(III) A cyclist intends to cycle up a 7.50^∘hill whose vertical height is 125 m. The pedals turn

(III) A cyclist intends to cycle up a 7.50^∘hill whose...

Key Concepts

Gravitational Potential Energy

This concept refers to the energy an object gains due to its position in a gravitational field. It is calculated by multiplying the object's mass by the gravitational acceleration and the vertical height raised (mgh). It is a key part of energy conservation and is used to determine the work required to elevate an object against gravity.

Work

Work is defined as the process of energy transfer when a force is applied over a distance. In the context of mechanical systems, work can be calculated by multiplying an appropriate force by the distance over which it acts. This concept connects the energy required to perform a task, such as moving an object up an incline, with the force that must be exerted.

Relationship Between Rotational and Translational Motion

This concept deals with how rotational movements, such as a pedal’s revolution, translate into linear displacement. It involves using the circumference of a circular path to relate the number of rotations to the distance covered. Understanding this relationship is crucial for converting the work done in a rotational motion, where forces are applied tangentially, into linear motion.

Transcript

00:02 All right, guys.

00:04 So today we will be solving problem 45 from section 6 of chapter 6 of physics principles with applications.

00:14 This problem asks us about the work done against gravity and the tangential force applied on a set of pedals on a bicycle.

00:26 So i'm going to start by setting up the problem that was given.

00:30 And the way i'm going to do that is draw the hill as a triangle, right? so let me go ahead and draw this hill right here.

00:44 We've got a theta, which is the incline angle that they have given us.

00:51 We've got, i'm going to mark this as h2 and mark this as h1.

00:55 And we know that h2 minus h1 is equal to 125.

01:03 So i'm going to go ahead and write down some of the values that they have given us, that the problem has given us to start out, which will help us out as we move along to solve parts a and b.

01:14 So we know that the change in height is 125 meters.

01:18 We know that the petals turn in a circle that has a diameter of 36, centimeters, which is equal to 0 .36 meters.

01:33 And i'm changing it all to meters to keep it easy as we move along.

01:37 We know that the mass of the cyclist and the bicycle is equal to 75 kilograms.

01:47 And i'm going to mark that down as m.

01:50 And a couple other things.

01:52 We know that theta is equal to 7 .5 degrees.

01:57 And later on, we know that the distance traveled in one revolution of the pedals on the bicycle.

02:07 So i'm going to mark that as x -rev is equal to 5 .1 meters.

02:12 So let's go ahead and start part a of this problem.

02:18 So we know that part a is looking for the work done against gravity.

02:24 And i'm going to denote that as w sub -g.

02:27 And we know that the work done against gravity is equal to the change in potential energy, which i'm going to mark as delta pe.

02:36 Now we know that the change in potential energy is equal to the potential energy at height 2, right? so i'm going to mark that as p subh2 minus p subh1, right? so the formula for potential energy in this case is going to be mass times the acceleration due to gravity multiplied by the height at that location.

03:12 Now we know that this, you know, if we go ahead and just transfer this formula down right here, right? we're going to get mgh2 minus mgh1.

03:24 So we can go ahead and factor out the mgs.

03:31 And now you'll notice that we know all of these values.

03:36 We know what m is equal to.

03:38 We know what g is equal to.

03:40 And we know that h2 minus h1 was given in the problem as well.

03:44 So i'm going to go ahead and say that once we substitute 75 kilograms, for mass and we know that once we substitute that and i'm gonna go ahead and actually just write this down all together so now we know that the work done against gravity is equal to mg times the change in height right and we know that this is equal to 75 kilograms for the mass and 9 .8 meters per second squared which is the acceleration due to gravity and we know that the change in height is 125 meters.

04:27 Now if we multiply all of these things out, then what we should get is 91 ,000 875 joules.

04:40 Now because of the sig figs that were given, so we were only given three sig figs in this value right here, which is the i'm going to go ahead and say that the work done to gravity is equal to 9 .19 times 10 to the 4th joules.

05:02 Okay.

05:04 Now let's move on to part b.

05:07 For part b we need to calculate the tangential force that are applied on the pedals.

05:13 All right, so for part b, we know that the work done on the pedals, which i'm going to denote as work or w subp, is equal to the change in potential energy during one revolution, which i'm going to mark as pe sub rev.

05:32 Right...

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