When a dancer on ice spins faster by folding her arms, the principle at play is related to angular momentum. The correct answer to your question is (c) Constant angular momentum and increase in kinetic energy. Let’s break this down to understand why this is the case.
The Basics of Angular Momentum
Angular momentum is a physical quantity that represents the amount of rotation an object has. For a dancer, it depends on two key factors: the mass of the dancer and the distribution of that mass relative to the axis of rotation. Mathematically, it can be expressed as:
L = Iω
- L is the angular momentum.
- I is the moment of inertia, which depends on how far the mass is from the axis of rotation.
- ω is the angular velocity, or how fast the dancer is spinning.
How Folding Arms Affects Spinning
When the dancer folds her arms, she is effectively reducing her moment of inertia (I). This is because her mass is being pulled closer to the axis of rotation. According to the law of conservation of angular momentum, if no external torque acts on the system, the angular momentum must remain constant.
Applying Conservation of Angular Momentum
If we denote the initial angular momentum as L_initial and the final angular momentum as L_final, we have:
L_initial = L_final
Since the dancer’s arms are folded, her moment of inertia decreases, which means that to keep the angular momentum constant, her angular velocity (ω) must increase. This means she spins faster.
Linking to Kinetic Energy
Kinetic energy in rotational motion is given by the formula:
K.E. = 1/2 I ω²
As the dancer reduces her moment of inertia while increasing her angular velocity, the kinetic energy can actually increase. This is because the increase in ω (since it's squared in the kinetic energy formula) can outweigh the decrease in I, leading to an overall increase in kinetic energy during the spin.
Illustration with a Real-Life Example
Think of a figure skater performing a spin. When her arms are extended, she spins at a slower rate. However, as she brings her arms in, she tightens her body, reduces her radius, and spins much faster. The visible effect of her spinning faster is a direct result of the physics principles we discussed, specifically the conservation of angular momentum and the relationship between moment of inertia and angular velocity.
So, the answer (c) captures the essence of what's happening: the angular momentum remains constant while the kinetic energy increases due to the changes in the dancer’s body position during the spin. This illustrates beautifully how physical principles govern the movements we see in sports and artistic performances.
