# Particle on inclined plane whose steepness is increasing

In summary, the problem involves determining the motion of a particle on a raised plane using Newtonian Mechanics. The equation of motion in polar coordinates is ##F_r = m \ddot{r} -mr\dot{\theta}^2,## and in order to solve it, the component of gravity directed down the plane must be taken into account, as well as the additional term in the radial acceleration due to varying coordinates. The same result can also be obtained using Lagrangian Mechanics.
Missing homework template due to originally being posted in other forum.
I am trying to solve a problem using Newtonian Mechanics. I was able to solve it using Lagrangian Mechanics, and got the correct answer from the textbook, but when I use Newtonian I get the wrong answer. I want to be able to do it both ways in order to understand/appreciate the difference between Newtonian and Lagrangian.

The problem
A particle of mass m rests on a smooth plane. The plane is raised to an inclination angle θ at a constant rate α (θ=0 at t=0), causing the particle to move down the plane. Determine the motion of the particle.

My attempt at a solution
If we designate the starting position of the particle to be r0 (distance from the axis around which the plane rises), and r(t) to be the distance traveled by the particle down the ramp at a given time t, then r(t)=∫r'(t)dt from 0→t, where r'(t)=∫r''(t)dt from 0→t.
Now r''(t) is just the component of gravity directed down the ramp, so r''(t)=g*sin(αt).
∴ r'(t)=∫g*sin(αt)dt=g/α (1-cos(αt)). Then r(t)=∫r'(t)dt=g/α∫(1-cos(αt))dt=g/α (t - 1/α sin(αt))

If you do it using Lagrangian mechanics, you get r0(1-cosh(αt))-(g/2α^2)*(sin(αt)-sinh(αt)).

Any insight on logical errors I have made with my Newtonian approach would be greatly appreciated. Thanks!

The equation of motion in polar coordinates in the radial direction is ##F_r = m \ddot{r} -mr\dot{\theta}^2.##

Last edited:
Dazed&Confused said:
The equation of motion in polar coordinates in the radial direction is ##F_r = m \ddot{r} -mr\dot{\theta}^2.##
Hi Dazed&Confused, thank you for your reply. Can you explain to me how you came up with that and why my assumption was incorrect? I thought F_r was just the component of gravity directed down the plane at a given time, so mg*sin(αt)

Hi Dazed&Confused, thank you for your reply. Can you explain to me how you came up with that and why my assumption was incorrect? I thought F_r was just the component of gravity directed down the plane at a given time, so mg*sin(αt)
Your assumption was that ar=r'' which is not true. As Dazed&Confused pointed out, there is another term in the radial acceleration (what people call centripetal acceleration).

In cartesian coordinates, the unit vectors are constant in time, so you can just differentiate twice to get ax=x''
But with polar coordinates, you are using coordinates which vary in time (if θ varies in time) and so you get additional terms.

If you look up 'acceleration in polar coordinates' then I'm sure there are many derivations of the full acceleration formula.
(Or you can find it yourself, but first you have to figure out ##\frac{d}{dt}\hat r## and ##\frac{d}{dt}\hat \theta## which are not zero unless dθ/dt is zero)

You can also derive this formula with the Lagrangian approach. With the Lagrangian $$\frac12m(\dot{r}^2 + r^2\alpha^2) -mgr\sin\alpha t$$ you will find the same formula with ##F_r = -mg\sin\alpha t.##

## 1. What is a particle on an inclined plane?

A particle on an inclined plane is a physical system in which a small object, called a particle, is placed on a ramp or slope that is inclined at an angle to the horizontal surface. The particle is then allowed to move and is affected by the forces of gravity and the slope of the plane.

## 2. How does the steepness of the inclined plane affect the particle's motion?

The steepness, or angle, of the inclined plane has a direct impact on the motion of the particle. As the angle increases, the force of gravity acting on the particle becomes greater, causing the particle to accelerate and move faster down the plane. On the other hand, a shallower angle will result in slower and less accelerated motion.

## 3. What is the relationship between the angle of the inclined plane and the force of gravity?

The angle of the inclined plane is directly proportional to the force of gravity acting on the particle. This means that as the angle increases, the force of gravity also increases, resulting in a stronger pull on the particle.

## 4. How does the mass of the particle affect its motion on the inclined plane?

The mass of the particle does not have a direct effect on its motion on the inclined plane. However, a heavier particle will experience a greater force of gravity and therefore may accelerate more quickly down the plane compared to a lighter particle.

## 5. What factors can affect the motion of a particle on an inclined plane?

Aside from the angle and mass of the particle, other factors that can affect its motion on an inclined plane include the coefficient of friction between the particle and the surface of the plane, the shape and size of the particle, and any external forces acting on the particle such as air resistance or applied forces.

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