A department-store escalator makes an angle of 30^∘ with the horizontal and takes 40 seconds to

A department-store escalator makes an angle of 30^∘ with the...

A department-store escalator makes an angle of $30^{\circ}$ with the horizontal and takes 40 seconds to transport a person from the first to the second floor with a vertical rise of $20 \mathrm{ft}$. At a certain instant, there are 10 people on the escalator averaging 150 lb per person and standing at rest relative to the moving steps. Additionally, three boys averaging $120 \mathrm{lb}$ each are running down the escalator at a speed of $2 \mathrm{ft} / \mathrm{sec}$ relative to the moving steps. Calculate the power output $P$ of the driving motor to maintain the constant speed of the escalator. The no-load power without passengers is 2.2 hp to overcome friction in the mechanism.

A department-store escalator makes an angle of $30^{\circ}$ with the horizontal and takes 40 seconds to transport a person from the first to the second floor with a vertical rise of $20 \mathrm{ft}$. At a certain instant, there are 10 people on the escalator averaging 150 lb per person and standing at rest relative to the moving steps. Additionally, three boys averaging $120 \mathrm{lb}$ each are running down the escalator at a speed of
$2 \mathrm{ft} / \mathrm{sec}$ relative to the moving steps. Calculate the power output $P$ of the driving motor to maintain the constant speed of the escalator. The no-load power without passengers is 2.2 hp to overcome friction in the mechanism.

Key Concepts

Inclined Plane Mechanics

Inclined plane mechanics involves analyzing the forces acting on objects moving along a sloped surface. By resolving the gravitational force into components parallel and perpendicular to the incline, one can determine the effective force needed to move or lift objects along the plane, a crucial step when dealing with the geometry of an escalator.

Relative Motion

Relative motion refers to the movement of objects as observed from different reference frames. In scenarios where individuals move on a moving escalator, their velocities relative to the escalator and the ground must be considered to accurately determine the net mechanical work and energy transfer in the system.

Mechanical Power

Mechanical power is the rate at which work is performed or energy is transferred in a system. It quantifies how much energy is used per unit time to overcome resistive forces, lift loads, or maintain motion, and is measured in units such as watts or horsepower.

Gravitational Potential Energy

Gravitational potential energy is the energy associated with an object’s position in a gravitational field. When a load is raised against gravity, this energy is increased by the product of its mass, the gravitational acceleration, and the change in height, making it a key parameter when calculating the work needed to lift passengers in systems like escalators.

Frictional Losses and No-Load Power

Frictional losses account for the energy dissipated due to resistive forces in a mechanical system, and no-load power represents the baseline power required to overcome these losses even when no additional load is present. Incorporating these factors is essential for calculating the total power output needed from a motor to maintain constant operation under various load conditions.

Transcript

00:01 So here on this problem we are given an accelerator that makes an angle of 30 degrees with the horizontal so this angle is 30 degrees and the time taken by the accelerator to move the person from the first floor to the second floor is 40 seconds and this vertical height we are given as 20 feet and the time taken is 40 seconds now at a given instance stand there are 10 person which are standing at rest relative to the exhalator and the weight of each person we are given as 150 pound and number of persons we have 10 and with this there are three boys which are traveling at a speed of two feet per second and the weight of each boys we are given as 120 pounds and we have to calculate the power required by the driving motor to maintain a constant speed of the accelerator and we are also given that no load power consumption is 2 .2 horsepower so this is for no load power consumption to overcome the friction so this power is given as 2 .2 horsepower now the power output of the driving motor to maintain a constant speed of the accelerator will be the combination of the power required by the persons plus the power required by the boys plus this frictional power this thing we are given as 2 .2 just two things we have to calculate now the power required to move 10 people it would be equals to np times weight times the velocity of the and this weight is for the weight for persons now number of persons we have ten and weight of every person is 150 pound times the velocity so velocity would be equal to the velocity of the accelerator which it would be equals to distance over the time the distance is 20 feet and time we are given us 40 second so from here the power would be equals to 750 and the unit would be pound feet per second.

02:54 Now we have to convert the unit into horsepower because this unit is given in horse power.

03:00 So we know that a one horsepower is equals to five 50 pound feet per second...

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