Euler's method for a mass sliding down a frictionless curve

[murrskeez]

Homework Statement

Consider a mass sliding down a frictionless curve in the shape of a quarter circle of radius
2.00 m as in the diagram. Assume it starts from rest. Use Euler’s method to approximate
both the time it takes to reach the bottom of the curve and its speed at the bottom. Hint: Define the position and acceleration off the mass in terms of the angle θ. Let Δt=0.2.

2. Homework Equations
s=rθ
t_n=t_n-1+Δt
x_n=x_n-1+v_n-1Δt
v_n=v_n-1+a_n-1Δt
a_n=ΣF(t_n)/m

The Attempt at a Solution

x(θ)=2θ
a(θ)=9.8cos(x(θ))

I know i need to find a way to make it so the angle is changing in small increments as I add the numbers up, but I'm not sure how.

 

[ehild]

Homework Statement

Consider a mass sliding down a frictionless curve in the shape of a quarter circle of radius
2.00 m as in the diagram. Assume it starts from rest. Use Euler’s method to approximate
both the time it takes to reach the bottom of the curve and its speed at the bottom. Hint: Define the position and acceleration off the mass in terms of the angle θ. Let Δt=0.2.
View attachment 76208

2. Homework Equations
s=rθ
t_n=t_n-1+Δt
x_n=x_n-1+v_n-1Δt
v_n=v_n-1+a_n-1Δt
a_n=ΣF(t_n)/m

The Attempt at a Solution

x(θ)=2θ
a(θ)=9.8cos(x(θ))

I know i need to find a way to make it so the angle is changing in small increments as I add the numbers up, but I'm not sure how.

The equation in red is wrong. The tangential acceleration is a(θ)=9.8cos(θ)
Does only gravity acting on the mass? What about normal force?
The objects moves along a circle. It has both radial and tangential acceleration. What is the radial acceleration in case of circular motion? What is the sum of the radial components of the applied forces?
a(θ) = R d²θ/dt². If you apply Newton's method, you get the angular velocity at the next step. But you need theta, so you need the other equation that relates the angular velocity with theta.

 

[murrskeez]

The equation in red is wrong. The tangential acceleration is a(θ)=9.8cos(θ)
Does only gravity acting on the mass? What about normal force?
The objects moves along a circle. It has both radial and tangential acceleration. What is the radial acceleration in case of circular motion? What is the sum of the radial components of the applied forces?
a(θ) = R d²θ/dt². If you apply Newton's method, you get the angular velocity at the next step. But you need theta, so you need the other equation that relates the angular velocity with theta.

Ok so for the sum of the radial components of the applied forces I am getting:
ΣF_r=m(v²/R)
N-mgsin(θ)=m(v²/R)
I'm not sure where to go from here though.

 

[ehild]

Ok so for the sum of the radial components of the applied forces I am getting:
ΣF_r=m(v²/R)
N-mgsin(θ)=m(v²/R)
I'm not sure where to go from here though.

You are right, it is not too useful, as you do not know N. I've just noticed that you wanted to find the the position (x(r) = θ r) and the speed at each subsequent step. Do that, show the first step.

 

[murrskeez]

You are right, it is not too useful, as you do not know N. I've just noticed that you wanted to find the the position (x(r) = θ r) and the speed at each subsequent step. Do that, show the first step.

V₀=0
V₁=0+9.8*0.2=1.96

X₀=0
X₁=0+0(0.2)=0
X₂=0+1.96(0.2)=.392

 

[ehild]

V₀=0
V₁=0+9.8*0.2=1.96

X₀=0
X₁=0+0(0.2)=0
X₂=0+1.96(0.2)=.392

OK, go ahead. What is v2?
Can you write a program, or use an Excel spreadsheet?

 

[ehild]

The problem would be much simpler to solve if you used conservation of energy. You can express v at a given theta as function of theta, and then you have the equation Rdθ/dt = f(θ), which you solve with the Euler method.

 

[murrskeez]

V₂=1.96+9.8(0.2)=3.92
V₃=3.92+9.8*cos(X₃/2)=5.842473
I can use excel to finish it off if I have the right pattern.

 

[ehild]

You can write the n-th step don't you? And you know v(0) and x(0).