The inclined plane depicted in long and rises 3.0 m. ( a )...

The inclined plane depicted in long and rises $3.0$ m. ( $a$ ) What minimum force $F$ parallel to the plane is required to slide a 20 -kg box up the plane if friction is neglected? ( $b$ ) What is the IMA of the plane? ( $c$ ) Find the AMA and efficiency if a $64-\mathrm{N}$ force is actually required. (a) There are several ways to approach this. Let's consider energy. Since there is no friction, the work done by the pushing force, (F) $(15 \mathrm{~m})$, must equal the lifting work done, $(20 \mathrm{~kg})(9.81$ $\mathrm{m} / \mathrm{s}^{2}$ ) $(3.0 \mathrm{~m})$. Equating these two expressions and solving for $\mathrm{F}$ gives $F=39 \mathrm{~N}$ $(b)$ $\mathrm{IMA}=\frac{\text { Distance moved by } F}{\text { Distance } F_{w} \text { is lifted }}=\frac{15 \mathrm{~m}}{3.0 \mathrm{~m}}=5.0$ (c) $\mathrm{AMA}=$ Force ratio $=\frac{F_{W}}{F}=\frac{196 \mathrm{~N}}{64 \mathrm{~N}}=3.06=3.1$ Efficiency $=\frac{A M A}{I M A}=\frac{3.06}{5.0}=0.61=61 \%$ $\mathrm{O} r$, as a check. Efficiency $=\frac{\text { Work output }}{\text { Work input }}=\frac{\left(F_{W}\right)(3.0 \mathrm{~m})}{(F)(15 \mathrm{~m})}=0.61=61 \%$

The inclined plane depicted in long and rises $3.0$
m. ( $a$ ) What minimum force $F$ parallel to the plane is required to slide a 20 -kg box up the plane if friction is neglected? ( $b$ ) What is the IMA of the plane? ( $c$ ) Find the AMA and efficiency if a $64-\mathrm{N}$ force is actually required.
(a) There are several ways to approach this. Let's consider energy. Since there is no friction, the work done by the pushing force,
(F) $(15 \mathrm{~m})$, must equal the lifting work done, $(20 \mathrm{~kg})(9.81$
$\mathrm{m} / \mathrm{s}^{2}$ ) $(3.0 \mathrm{~m})$. Equating these two expressions and solving for $\mathrm{F}$ gives $F=39 \mathrm{~N}$
$(b)$
$\mathrm{IMA}=\frac{\text { Distance moved by } F}{\text { Distance } F_{w} \text { is lifted }}=\frac{15 \mathrm{~m}}{3.0 \mathrm{~m}}=5.0$
(c) $\mathrm{AMA}=$ Force ratio $=\frac{F_{W}}{F}=\frac{196 \mathrm{~N}}{64 \mathrm{~N}}=3.06=3.1$
Efficiency $=\frac{A M A}{I M A}=\frac{3.06}{5.0}=0.61=61 \%$
$\mathrm{O} r$, as a check.
Efficiency $=\frac{\text { Work output }}{\text { Work input }}=\frac{\left(F_{W}\right)(3.0 \mathrm{~m})}{(F)(15 \mathrm{~m})}=0.61=61 \%$

Key Concepts

Efficiency

Efficiency in the context of mechanical systems is defined as the ratio of useful work output to the total work input. It measures how well a machine converts input energy into useful output energy, with losses often due to friction, heat, or other resistive forces. High efficiency indicates that most of the input work is effectively used for the desired output.

Actual Mechanical Advantage (AMA)

Actual Mechanical Advantage represents the real-world force multiplication observed in a machine, taking into account energy losses due to factors like friction. It is calculated as the ratio of the output force to the actual input force used, and typically, AMA is less than or equal to the IMA because of inherent inefficiencies in practical systems.

Ideal Mechanical Advantage (IMA)

The Ideal Mechanical Advantage is the theoretical maximum force multiplication a machine can provide, calculated under the assumption of no friction or energy losses. It is determined by the geometric or physical characteristics of the machine, such as the ratio of distances over which forces act, and serves as a benchmark for evaluating the mechanical efficiency.

Work and Energy

Work in a physics context is defined as the force applied over a distance. When moving an object against gravity, energy is transferred to increase the object’s gravitational potential energy. This relationship helps in understanding how forces in systems like inclined planes convert work into changes in energy, which is essential in analyzing mechanical systems.

Inclined Plane

An inclined plane is a flat surface set at an angle to the horizontal. It is used to raise or lower a load with less effort than lifting it vertically. This simple machine works by increasing the distance over which a force must be applied, effectively reducing the magnitude of force needed to overcome gravity.

Mechanical Advantage

Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device, or machine system. It compares the output force (load) to the input force (effort). This concept is central in evaluating how machines make work easier by allowing a smaller input force to move a larger load.

Transcript

00:05 In this lesson, i'm looking at inclined planes as simple machines.

00:10 All right.

00:11 So we have a plane here.

00:14 And it's going to be used to lift objects.

00:17 We have some parameters here.

00:18 We have the length of the plane here is 15 meters.

00:23 And the height is 3 meters.

00:27 And we have a box with a mass of 20 kilograms.

00:35 And we're going to be applying some force here to raise the box 3 meters.

00:41 And what we want to do is find the minimum force we need to apply to get to the three meters height.

00:48 All right.

00:49 So how do we do this? we know the work done by the force is going to be equal to the change in mechanical energy of the box to do the work energy theorem.

01:01 All right.

01:02 So we know the work done is going to be our force applied times our distance as it's applied over, so our 15 meters.

01:10 And that just equals to mgh for the box.

01:16 So we have force applied times 15 meters equals 20 kilograms times acceleration due to gravity is 9 .8 meters per second.

01:33 Times our change in height are 3 meters.

01:35 And that equals 588 jules.

01:43 We divide that by 15 to get our applied force of 39 newtons, right? and that is our minimum force needed.

01:56 That's when the plane is operating at 100 % efficiency, meaning no work or energy was lost to friction or heat or any other factor.

02:07 But we don't live in a 100 % efficient world...

Please give Ace some feedback