You and your bicycle have combined mass 80.0kg.When you reach the base of the bridge, you are
travelling along the road at 5.00m/s. (See figure below). At the top of the bridge, you have climbed a
vertical distance of 5.20m and have slowed to 1.50m/s. You can ignore work done by friction and any
efficiency in the bike or your legs.
(a) What is the total work done on you and your bicycle when you go from base to the top of
the bridge? (3 mks)
(b) How much work have you done with the force you apply to the pedals?(6 mks)
(c) Does this imply that energy is conserved? Briefly explain with respect to your findings above
(a) Well, let's calculate the work done on you and your bicycle. Work is equal to force multiplied by distance. In this case, the force is equal to the combined mass of you and your bicycle multiplied by the acceleration due to gravity (9.8 m/s²) since it's against gravity. The distance is the vertical distance you climbed, which is 5.20m.
So, the work done on you and your bicycle is:
Work = Force x Distance
Work = (80.0kg) x (9.8 m/s²) x (5.20m)
Work = 4038.4 J
Therefore, the total work done on you and your bicycle when you go from the base to the top of the bridge is 4038.4 Joules.
(b) Now let's calculate the work you have done with the force you apply to the pedals. Since the force you apply is in the horizontal direction, and the distance is also in the horizontal direction, we can use the formula:
Work = Force x Distance
Work = Force applied x Distance traveled horizontally
Since there is no friction or efficiency involved, the work you do is simply equal to the force you apply to the pedals multiplied by the distance traveled horizontally.
Unfortunately, we don't have enough information about the force you apply or the exact distance traveled horizontally to calculate this work. So, let's move on to the next part.
(c) Based on our findings above, we can see that the total work done on you and your bicycle is not equal to the work you have done with the force you apply to the pedals. This implies that energy is not conserved in this case.
Energy conservation states that the total energy of an isolated system remains constant if no external forces or work are involved. In this scenario, there are external forces involved such as gravity and the force applied to the pedals, which result in a difference in the work done. Hence, energy is not conserved.
Remember, these calculations are not meant to clown around, but to provide a better understanding of the concepts involved.
(a) The total work done on you and your bicycle can be calculated using the work-energy theorem. The work done on an object is equal to the change in its kinetic energy. The change in kinetic energy can be calculated as:
ΔK = 1/2 * m * (vf^2 - vi^2)
Where ΔK is the change in kinetic energy, m is the mass of the bicycle and you, vi is the initial velocity, and vf is the final velocity.
Given:
m = 80.0 kg
vi = 5.00 m/s
vf = 1.50 m/s
ΔK = 1/2 * 80.0 kg * (1.50 m/s)^2 - 1/2 * 80.0 kg * (5.00 m/s)^2
ΔK = 1/2 * 80.0 kg * (2.25 m^2/s^2) - 1/2 * 80.0 kg * (25.0 m^2/s^2)
ΔK = 1/2 * 80.0 kg * (-22.75 m^2/s^2)
ΔK = -918 J
The total work done on you and your bicycle is -918 J.
(b) The work you do with the force you apply to the pedals can be calculated using the equation:
W = F * d * cos(θ)
Where W is the work done, F is the force applied, d is the displacement, and θ is the angle between the force and the displacement.
Since the force applied and the displacement are in the same direction, the angle between them is 0 degrees, and the cosine of 0 degrees is 1.
Therefore, W = F * d
To calculate the work done with the force you apply to the pedals, we need to know the force applied and the distance traveled.
(c) The fact that the work done on you and your bicycle is negative (-918 J) indicates that energy is lost during the ascent of the bridge. This loss of energy could be due to various factors such as air resistance, friction, or inefficiency in converting muscle energy to mechanical energy. Therefore, in this scenario, energy is not conserved.
It is important to note that energy conservation is a fundamental principle in physics, but in real-world situations, there are often external factors that can cause energy losses.
To find the answers to these questions, we need to understand the concept of work and energy.
(a) The total work done on you and your bicycle when you go from the base to the top of the bridge can be calculated using the work-energy principle. The principle states that the work done on an object is equal to the change in its kinetic energy.
Since there is no work done by friction or any other external force, the only work done on you and your bicycle is done against gravity. The work done against gravity is equal to the gravitational potential energy gained. The equation for gravitational potential energy is given by:
Potential energy = mass * gravitational acceleration * height
In this case, the mass of you and your bicycle is 80.0 kg, the acceleration due to gravity is approximately 9.8 m/s^2, and the height climbed is 5.20 m.
Therefore, the total work done on you and your bicycle is:
Work = mass * gravitational acceleration * height
Work = 80.0 kg * 9.8 m/s^2 * 5.20 m
Solving this equation will give you the answer in joules (J).
(b) The work done with the force you apply to the pedals can be calculated using the simple work formula:
Work = force * distance * cos(theta)
In this case, the force you apply to the pedals is the force required to overcome any resistance (such as air resistance or friction) and propel you and your bicycle. The distance is the distance traveled while applying this force, and theta is the angle between the force applied and the direction of motion.
The problem doesn't provide any specific values for force, distance, or angle, so you would need those values to calculate the work done.
(c) The conservation of energy states that energy cannot be created or destroyed; it can only change forms. In this situation, energy is conserved because the total work done on you and your bicycle (as calculated in part a) is equal to the work done with the force you apply to the pedals (as calculated in part b).
Briefly put, the energy gained in climbing the bridge is equal to the energy supplied by the force applied to the pedals. This shows that energy is conserved in this scenario.