If we multiply all the design dimensions of an object by a scaling factor f, its volume and mass will be multiplied by f^3. (b) If a 1/48 scale model has a rotational kinetic energy of 2.5 J, what will be the kinetic energy for the full-scale object of the same material rotating at the same angular velocity?

Verified step by step guidance

1. Identify the scaling factor between the model and the full-scale object. Since the model is a 1/48 scale, the scaling factor from the model to the full-scale object is 48.
2. Understand that when dimensions are scaled by a factor of f, the volume and hence the mass (assuming constant density) scale by a factor of f^3. Therefore, the mass of the full-scale object is 48^3 times the mass of the model.
3. Recall the formula for rotational kinetic energy, KE = 0.5 * I * \\omega^2, where I is the moment of inertia and \\omega is the angular velocity. The moment of inertia I is proportional to the mass of the object.
4. Since the mass of the full-scale object is 48^3 times the mass of the model, and the moment of inertia is directly proportional to the mass, the moment of inertia of the full-scale object is also 48^3 times that of the model.
5. Calculate the rotational kinetic energy of the full-scale object by multiplying the kinetic energy of the model by 48^3, because the kinetic energy is proportional to the moment of inertia, and both the angular velocity and the shape of the object remain constant.

Key Concepts

Scaling Laws

Scaling laws describe how physical properties of an object change when its dimensions are scaled by a factor. For example, if all linear dimensions of an object are multiplied by a factor 'f', its volume increases by 'f^3' and surface area by 'f^2'. This principle is crucial for understanding how properties like mass and energy relate to size.

Rotational Kinetic Energy

Rotational kinetic energy is the energy an object possesses due to its rotation, calculated using the formula KE = 1/2 I ω^2, where 'I' is the moment of inertia and 'ω' is the angular velocity. When scaling an object, its moment of inertia changes with the square of the scaling factor, affecting the kinetic energy proportionally.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion, depending on the mass distribution relative to the axis of rotation. For similar shapes, the moment of inertia scales with the mass and the square of the scaling factor, which is essential for calculating the kinetic energy of scaled models.