A piece of ice slides from rest down a rough 33.0^∘ incline in twice the time it takes to slide

A piece of ice slides from rest down a rough 33.0^∘ incline...

Key Concepts

Coefficient of Kinetic Friction

The coefficient of kinetic friction is a dimensionless constant that quantifies the frictional resistance between moving surfaces. It is calculated by comparing the frictional force to the normal force. Determining its value is critical in problems where friction affects the motion, as it directly influences the acceleration and, consequently, the time of travel along the incline.

Frictional Forces on Inclined Planes

On an inclined surface, friction acts opposite to the direction of motion and depends on the normal force and the coefficient of kinetic friction. This force reduces the net acceleration along the incline when compared with a frictionless surface. Recognizing how friction interacts with gravitational components is key to solving problems involving motion on slopes.

Comparative Analysis of Frictionless and Frictional Motion

This concept involves comparing the motion of an object down an incline when friction is absent versus when friction is present. By analyzing how friction reduces acceleration and increases travel time, one can derive relationships between the temporal measures in both cases. This comparative analysis is vital in setting up equations that eventually lead to the determination of the coefficient of kinetic friction.

Newton's Second Law on an Incline

This principle relates the net force acting on an object to its mass and acceleration. On an inclined plane, the weight of the object is decomposed into components parallel and perpendicular to the plane, which helps in determining the net force acting along the incline. This concept is essential for analyzing how friction alters the acceleration compared to a frictionless scenario.

Kinematic Equations for Constant Acceleration

The kinematic equations relate displacement, initial velocity, time, and acceleration when acceleration is constant. They are used to calculate the time taken for an object to travel a certain distance along the incline, both in the presence and absence of friction. Understanding these equations is fundamental for comparing motion under different conditions.

Transcript

00:01 So we will be looking at the problem 21 from chapter 5 of the physics fifth book.

00:08 The question says a piece of ice slides down from rest down a rough 33 .0 degree incline in twice the time it takes to slide down a frictionless 33 .0 incline of the same length.

00:24 Find the coefficient of kinetic friction between the ice and the rough incline.

00:28 So this is a bit of a different style of question to what we've had before.

00:33 It involves a bit more algebra.

00:35 So we're still going to have exactly the same setup.

00:38 We have our weight mg and we have our component in this direction perpendicular is mg cos 33.

00:47 Our component parallel to the slope is mg side 33 and we have our n and f in the opposite directions to those two.

00:55 And obviously our angle is 33 degrees as shown on this slope.

01:00 Now, so this is wrong and i will mark this as, we're going to mark our two things as, we're going to mark them as mu v1 and mu v2.

01:16 Now mu v1 is the mu when we're sliding down a friction in the slope.

01:24 So in that case, a mu for a frictionless slope is zero.

01:30 And we want to find out what mu v2 is.

01:34 So the first thing to do is to try and relate to the accelerations, so a1 and a2 from each of these.

01:41 So we need to find some equation where a1 is equal to x, a2.

01:47 And the only way to do this is, of course, finding the accelerations.

01:51 So we're going to use our suvat equations.

01:53 S1 is equal to ut plus a half a t a 1t squared.

02:00 So here t is equal to t and here s2 is equal to ut plus a half a t times 40 squared here t is obviously two t it takes twice as long to get down the slope so therefore we have 2t squared being equal to 4t squared.

02:26 2t equals squared is equal to 4 squared.

02:28 And we also know that it took twice the time to slide down the exact same distance.

02:33 So s1 is equal to s2.

02:35 So we get obviously u is zero because they both start at zero velocity here.

02:43 So we have a half times a1 times t squared is equal to so this is equal to this, a half times a2 times 4t squared, which is 2 times a2 times t squared.

03:00 So we get that a1 is equal to 4 times a2.

03:04 Now to calculate the coefficient of kinetic friction in each case, we do, as we always do, the f, which is, well, we do the mg sign angle, minus the mass times the acceleration of the object...

Please give Ace some feedback