Thomas Li

Mar 05, 2022

Figure Skating - Uniform Circular Motion & Mechanical Energy

Not only motion and forces when moving across the ice, but I also want to investigate Yuzu’s jump, which is an important technique and skill in figure skating. A jump requires the athletes to jump as high as possible and spin a certain number of rotations in the air. The athletes also need to choose what types of jump they want to do, knowing that different jumps have different difficulty levels. The type and quality of the jumps are the main sources of technical score that is used to rank the athletes. Hence, I now put my focus into Yuzu’s free skating, where the most jumps take place.

Jumping also involves lots of physics! When you jump on ice, you have to take gravity and its force into consideration. Since you are spinning, uniform circular motion plays a big role in it. In my question, I am also asking about energy because I noticed the change in mechanical energy (kinetic and gravitational) as Yuzu completes his jump. The concepts that are involved and their definitions are listed below.

Uniform Circular Motion: the motion of an object in a circle at a constant speed.

Tangential Velocity: the component of motion along the edge of a circle measured at any arbitrary instant. Yuzu’s velocity when spinning in the air is tangential velocity.

Conservation of Mechanical Energy: the total mechanical energy in a system (i.e., the sum of the potential plus kinetic energies) remains constant as long as the only forces acting are conservative forces. Yuzu is not experiencing outside work and therefore conserves his energy.

Conservative Force: a force with the property that the total work done in moving a particle between two points is independent of the path taken. Yuzu doesn’t touch the ice when he’s in the air, thus it’s a conservative force.

Kinetic Energy: a form of energy that an object or a particle has by reason of its motion. Yuzu is moving, meaning he has kinetic energy.

Gravitational Potential Energy: the potential energy a massive object has in relation to another massive object due to gravity. Yuzu jumps up and then falls down, meaning his gravitational potential energy is changing.

→ Tangential Velocity Equation

- v is Yuzu’s tangential velocity when spinning

→ Conservation of Energy Equation

- Eki is the initial kinetic energy

- Egi is the initial gravitational energy

- Ekf is the final kinetic energy

- Egf is the final gravitational energy

→ Kinetic Energy Equation

- Ek is the kinetic energy

- m is Yuzu’s mass

- v is Yuzu’s velocity (vertical since this question involves vertical jumping)

→ Gravitational Energy Equation

- Eg is the gravitational energy

- m is Yuzu’s mass

- g is the gravitational constant 9.8 m/s2

- h is the height of Yuzu’s jump

Assumption: Ignore air resistance!

Thomas is happy that he found the mass of Yuzu! Now he wants to solve more things! In Yuzu’s free skating, he successfully completes a triple Lutz jump (3Lz) by spinning 3 rotations in the air for 0.64 seconds in total. Assume this jump is in uniform circular motion, and air resistance is ignored.

a) The radius of Yuzu’s rotation is 17.5 cm. What is his tangential velocity in this jump?

b) Yuzu reaches a maximum height of 0.52 metres in this jump. What is his vertical landing velocity?

To solve part a, we need to first determine the given and required information, and draw out a diagram to model Yuzu’s spin in the air (since no forces are involved, no FBD is needed).

Now, let’s start solving!

As you can see, I used the tangential velocity equation and plugged in values for r and T. Finally, I got 5.16 m/s for tangential velocity and wrote a therefore statement.

Now, let’s solve part b!

I drew out a diagram to model Yuzu’s jump across both horizontal and vertical directions. But since the question is only asking for the final vertical velocity, I only care about vertical jumping distance. I set the highest point of Yuzu’s jump to be my initial point and his landing point to be the final point for easier calculations.

As you can see, I set up the equation Emi = Emi, then expanded it to result in 4 terms. I cancelled Eki and Egf because both vi and hf are zero.

I could cancel out m after further expanding the equation. I isolated for vf using this expanded equation.

Finally, I plugged in all the values and got vf = 3.19 m/s. The direction is down because Yuzu is moving down when landing on the ice.