# How Do You Derive the Constraint Equation for a Disc Rolling Along a Parabola?

• QuantumDuality
In summary: Also:##ds = \sqrt{(dx)^2 + (dy)^2} = \sqrt{1 + 4a^2 x^2} dx (2)##The Attempt at a SolutionI think this equation is saying that the distance from the contact point to the center of the disc is the same as the distance from the center of the disc to the point where the x-axis and y-axis intersect.
QuantumDuality

## Homework Statement

A disc of radius R rolls without slipping along the parabola y= ax2. Obtain the constrain equation

## Homework Equations

Because there's no slipping, then:
##R d \theta = ds (1)##
Where ##\theta ## is the angle between the line from the center of the disc to a fixed point and the line from the center of the disc to the contract point with the parabola
Also:
##ds = \sqrt{(dx)^2 + (dy)^2} = \sqrt{1 + 4a^2 x^2} dx (2)##

## The Attempt at a Solution

So I just have to equate (1) and (2) to get the constraint equation?, Because I have seen a generalization of the constraint equation for an arbitrary curve that use

##ds = R(d\theta + d\phi) ##

Where ##\theta## is an angle between a radius to a fixed point and the radius parallel to the y axis, while ##\phi## is an angle between the radius parallel to the y-axis and the radius to the contact point
Here's the generalization:

https://campus.mst.edu/physics/courses/409/Problem-Solutions/HW#4/HW4_prob3_ hoop on curve.pdf

Your problem statement says, "... rolls along ... the parabola..." Is the disk in the plane of the parabola, perhaps normal to the plane of the parabola, or something else again?

QuantumDuality said:
Because there's no slipping, then:
##R d \theta = ds (1)##
Where ##\theta ## is the angle between the line from the center of the disc to a fixed point and the line from the center of the disc to the contract point with the parabola
Took me a while to realize you meant a point fixed on the perimeter of the disc.
QuantumDuality said:
Where θ is an angle between a radius to a fixed point and the radius parallel to the y axis, while ϕ is an angle between the radius parallel to the y-axis and the radius to the contact point
I think if you draw a diagram you will find that your θ equals their θ+φ+constant.

QuantumDuality

## 1. How does a disc go along a parabola?

A disc goes along a parabola due to the forces acting on it, specifically gravity and the initial velocity given to the disc. When a disc is thrown at an angle, it follows a curved path known as a parabola due to the combination of its horizontal and vertical motions.

## 2. What factors affect the disc's path along a parabola?

The factors that affect the disc's path along a parabola include the initial velocity given to the disc, the angle at which it is thrown, and external forces such as air resistance. These factors determine the shape and distance of the parabolic path.

## 3. Can a disc go along a perfect parabola?

In theory, a disc can go along a perfect parabola if there are no external forces acting on it and the initial velocity and angle are precisely calculated. However, in real-world situations, there are always external factors that may alter the disc's path and make it slightly deviate from a perfect parabola.

## 4. How is the disc's speed affected along a parabola?

The disc's speed along a parabola is not constant, as it is constantly changing due to the forces acting on it. The disc's speed is highest at the beginning when it is first thrown and gradually decreases as it moves along the curved path. The speed also depends on the initial velocity and angle at which the disc is thrown.

## 5. What is the importance of understanding a disc's path along a parabola?

Understanding a disc's path along a parabola is essential for various fields such as physics, sports, and engineering. It helps in predicting the trajectory of a thrown object and allows for accurate calculations and measurements. This understanding also helps in optimizing the performance of sports equipment, such as frisbees and discs, and in designing structures such as bridges and roller coasters.

Replies
1
Views
2K
Replies
2
Views
2K
Replies
28
Views
2K
Replies
8
Views
2K
Replies
2
Views
1K
Replies
13
Views
1K
Replies
60
Views
1K
Replies
13
Views
1K
Replies
17
Views
2K
Replies
7
Views
2K