D’Alembert Vs Conservation of Energy

[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

 

[tnich]

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:

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]

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.

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?

 

[tnich]

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.