You and your bicycle have combined mass 80.0 kg. When you reach the base of a bridge, you are t

You and your bicycle have combined mass 80.0 kg. When you...

You and your bicycle have combined mass $80.0 \mathrm{~kg}$. When you reach the base of a bridge, you are traveling along the road at $5.00 \mathrm{~m} / \mathrm{s}$ (Fig. $\mathrm{P} 6.74$ ). At the top of the bridge, you have climbed a vertical distance of $5.20 \mathrm{~m}$ and slowed to $1.50 \mathrm{~m} / \mathrm{s}$. Ignore work done by friction and any inefficiency in the bike or your legs. (a) What is the total work done on you and your bicycle when you go from the base to the top of the bridge? (b) How much work have you done with the force you apply to the pedals?

You and your bicycle have combined mass $80.0 \mathrm{~kg}$. When you reach the base of a bridge, you are traveling along the road at $5.00 \mathrm{~m} / \mathrm{s}$ (Fig. $\mathrm{P} 6.74$ ). At the top of the bridge, you have climbed a vertical distance of $5.20 \mathrm{~m}$ and slowed to $1.50 \mathrm{~m} / \mathrm{s}$. Ignore work done by friction and any inefficiency in the bike or your legs.
(a) What is the total work done on you and your bicycle when you go from the base to the top of the bridge?
(b) How much work have you done with the force you apply to the pedals?

Key Concepts

Applied Work versus Net Work

This concept differentiates between the total work done on an object (net work) and the work provided by a specific force. In system analysis, net work considers the sum of all forces acting (including conservative forces like gravity), while the work applied (such as by the rider at the pedals) is usually calculated based on the energy input necessary to achieve the observed changes in the system.

Energy Conservation

The principle of energy conservation asserts that in an isolated system, energy can neither be created nor destroyed; it only transforms from one form to another. In this scenario, ignoring friction, the work done by one force, such as the applied force at the pedals, is manifested as changes in both kinetic and potential energy.

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object due to its position in a gravitational field. In the context of this problem, the change in gravitational potential energy is calculated based on the vertical displacement of the object, reflecting the energy required to elevate the object against gravity.

Work–Energy Theorem

This theorem states that the net work done on an object is equal to the change in its kinetic energy. In problems like this, it is used to relate the work acting on the system to the modification of the object's speed, taking the difference between the initial and final kinetic energies.

Kinetic Energy

Kinetic energy is the energy of motion and is given by one?half the product of mass and the square of the velocity. It is a key factor in determining the energy state of an object at any point and is directly involved in the work–energy calculations when an object accelerates or decelerates.

Transcript

00:01 Now in this question we are given that you and your bicycle have a combined mass of 80 kilogram okay, and here this figure is also given now when you reach at the base of the bridge you are moving with a speed of 5 .0 meter per second, okay, and when you reach to the top of the bridge your speed is slowed down to 1 .50 meter per second and the vertical height gained by you is 5 .20 meter from the reference.

00:37 This is a reference we have taken now there are two parts in this question that we need to find in the a part we need to find that what is the total work done on you and your bicycle when you move from base to the top of? the bridge and we can ignore the friction and any inefficiencies in the bike or on your legs okay, so to find the total work done we can use the most powerful theorem in physics the work energy theorem the total work done is equals to the change in its kinetic energy total work done on the system here system is you and your bicycle.

01:13 So total work done is equals to the change in your kinetic energy this thing is what we can write from the work energy theorem so the total work done would be change in kinetic energy is half m the final speed square minus the initial speed square, right? so 1 over 2 mass is given to be 80 vf square is this is the final speed 1 .5 and this is the initial speed, right? so 1 .5 square minus 5 .0 square now simplify this by using your calculator.

01:52 You'll get minus 910 joules so there's a total work done on you and the bicycle now next and the b part we need to find that how much work you have done with the force you apply to the pedals okay so the total work done is what because we don't have to consider friction total work done would be the work done by the person plus the work done by the gravity total work done we have already calculated now if we are able to calculate this work done by gravity we can tell that what is the work done you have done the work done by the person so the gravity work done first, let us calculate the force of gravity this is mass multiplied with gravity mg.

02:39 So this is a force which is always acting downward and you are moving upward the displacement is in the upward direction.

02:47 So this angle is what? angle is 180 degree.

02:51 So work done by the gravity would be equals to the force multiplied with the displacement times cosine theta so fg we have m into g...

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