## Simulation experiments

Some suggested experiments for teachers to share with students. Aimed at A level / Cert HE / 1st year undergraduate students studying classical mechanics. Suitable for group work or for home study if the students have access to a Windows PC.

• EXPERIMENT 1: Demonstration that more work is done against friction on a u-shaped (convex downward) quadratic ramp than on a linear ramp.

Steepness: 1.50
Y-intercept: 3.00 m
Co-efficient of friction: 0.470
X0: 0.00 m
V0: 0.00 m/s

• EXPERIMENT 2: Demonstration that less work is done against friction on an n-shaped (concave downward) quadratic ramp than on a linear ramp.

Steepness: 2.00
Y-intercept: 3.00 m
Co-efficient of friction: 0.470
X0: 0.00 m
V0: 0.00 m/s

• EXPERIMENT 3: Demonstration that path length is not the important factor in determining the work done against friction.

Ramp shape: Cubic
Steepness: 0.75
Y-intercept: 1.90 m
Co-efficient of friction: 0.200
X0: 0.00 m
V0: 0.00 m/s

The work done against friction along this cubic ramp is equal to the work done against friction on an equivalent straight line ramp, showing that a longer path length does not necessarily mean more work being done against friction.

• EXPERIMENT 4: Demonstration that work done against friction depends on velocity (on a non-linear ramp).

Steepness: 0.50
Y-intercept: 1.00 m
Co-efficient of friction: 0.050
X0: 0.00 m

Vary the velocity as follows to see the effect it has on the work done against friction:

V0: 4.00 m/s
V0: 5.00 m/s
V0: 6.00 m/s
V0: 7.00 m/s

This experiment shows that on a concave downward parabolic ramp it is possible to do almost no work against friction if the particle begins with just the right initial velocity.

Additional observations: What happens if the initial velocity (V0) is 3.30 m/s or less? Can you explain why? What happens if the initial velocity (V0) is greater than 7.00 m/s? Can you explain why?

• EXPERIMENT 5: Demonstration that the work done against friction on a linear path is independent of vertical distance travelled.

Ramp shape: Linear
Co-efficient of friction: 0.200
X0: 0.00 m
V0: 0.00 m/s

Vary the Steepness and Y-intercept as follows to see the effect the gradient of the ramp has on the work done against friction:

Steepness: 1.00 / Y-intercept: 3.00 m
Steepness: 0.75 / Y-intercept: 2.25 m
Steepness: 0.50 / Y-intercept: 1.50 m
Steepness: 0.25 / Y-intercept: 0.75 m

Additional observations: What happens if you vary the initial velocity (V0)? Why does this differ to the result obtained in experiment 4? What happens if you start the particle at X0 = 3 m with an initial velocity (V0) of 10 m/s?

• EXPERIMENT 6: Comparing a reciprocal-shaped ramp and an exponential-shaped ramp.

The following two ramps both start and finish at the same x and y co-ordinates; hence in both cases the particle starts and finishes with the same gravitational potential energy. On which ramp do you think the particle will finish with the most kinetic energy? Why? What actually happens?

Ramp shape: Reciprocal
Steepness: 1.40
Y-intercept: 3.00 m
Co-efficient of friction: 0.287
X0: 0.00 m
V0: 0.00 m/s

Ramp shape: Exponential
Steepness: 1.35
Y-intercept: 3.00 m
Co-efficient of friction: 0.287
X0: 0.00 m
V0: 0.00 m/s

Additional observation: What happens if you increase the initial velocity (V0)? Can you explain this behaviour?

• EXPERIMENT 7: Looking at stable and unstable stationary points / potential wells.

Ramp shape: Quartic
Steepness: 0.70
Y-intercept: 2.00 m
Co-efficient of friction: 0.150
X0: 0.00 m

The particle stays on the ramp for velocities between 0.00 m/s and 3.87 m/s. It comes to rest at different points on the ramp depending on its initial velocity (V0). Investigate this behaviour, and relate your observations to the shape of the ramp. Where is the particle most likely to end up? Why?

• EXPERIMENT 8: The effect of varying dt in the numerical solution to the particle's equation of motion.

Ramp shape: Quartic
Steepness: 0.70
Y-intercept: 2.00 m
X0: 0.00 m
V0: 2.19 m/s
Co-efficient of friction: 0.150

dt is the tiny increment in time used in the numerical solution to the particle's equation of motion. The smaller the value, the closer the numerical approximation approaches the exact solution. Vary dt between 0.000050 s and 0.000100 s in steps of 0.000010 s What happens? Can you explain this behaviour?

• EXPERIMENT 9: Just for fun!

Can you get the particle to the other end of the ramp without it leaving the surface?! Vary the initial velocity (V0) to see if you can get the particle across the full length of the ramp.

Ramp shape: Sine wave
Steepness: 2.00
Y-intercept: 1.00 m
X0: 0.00 m
Co-efficient of friction: 0.015