(a) A 120-lb woman rides a 15-lb bicycle up a 3-percent...

(a) A 120-lb woman rides a 15-lb bicycle up a 3-percent slope at a constant speed of $5 \mathrm{ft} / \mathrm{s}$. How much power must be developed by the woman? (b) A 180-lb man on an 18-lb bicycle starts down the same slope and maintains a constant speed of $20 \mathrm{ft} / \mathrm{s}$ by braking. How much power is dissipated by the brakes? Ignore air resistance and rolling resistance.

Key Concepts

Inclined Plane Mechanics

Inclined plane mechanics involves analyzing the motion and forces acting on objects moving along a slope. A key aspect is resolving the gravitational force into components parallel and perpendicular to the incline, often using the sine (or tangent, for small angles) of the slope angle. The parallel component is critical for calculating the work done against gravity during ascent or the energy dissipated during controlled descent.

Energy Dissipation

Energy dissipation refers to the process of converting mechanical energy—such as kinetic or potential energy—into other forms, usually heat, through non-conservative forces like friction or braking. In contexts where a vehicle maintains constant speed by braking on a slope, the gravitational energy is transformed by the brakes, and the power dissipation represents the rate at which this energy conversion occurs.

Power

Power is the rate at which work is done or energy is transferred over time. In mechanics, it is commonly calculated as the product of force and velocity, indicating how quickly energy is being expended or generated in a system.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position within a gravitational field. For an object in an inclined motion, the increase in vertical height directly corresponds to an increase in gravitational potential energy, calculated as the product of the object's weight and the vertical distance climbed.

Transcript

00:01 Hi everyone.

00:01 Thanks for joining me today.

00:02 So what we're going to be looking at is a problem dealing with power.

00:06 So what we have in the first case is a biker going up a 3 % slope.

00:13 And the biker is going at a velocity of v equal to 5 feet per second.

00:19 The biker and the bike have a mass or a weight equal to 135 pounds and then going up a 3 % slope.

00:27 In situation 2, we have another biker going down that same 3 % slope at a velocity v.

00:34 That v is equal to 20 feet per second, and they have a total mass or weight of 198 pounds.

00:43 What we're asked to solve for is the power that is applied by the biker in situation 1 and the power that's dissipated in situation 2 by the brakes.

00:52 So to find the power developed by the biker in situation one, we draw a free body diagram.

01:01 And what we have is we have the biker.

01:05 We have the normal force from the ground.

01:08 We have the force applied by the biker.

01:12 And then we have the weight force that is due to gravity.

01:18 And what we're going to see is we're also going to come up with our convention for our direction.

01:25 So that's positive x and this is positive y.

01:30 And as we see that when we resolve our forces, that the force of the biker is actually overcoming the force of w -sign theta.

01:42 And theta is the angle between the horizon and the slope itself.

01:49 Remember that we're given that the slope is a 3 % slope...