CALC A hollow, thin-walled sphere of mass 12.0 kg and diameter 48.0 cm is rotating about an axle through its center. The angle (in radians) through which it turns as a function of time (in seconds) is given by θ(t) = At^2 + Bt^4, where A has numerical value 1.50 and B has numerical value 1.10. (a) What are the units of the constants A and B?

Verified step by step guidance

1. Identify the units of the given function θ(t) = At^2 + Bt^4. Since θ(t) represents an angle in radians, the units of θ(t) are radians (rad).
2. Analyze the units of each term in the equation separately. For the term At^2, the units of A must be such that when multiplied by time squared (t^2), the result is in radians. Since time (t) is typically measured in seconds (s), the units of t^2 are seconds squared (s^2).
3. Determine the units of A by setting up the equation for the units: [A] * [t^2] = radians. Here, [A] represents the units of A and [t^2] is s^2. Solving for [A] gives [A] = radians/s^2.
4. Similarly, analyze the units for the term Bt^4. The units of B must be such that when multiplied by time to the fourth power (t^4), the result is in radians. The units of t^4 are seconds to the fourth power (s^4).
5. Determine the units of B by setting up the equation for the units: [B] * [t^4] = radians. Here, [B] represents the units of B and [t^4] is s^4. Solving for [B] gives [B] = radians/s^4.

Key Concepts

Angular Displacement

Angular displacement refers to the angle through which an object rotates about a fixed point or axis. It is typically measured in radians, which is a dimensionless unit that represents the ratio of the arc length to the radius of the circle. In the context of the given equation θ(t) = At^2 + Bt^4, θ(t) represents the angular displacement as a function of time.

Units of Measurement

Units of measurement are standard quantities used to express physical quantities. In this problem, we need to determine the units of the constants A and B in the equation for angular displacement. Since θ(t) is measured in radians, the units of A and B must be such that when multiplied by the time terms (t^2 and t^4, respectively), the result is in radians.

Dimensional Analysis

Dimensional analysis is a method used to check the consistency of equations by analyzing the dimensions of the quantities involved. It helps in determining the units of unknown constants by ensuring that both sides of an equation have the same dimensions. In this case, we will apply dimensional analysis to find the appropriate units for A and B based on their roles in the equation for angular displacement.