An object is sliding down a ramp inclined at an angle of θradians and attains a terminal velocit

An object is sliding down a ramp inclined at an angle of...

An object is sliding down a ramp inclined at an angle of $\theta$ radians and attains a terminal velocity before reaching the bottom. Assume that the drag force caused by the air is proportional to $S v^{2}$, where $S$ is the cross-sectional area perpendicular to the direction of motion and $v$ is the speed. Further assume that the sliding friction between the object and the ramp is proportional to the normal weight of the object. Determine the relationship between the terminal velocity and the mass of the object. If two different boxes, weighing 600 and $800 \mathrm{lb}$, are pushed down the ramp, find the relationship between their terminal velocities.

Key Concepts

Frictional Force on an Inclined Plane

Friction on an inclined plane is directly proportional to the normal force (the component of weight perpendicular to the surface). This frictional force opposes the motion of the object and is central to the force balance that establishes the terminal velocity.

Drag Force (Quadratic Resistance)

The drag force is a resistive force exerted by air and, in many circumstances, is proportional to the square of the object's speed and its cross-sectional area. This quadratic relationship is particularly important at higher speeds and plays a key role in determining the terminal velocity of an object.

Mass Dependence in Terminal Velocity

For scenarios where friction is proportional to the normal weight and drag is proportional to the square of the speed, the terminal velocity is found to be proportional to the square root of the mass. This relationship implies that if two objects have the same geometric properties and are subject to the same conditions, the object with greater mass will have a higher terminal velocity, with their speeds scaling as the square root of their masses.

Force Equilibrium

Force equilibrium occurs when the forces acting on an object cancel each other, leaving a net force of zero. In the context of the problem, the gravitational force component down the incline is balanced by both frictional forces (proportional to the normal force) and drag forces (proportional to the square of the velocity).

Terminal Velocity

Terminal velocity is the constant speed that an object eventually reaches when the net accelerating force acting on it is zero. This happens when the sum of all forces—such as gravitational, frictional, and drag forces—balances out, resulting in no further acceleration.

Component Analysis on an Incline

When an object is on an inclined plane, its weight can be decomposed into two components: one parallel to the surface that drives motion down the plane, and one perpendicular to the surface that determines the normal force, which in turn influences friction.

Transcript

00:01 In this question, we have a block sliding down a frictionless ramp onto a different frictionless ramp.

00:07 And we want to know a couple things.

00:08 First, half far along the second ramp does the object slide before coming to a stop.

00:12 And on its way back down, what is the speed when the block is at seven meters off the ground? well, so the block is starting with gravitational potential energy, and it's ending with gravitational potential energy.

00:25 So we'll call that gpe prime.

00:28 So m .g .h has to equal m .g .h.

00:32 Prime and since the m gs go away h has to equal h prime so the height that it's going to reach on the new ramp is the same as the height that it left the first ramp with which is 12 meters off the ground or above the base of the ramp now this isn't how far it's set of the ramp what we want to know so we know that this side is 12 meters we're looking for x and we know that this angle in here is 37 degrees.

01:02 So we have opposite and hypotenuse here.

01:06 So we're going to use sine theta equals opposite over hypotenuse.

01:14 So we got sign of 37 degrees equals 12 meters over x.

01:27 So our length up the ramp, our x value here is going to be 19 .9 meters.

01:36 So we have part a.

01:41 So now take a look at part b.

01:42 Part b, we want to know what will be the speed when it comes back down to seven meters off the ground...