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Physics 1:  Mechanics:  Conservation of Energy, Energy invested against friction 

 

Javier Montenegro Joo

jmj@VirtualDynamics.Org

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Conservation of Energy.-

 
     
 

(1) In the fig. , a block of 1 kg is at a height of 52 cm on a ramp by which it slides down. Determine: (a) the speed that acquires the block in the horizontal sector before hitting the spring. (b) The maximum distance the spring shrinks when pushed by the block. It is known that the spring stretches 60 cm when it is placed vertical and a trunk of 10 kg is hanged from it.

 

Solution.-

Initially as the block is at a certain height, there is only gravitational potential energy. Seeing that there is no friction, the initial potential energy is entirely turned into kinetic energy of the block while it slides on the horizontal before reaching the spring. Finally the kinetic energy of the block is invested in compressing the spring, so that at the end, the gravitational potential energy has been converted into elastic potential energy of the compressed spring.

 

 
 

(2) When a ball of 10 kg hangs from an upright spring, the spring stretches 60 cm. In the fig., the same spring rests on the floor, with one of its ends against the wall and the other end, which is compressed 25 cm, touches a block of 1 kg.  When the spring is freed the block slides on the floor and then it goes up the ramp. Determine (a) the speed of the block when it slides between the spring and the ramp (b) the maximum height the block reaches on the ramp.

Solution.-

In this case at the beginning there is only elastic potential energy of the compressed spring. Since there is no friction this potential energy is totally invested into kinetic energy while the block displaces on the horizontal. Finally the kinetic energy becomes gravitational potential energy.

 
 

(3) In the figure, a small block of mass m is initially at rest at the top of a semi-circular track of radius R. Next, the block slides down without friction on the track. For the moment the block passes by point B, determine: (a) the speed of the block (b) the force that exerts the track on the block.

Solution.-

If there is no friction, there is no rolling; hence the block slides down over the curved surface without rolling. Additionally since there is no fiction the energy is conserved. At point A all the energy is potential and at point B the energy has been transformed into kinetic and potential energies:

 
 

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Energy invested against friction.-

 
     
 

(1) A block of mass m initially at rest at the top of a semi-circular track of radius R, slides down over the frictionless surface from A to B. The surface from B to C has friction and the block reaches a maximum height h there. Determine: (a) Velocity at the bottom (b) the energy lost due to friction.

Solution.-

The total available energy is the potential energy at the top at point A:  E = Ep = mgR

((a)  (a) At the bottom at point B, the potential energy is zero and all the available energy has been converted to kinetic energy, then

 
  (2) A spring of elastic constant k is placed at the end of a horizontal track. There exists friction Uk   along the distance L between A and B and, there is no friction from B to C. The block of mass m slides over the track towards the spring and, when passing by point A it has a velocity Vo.  Determine: (a) Velocity of the block when passing by point B. (b) The maximum deformation (shrinkage) of the spring when it is pressed by the block.

Solution.-

Since there is friction, the block loses part of its original kinetic energy when displacing from A to B. The remaining of the energy, this is, the (kinetic) energy at point B is used to compress the spring.

 
 

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