlockwise moments about the same pivot.

The book is saying that if an object is in equilibrium, then the resultant moment about a pivot (in fact, about any point) is zero. However, the converse is not true. In other words, an object having zero resultant moment about the pivot might not be in equilibrium. Why is that so?

There are two types of equilibrium: translational equilibrium and rotational equilibrium. An object is in translational equilibrium when the resultant force is zero. An object is in rotational equilibrium when the resultant moment about the pivot is zero. An object having zero resultant moment might not have zero resultant force (see diagram below).

Acknowledgement: The diagram is obtained from University Physics with Modern Physics (13th edition) by H. Young and R. Freedman.

In general, students are able to understand the principle of moments. However, they might not know when they should apply the principle. One way is to emphasize that we often need to apply the principle of moments when we are dealing with extended bodies (for examination, objects with dimensions provided).

Since I am on the topic of principle of moment, let me end off by posing a challenge to you: Is the stationary uniform metre rule shown in the diagram below in equilibrium? =P

How about the following question, which is obtained from Cambridge International A-level?

Oil drop challenge

A car moving with constant acceleration drips oil on the road at a rate of one drop every second. The diagram below shows the position of the oil drops along the road.

(a) Calculate the average speed of the car between position A and position C.

(b) Calculate the instantaneous speed of the car when it was at B.

Answers: (a) 3 m/s; (b) 4 m/s

What is the difference between deceleration and negative acceleration?

According to Five Easy Lessons: Strategies for Successful Physics Teaching by R. Knight,

Students interpret a positive acceleration as always meaning "speeding up" and a negative acceleration as "slowing down", rather than associating the sign with the direction of the acceleration vector. This is a difficult idea to change, and for many students it becomes a serious difficultly when they get to Newton's second law.

The sign of the acceleration indicates the direction of the acceleration vector. However, it does not indicate how the speed of the object will change. When the acceleration vector is parallel to the velocity vector, the speed of the object will increase. Conversely, when the acceleration vector is anti-parallel (in opposite directions) to the velocity vector, the speed of the object will decrease.

According to Physics for Scientists and Engineers (6th edition) by R. Serway and J. Jewett,

The word deceleration has the common popular connotation of slowing down. We will not use this word in this text, because it further confuses the definition we have given for negative acceleration.

In other words, deceleration is NOT equivalent to negative acceleration.

NUS PC1431 (AY11/12 SEM 2) FINAL EXAM SUGGESTED SOLUTIONS

I don't have time to draw any nice diagrams so you may have to spend some time reading my long answers. Hopefully, I don't have too many typos. Just leave a comment if you have found one and alert me if you think I am talking rubbish anywhere in the solution. Anyway, the examination paper can be obtained from http://libportal.nus.edu.sg/frontend/index. =)

(Disclaimer: The author has tried his best to ensure the quality of the solution. However, the author will not responsible for any liabilities that arise directly or indirectly from the usage of these solutions.)

NUS PC1431 (AY11/12 SEM 1) FINAL EXAM SUGGESTED SOLUTIONS

I don't have time to draw any nice diagrams so you may have to spend some time reading my long answers. Hopefully, I don't have too many typos. Just leave a comment if you have found one and alert me if you think I am talking rubbish anywhere in the solution. Anyway, the examination paper can be obtained from http://libportal.nus.edu.sg/frontend/index. =)

(Disclaimer: The author has tried his best to ensure the quality of the solution. However, the author will not responsible for any liabilities that arise directly or indirectly from the usage of these solutions.)

Mass, Weight, Density

Mass

• Mass is the measure of the amount of substance in a body.
• Unit of mass: kilogram (kg), gram (g)
• Scalar quantity
• Measured using a beam balance
• Fixed value
• Inertia is directly proportional to mass. Inertia is the object’s reluctance to start/stop moving.

Volume

• Volume is amount of space the object occupies.
• The volume of the solid/liquid can be obtained using a measuring cylinder
•  The volume of a regularly shaped solid or an irregularly shaped solid which sinks in water can be determined using displacement method.

Gravitational field

• Gravitational field is a region in which a mass experiences a force due to gravitational attraction.
• The strength of the gravitational field is due to the mass of the planet
• Gravitational field strength, g, measures the strength of the gravitational field
• Unit of gravitational field strength: Nkg-1
• g = 10 Nkg-1 at the surface of the Earth
• The strength of the gravitational field decreases as we get further away from the planet

Weight

• Weight is the gravitational force on the body.
• Units: Newton (N)
• Vector quantity. Points downward.
• Measured using a spring balance
• Weight = mass × gravitational field strength
• Weight depends on the distance of the object from the planet

Density

• Density is mass per unit volume
• Units of density: kgm-3, gcm-3 (1000kgm-3 = 1gcm-3)
• density = mass/volume.
• Scalar quantity
• Depends on type of material. Objects of different sizes made up of the same material will have the same density.
• The density of object can be obtained by measuring both the mass (beam balance) and volume of the object (displacement method). Then, use the equation (density = mass/volume)
• Average density = total mass/total volume
• Generally, heating causes an increase in volume without an increase in the mass. Density decreases. (Convection)
• If the object has a higher density than the surrounding medium, the object will sink. If the object has a lower density than the surrounding medium, the object will float. If the object has the same density as the surrounding medium, the object will suspend in the medium