# D’Alembert Vs Conservation of Energy

• GrimUpNorth
In summary, a 5kg block is pulled up an incline with a 20 degree angle, a coefficient of friction of 0.2, and a distance of 10m. Using D'Alembert's principle, the net force on the block is found to be 12N. Using conservation of energy, the work done on the block is calculated to be 198.28J, resulting in a pulling force of 19.828N. Both principles are used to compare and contrast the results for the assignment.
GrimUpNorth

## Homework Statement

A block with a mass of 5kg is pulled up an incline with a 20 degree incline, the coefficient of friction is 0.2 and the distance 10 metres. The block accelerates from 1m/s at point A to 5m/s at point B.
I have to use both D’Alembert’s and Conservation of energy principles.

Does the following look correct? I only have one chance to get it right so want to make sure it’s correct.

## Homework Equations

Frictional force=µ*M*g*cosӨ
a=(v2-u2)/2s
Net force=Ma
t=(v-u)/a

W + PE(initial) +KE(initial) =PE(final) +KE(final) +Heat lost due to friction
3. The Attempt at a Solution [/B]
Mass of the block=5kg
Angle of inclination=20 ̊
Displacement, s=10m
Initial velocity=1m/s
Final velocity=5m/s
Coefficient of friction, µ=0.2
From Alembert’s principle, net force-Ma=0 (Ghosh, 2016, pp.9)
Net force=pulling force -frictional force
Frictional force=µ*M*g*cosӨ
=0.2*5*9.81*cos (20)
=9.218 N
From Newtons laws, v2=u2+2as
a=(v2-u2)/2s
= (25-1)/2*10
=2.4m/s2
Therefore, Ma=5*2.4=12N
From the above principle,
Net force=Ma
=12N
Pulling force=Net force + frictional force
=12N + 9.218N
=21.218N
Work done in moving the block from point A to B equals pulling force*displacement(AB)
=21.218*10
=212.18J
From Newtons laws=U+ at
Therefore, t=(v-u)/a
= (5-1)/2.4
=1.66sec
Power=work done/time
=212.18/1.66 =128.2 Watts
Principle of conservation of energy
From this principle, energy can only be transformed into other forms but cannot be created or destroyed. Therefore, the net work done by the forces is zero as mechanical energy is conserved.
Work input=W
potential energy=PE
Kinetic energy =KE
Using this principle, we obtain the following equation;
W + PE(initial) +KE(initial) =PE(final) +KE(final) +Heat lost due to friction……(i)
PE(initial)=0, since initial height is zero
KE(initial)=(m*u2)/2= 2.5J
PE(final) =(mgh)=5*9.81*cos (20)
=46.1J
KE(final)=(m*v2)/2= (5*52)/2
=62.5J
Heat lost due to friction=µMg*cosӨ*d=0.2*5*9.81*cos20*10
=92.18J
Equation (i) above reduces to:
W+KE=PE +KE+Energy lost as heat, since initial PE equals zero
Substituting the corresponding values,
W+2.5=46.1+62.5+92.18
Making W the subject of the formula,
W=-2.5+46.1+62.5+92.18
=198.28J
From work done obtained above(W=198.28J), pulling force will be
F=w/d
=198.28/10=19.828N

Though you do not say so, I assume you are trying to solve for the force applied to pull the block up the ramp.

You have an arithmetic error here:
GrimUpNorth said:
a=(v2-u2)/2s
= (25-1)/2*10
=2.4m/s2

I would not use conservation of energy here since friction is a non-conservative force. You do account for the heat lost to friction, but I think you are making things unnecessarily complicated. I suggest that you write the equation for the net acceleration up the ramp, including the pulling force, friction and component of the force of gravity pulling down the ramp. You can also find the net acceleration in terms of the distance traveled and initial and final velocities of the block. Set the two expressions for net acceleration equal and solve for the pulling force.

GrimUpNorth
tnich said:
Though you do not say so, I assume you are trying to solve for the force applied to pull the block up the ramp.

You have an arithmetic error here:I would not use conservation of energy here since friction is a non-conservative force. You do account for the heat lost to friction, but I think you are making things unnecessarily complicated. I suggest that you write the equation for the net acceleration up the ramp, including the pulling force, friction and component of the force of gravity pulling down the ramp. You can also find the net acceleration in terms of the distance traveled and initial and final velocities of the block. Set the two expressions for net acceleration equal and solve for the pulling force.
Thank you, I have to use both principles and then compare and contrast the results for the assignment.
tnich said:
Though you do not say so, I assume you are trying to solve for the force applied to pull the block up the ramp.

You have an arithmetic error here:I would not use conservation of energy here since friction is a non-conservative force. You do account for the heat lost to friction, but I think you are making things unnecessarily complicated. I suggest that you write the equation for the net acceleration up the ramp, including the pulling force, friction and component of the force of gravity pulling down the ramp. You can also find the net acceleration in terms of the distance traveled and initial and final velocities of the block. Set the two expressions for net acceleration equal and solve for the pulling force.

Thank you, I need to use both principles for the assignment.
Does this look a better equation to use?

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GrimUpNorth said:
Thank you, I have to use both principles and then compare and contrast the results for the assignment.Thank you, I need to use both principles for the assignment.
Does this look a better equation to use?
Yes, I think you have the right numerical answer now. That will give you something to check the other methods against.

## What is D’Alembert’s Principle?

D'Alembert's Principle is a fundamental concept in classical mechanics that states that the net force acting on a body is equal to the product of its mass and acceleration.

## What is the Conservation of Energy?

The Conservation of Energy is a fundamental principle in physics that states that energy cannot be created or destroyed, only transferred or converted from one form to another.

## How are D’Alembert's Principle and Conservation of Energy related?

D'Alembert's Principle and the Conservation of Energy are related in that they both deal with the fundamental concepts of forces and energy in classical mechanics. D'Alembert's Principle is used to analyze systems in motion, while the Conservation of Energy ensures that the total energy in a closed system remains constant.

## What are some real-world applications of D’Alembert's Principle and Conservation of Energy?

D'Alembert's Principle and the Conservation of Energy have numerous real-world applications, such as analyzing the motion of objects in space, designing structures and machinery, and understanding the behavior of fluids.

## Are there any limitations to D’Alembert's Principle and Conservation of Energy?

While D'Alembert's Principle and the Conservation of Energy are widely used and well-established principles, they do have limitations. For example, they do not apply to systems that involve non-conservative forces, such as friction or air resistance. Additionally, they are based on classical mechanics and do not fully account for the principles of quantum mechanics.

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