Kinetics of a Particle: Work and Energy

[Suitengu]

Homework Statement

The cyclist travels to a point A, pedaling until he reaches a speed Va = 4m/s. He then coasts up freely the curved surface. Determine how high he reaches up the surface before he comes to a stop. Also, what are the resultant normal force on the surface at this point and his acceleration? The total mass of the bike and man is 75 kg. Neglect friction, the mass of the wheels and the size of the bicycle.

Homework Equations

a is 4m from diagram.

The Attempt at a Solution

I found y first using the work-energy theorem and solve for x using the equation of the path given. I was then stuck at calculating the angle at this coordinate so I sought out help. The individual told me to take the square root of y/x and take the arctan of that. To my surprise, I worked with this value using normal and tangent coordinates and it worked. I would like if someone could explain to me why it was done like this as I am still lost. Thanks in advance.

 

[KataKoniK]

Dear forum post author,

Thank you for sharing your problem with us. After reviewing your attempt at a solution, I believe the individual who helped you may have made a mistake. The square root of y/x is not the correct way to calculate the angle in this scenario.

To determine the angle, we can use the inverse tangent function, also known as arctan, which takes the ratio of the opposite and adjacent sides of a right triangle. In this case, the opposite side is the change in height (y) and the adjacent side is the distance traveled (x). So, the correct way to calculate the angle would be to take the inverse tangent of y/x, not the square root.

I hope this helps clarify the issue for you. If you have any further questions or need any additional assistance, please don't hesitate to reach out. it is important to always double check our calculations and seek out help when needed. Keep up the good work!

 

[piercas]

I would like to clarify some of the concepts and equations used in this problem.

First, the work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done by the cyclist in pedaling is equal to the change in his kinetic energy, which is given as Va = 4m/s. However, the cyclist is not pedaling while coasting up the curved surface, so there is no work done and no change in kinetic energy.

To determine the height the cyclist reaches before coming to a stop, we can use the conservation of energy principle. The total energy of the system (cyclist and bike) at the top of the curve is equal to the sum of its kinetic and potential energy. Since the cyclist is at rest at the top of the curve, its kinetic energy is zero. The potential energy at this point is equal to the height reached (h) multiplied by the mass of the system (75 kg) and the acceleration due to gravity (9.8 m/s^2). Therefore, we can write the equation:

Potential energy = mgh = 0

Solving for h, we get h= 0 m. This means that the cyclist does not reach any height before coming to a stop.

Next, let's look at the resultant normal force on the surface at this point. The normal force is the force that the surface exerts on the cyclist in a direction perpendicular to the surface. In this case, since the cyclist is at rest, the normal force is equal in magnitude and opposite in direction to the force of gravity acting on the cyclist. This is given by the equation:

Normal force = mg = (75 kg)(9.8 m/s^2) = 735 N

Finally, the acceleration of the cyclist is also zero at this point since he is at rest.

In summary, the cyclist does not reach any height before coming to a stop, the resultant normal force on the surface is 735 N, and the acceleration is 0 m/s^2. It is important to note that these values are only true if we neglect friction and the mass of the wheels and size of the bicycle, as stated in the problem. If these factors were taken into account, the values would be different.

As for the suggestion to take the square root of y/x and take the arctan of that, it