Oscillating or Vibrating Object in a Wave Motion

When an object is oscillating or vibrating in a wave motion, the speed of the object varies along the path.

In this video, there are 3 examples of vibrating object.

1) Mass vibrating vertically from a spring

2) Pendulum bob oscillating

3) A particle vibrating up and down on a transverse wave

In general, when the object is at the extreme ends of the oscillation or vibration, it is momentarily at rest. Hence its KE at these points is minimum or 0 J.

And in the middle that is where the object is travelling the fastest, hence the KE is the maximum.

Understanding Conservation of Energy and applying in calculation questions

Page 4 of Work Energy and Power: Understanding COE

Energy calculation involving Work Done against friction

A box of mass 2 kg has an initial speed of 10 m/s at the foot of the ramp. Given that the friction along the ramp is 2 N, calculate the height h that it reaches when the speed of the box is 5 m/s. (g = 10 m/s² )   

Solutions:
For this question, the additional thing to note is the work done against friction. All the energy possesses by the box at bottom is KE. This KE will decrease and be converted to remaining KE at height h + gain in GPE + work done against friction.

To find the work done against friction, we need to find the distance moved by the box on the inclined ramp, d.

sin60 = h / d
d = h / sin60

Conservation of energy,
KE at bottom = KE at h + GPE at h + friction force
1/2mv² = 1/2mv² +   mgh  +   F x d
(1/2 x 2 x 10²) = (1/2 x 2 x 5²) + (2 x 10 x h) + (2 x h/sin60)
h     = 3.36 m

Ratio of KE and GPE at various positions

A ball in thrown vertically up. D is the highest point the ball reached. Find the ratio of KE at B to the PE at C.   

Solutions: 1 : 1

To solve such question, you have to apply conservation of energy. In other words, total energy of ball at any positions (A, B, C and D) is the same (no air resistance etc). There is no way to find KE as no speed is given. So to find KE, you need to find it indirectly with the help of PE.

Understand the concept here as there can be many variations of question asked.

Work Done, Energy and Power are Scalar Quantities

Work Done, Kinetic Energy (KE), Gravitational Potential Energy (GPE) and Power are all SCALAR quantities.

Many have asked if Work Done is scalar or vector? Though from our textbook, Work Done = Force x distance in the direction of the force.

In other words, Work Done = Force x Displacement

We already know that Force is a vector. Weight being a force, is also a vector quantity. Displacement is distance in a specific direction, hence it is a vector quantity too. In tertiary education, you will learn in details that the product (multiplication) of two vectors will result in a scalar or dot product.

Since it is out of syllabus, it will be good to use other ways to help you to understand and recall.

Method 1 Remember vector x vector will end up a scalar. Similar to (-)x(-) = (+) Method 2 Consider moving a box up to a certain height, work done or gain in GPE is constant regardless of the path it takes (diagonal or vertical). Hence direction is not important, therefore GPE  is a scalar.

When considering KE, since KE = 1/2mv² , similar to Method 1, v² is actually velocity x velocity. Hence it product of 2 vectors, resulting in KE being a scalar.

Power = Energy Conversion/ time    or     Work Done / time,
since energy, work done and time are scalar, Power is a scalar quantity.

Other important note: Power = Rate of work done = Rate of energy conversion.

Work Done, Energy and Power are Scalar Quantities

Work Done, Kinetic Energy (KE), Gravitational Potential Energy (GPE) and Power are all SCALAR quantities.

Many have asked if Work Done is scalar or vector? Though from our textbook, Work Done = Force x distance in the direction of the force.

In other words, Work Done = Force x Displacement

We already know that Force is a vector. Weight being a force, is also a vector quantity. Displacement is distance in a specific direction, hence it is a vector quantity too. In tertiary education, you will learn in details that the product (multiplication) of two vectors will result in a scalar or dot product.

Since it is out of syllabus, it will be good to use other ways to help you to understand and recall.

Method 1 Remember vector x vector will end up a scalar. Similar to (-)x(-) = (+) Method 2 Consider moving a box up to a certain height, work done or gain in GPE is constant regardless of the path it takes (diagonal or vertical). Hence direction is not important, therefore GPE  is a scalar.

When considering KE, since KE = 1/2mv² , similar to Method 1, v² is actually velocity x velocity. Hence it product of 2 vectors, resulting in KE being a scalar.

Power = Energy Conversion/ time    or     Work Done / time,
since energy, work done and time are scalar, Power is a scalar quantity.

Other important note: Power = Rate of work done = Rate of energy conversion.

Work done – Man jumps from height 3 m, what is the force exerted by his legs?

A boy of mass 40 kg jumps from rest from a platform of height 3 m. He lands by bending his knees and stops his body in 0.5 s after landing. What is the force exerted by his legs? 

Solutions: 620 N

To solve this question, you need to apply conservation of energy (COE) to find the kinetic energy (KE) that the boy possesses just before he reaches the ground.
After which, you can solve the question using (1) Conservation of energy or (2) Kinematics to solve.

Using COE to find the KE just before he reaches the ground.

Method 1: Using Conservation of Energy (COE)

Method 2: Using Kinematics

You must know these 2 methods and always think along this 2 directions when dealing with such questions.