If the angular quantities θ,ω,α were specified in terms of degrees rather than radians, how would the kinematics equations for uniformly accelerated rotational motion have to be altered?
The angle, angular velocity, angular momentum and angular acceleration terms would have to be multiplied by 180/pi (degrees per radian).
When dealing with angular quantities in degrees instead of radians, the kinematics equations for uniformly accelerated rotational motion would have to be altered accordingly. The alterations can be summarized as follows:
1. Angular Displacement (θ):
- In the unit of degrees, the angular displacement is denoted as θ (in degrees) instead of θ (in radians).
- To convert between degrees and radians, you can use the conversion factor: 1 radian = π/180 degrees.
- Therefore, to convert an angle from degrees to radians, multiply it by π/180.
2. Angular Velocity (ω):
- The representation of angular velocity remains the same as ω (in radians per second).
- To convert angular velocity from degrees per second to radians per second, again use the conversion factor: 1 radian = π/180 degrees.
- Multiply the angular velocity (in degrees per second) by π/180 to obtain it in radians per second.
3. Angular Acceleration (α):
- The angular acceleration representation remains the same as α (in radians per second squared).
- To convert angular acceleration from degrees per second squared to radians per second squared, again multiply by the conversion factor: 1 radian = π/180 degrees.
- Multiply the angular acceleration (in degrees per second squared) by π/180 to obtain it in radians per second squared.
Note: While the symbols used for angular displacement, velocity, and acceleration remain the same, the numerical values may need to be converted accordingly.
By adapting these alterations, you can work with degrees in the kinematics equations for uniformly accelerated rotational motion.
To understand how the kinematic equations for uniformly accelerated rotational motion would be altered if the angular quantities were specified in degrees instead of radians, let's start by reviewing the standard equations.
In the standard form, the kinematic equations for uniformly accelerated rotational motion are:
θ = θ₀ + ω₀t + 0.5αt² (Equation 1)
ω = ω₀ + αt (Equation 2)
ω² = ω₀² + 2α(θ - θ₀) (Equation 3)
where:
θ, θ₀ are the final and initial angular positions in radians,
ω, ω₀ are the final and initial angular velocities in radians per second,
α is the angular acceleration in radians per second squared, and
t is the time in seconds.
If we want to express these quantities in terms of degrees instead of radians, we need to consider the conversions between radians and degrees.
To convert radians to degrees, we use the formula:
degrees = radians × (180/π)
To convert degrees to radians, we use the formula:
radians = degrees × (π/180)
Knowing these conversion formulas, we can proceed to alter the kinematic equations for uniformly accelerated rotational motion.
Alteration of Equation 1:
θ (in degrees) = θ₀ (in degrees) + ω₀t + 0.5 α (t² in seconds)
Alteration of Equation 2:
ω (in degrees per second) = ω₀ (in degrees per second) + αt
Alteration of Equation 3:
ω² (in degrees squared per second squared) = ω₀² (in degrees squared per second squared) + 2α (θ - θ₀) (in degrees)
By making these alterations and applying the appropriate conversion formulas, you can express the kinematic equations for uniformly accelerated rotational motion in terms of degrees.