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Solving problems is an essential part of the understanding process.

Questions and their answers are presented here in the module text format as if it were an extension of the treatment of the topic. The idea is to provide a verbose explanation, detailing the application of theory. Solution presented is, therefore, treated as the part of the understanding process – not merely a Q/A session. The emphasis is to enforce ideas and concepts, which can not be completely absorbed unless they are put to real time situation.

Representative problems and their solutions

We discuss problems, which highlight certain aspects of the study leading to the conservation of energy. The questions are categorized in terms of the characterizing features of the subject matter :

  • Particle
  • Projectile
  • Pulley - block system
  • Incline – block - spring system
  • Vertical circular motion

Particle

Problem 1: The potential energy of a particle of mass "m" is given by " 1 2 A x 2 ". If non-conservative force is not involved and total mechanical energy is "E", then find its speed at "x".

Solution : External force and non-conservative force are absent here. Hence, we can apply conservation of mechanical energy. Total mechanical energy of the particle is constant,

E = U + K

K = E U

K = E 1 2 A x 2

1 2 m v 2 = E 1 2 A x 2

v = { 2 E A x 2 m }

Projectile

Problem 2: A ball of 10 gm is fired from a spring gun of spring constant 500 N/m. The spring is initially compressed by 5 cm and is, then, released. Find the maximum horizontal distance (in meter) traveled by the bullet.

Solution : We are required to find maximum horizontal distance. The horizontal range of a projectile from the ground at an angle “θ” is given by :

Projectile

A ball is fired from spring gun.

R = v 2 sin 2 θ g

For maximum horizontal range, the angle of projection should be 45°. The maximum range is :

R = v 2 g

In order to find speed of projection, we shall employ conservation of mechanical energy. The gun and ball forms an isolated system. Initially, kinetic energy of the bullet is zero, whereas initial elastic potential energy of the spring is “ 1 2 k x 2 ”. Let “v” be the speed when ball breaks the contact with spring. Now, applying conservation of mechanical energy,

K i + U i = K f + U f

Here,

K i = 0

U i = 1 2 k x 2

K f = 1 2 m v 2

U f = 0

Thus,

1 2 k x 2 = 1 2 m v 2

v 2 = k x 2 m

Putting this expression in the expression of horizontal range, the maximum horizontal range is :

R = k x 2 m g

R = 500 X 0.05 2 0.01 X 10 = 500 x 0.0025 0.1 = 1.25 0.1 = 12.5 m

Pulley - block system

Problem 3: In the arrangement shown, the lighter block ascends a height of 2 m after being released. Find the speed of the blocks. Consider string to be mass-less and no friction at the contact.

Pulley - block system

The blocks move vertically by 2m.

Solution : Since friction is absent, only gravity interacts with the elements of the system. We can consider arrangement as isolated system. No external force acts on the system. Remember that Earth is part of the system and gravitational force is internal to the system. Hence, the situation fulfills all conditions for applying conservation of mechanical energy.

Now, two blocks are constrained by a taut string. It means that both blocks move with same speed. Let us denote 4 kg and 10 kg blocks with subscript “1” and “2”. Note that data is given for the change in height. Hence, it is easier to determine change in the potential energy.

Questions & Answers

water boil at 100 and why
isaac Reply
what is upper limit of speed
Riya Reply
what temperature is 0 k
Riya
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
Mustapha
How MKS system is the subset of SI system?
Clash Reply
which colour has the shortest wavelength in the white light spectrum
Mustapha Reply
how do we add
Jennifer Reply
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
Abhyanshu Reply
x=5.8-3.22 x=2.58
sajjad
what is the definition of resolution of forces
Atinuke Reply
what is energy?
James Reply
Ability of doing work is called energy energy neither be create nor destryoed but change in one form to an other form
Abdul
motion
Mustapha
highlights of atomic physics
Benjamin
can anyone tell who founded equations of motion !?
Ztechy Reply
n=a+b/T² find the linear express
Donsmart Reply
أوك
عباس
Quiklyyy
Sultan Reply
Moment of inertia of a bar in terms of perpendicular axis theorem
Sultan Reply
How should i know when to add/subtract the velocities and when to use the Pythagoras theorem?
Yara Reply
Centre of mass of two uniform rods of same length but made of different materials and kept at L-shape meeting point is origin of coordinate
Rama Reply
A balloon is released from the ground which rises vertically up with acceleration 1.4m/sec^2.a ball is released from the balloon 20 second after the balloon has left the ground. The maximum height reached by the ball from the ground is
Lucky Reply

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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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