The figure shows a 0.41-kg block sliding from A to B along a frictionless surface. When the bloc

The figure shows a 0.41-kg block sliding from A to B along a...

The figure shows a $0.41-\mathrm{kg}$ block sliding from $\mathrm{A}$ to $\mathrm{B}$ along a frictionless surface. When the block reaches $B,$ it continues to slide along the horizontal surface $\mathrm{BC}$ where the kinetic frictional force acts. As a result, the block slows down, coming to rest at C. The kinetic energy of the block at A is $37 \mathrm{J},$ and the heights of $\mathrm{A}$ and $\mathrm{B}$ are 12.0 and 7.0 $\mathrm{m}$ above the ground, respectively. Concepts: (i) Is the total mechanical energy of the block conserved as the block goes from $A$ to $B ?$ Why or why not? (ii) When the block reaches point $\mathrm{B},$ has its kinetic energy increased, decreased, or remained the same relative to what it was at A? Give a reason for your answer. (iii) Is the total mechanical energy of the block conserved as the block goes from B to C? Justify your answer. Calculations: (a) What is the value of the kinetic energy of the block when it reaches $\mathrm{B} ?$ (b) How much work does the kinetic frictional force do during the BC segment of the trip?

The figure shows a $0.41-\mathrm{kg}$ block sliding from $\mathrm{A}$ to $\mathrm{B}$ along a frictionless surface. When the block reaches $B,$ it continues to slide along the horizontal surface $\mathrm{BC}$ where the kinetic frictional force acts. As a result, the block slows down, coming to rest at C. The kinetic energy of the block at A is $37 \mathrm{J},$ and the heights of $\mathrm{A}$ and $\mathrm{B}$ are 12.0 and 7.0 $\mathrm{m}$ above the ground, respectively. Concepts:
(i) Is the total mechanical energy of the block conserved as the block goes from $A$ to $B ?$ Why or why not?
(ii) When the block reaches point $\mathrm{B},$ has its kinetic energy increased, decreased, or remained the same relative to what it was at A? Give a reason for your answer. (iii) Is the total mechanical energy of the block conserved as the block goes from B to C? Justify your answer. Calculations: (a) What is the value of the kinetic energy of the block when it reaches $\mathrm{B} ?$ (b) How much work does the kinetic frictional force do during the BC segment of the trip?

Key Concepts

Nonconservative Forces and Energy Dissipation

Nonconservative forces, such as friction, remove mechanical energy from a system by converting it into thermal energy or other forms that are not useful for mechanical work. This means that when such forces are acting, the mechanical energy of the system will not be conserved.

Mechanical Energy Conservation

This concept refers to the principle that, in the absence of non-conservative forces (like friction), the total mechanical energy (the sum of kinetic and potential energy) remains constant. It is key to analyzing systems where energy transformation occurs without energy loss.

Energy Transformation Between Potential and Kinetic Energy

This concept explains how an object’s energy can change form, such as converting gravitational potential energy into kinetic energy when moving from a higher to a lower elevation. The total energy remains the same if only conservative forces are present.

Work-Energy Principle

The work-energy principle states that the net work done on an object is equal to the change in its kinetic energy. This principle is essential for understanding how forces, including friction, affect the motion of an object by increasing or decreasing its kinetic energy.

Transcript

00:01 Okay, so we're giving a problem where we have an object of mass of 0 .41 kilograms as an initial kinetic energy at state a of 37 joules, and it will slide along this frictionless slope going down to state b from a height of 12 meters to 7 meters.

00:25 And then after that, from state b to state c, it would start to slow down from this very hard surface with friction.

00:34 And it will stop with a velocity of zero.

00:43 Okay, first let's look at this conceptually.

00:48 From a to b, energy will be conserved because no outside forces are acting on the object.

00:55 So no air resistance, no friction to slow it down.

00:59 So therefore, energy is indeed conserved.

01:06 As the block reaches b, kinetic energy has increased because since energy is conserved, potential energy has decreased at b.

01:16 So all that extra energy, actually all that energy has been transferred into kinetic energy from the potential energy, because height has decreased over here from 12 meters to 7 meters.

01:36 Now, the total mechanical energy is not conserved from b to c because friction acts on the object.

01:45 So there's actually outside forces.

01:47 That acts on the object that makes the mechanical energy not conserved.

01:59 Okay, now let's look at the calculation over here.

02:07 We need to find the kinetic energy as it goes, the final kinetic energy as it goes from state a to state b...

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