`●` Let angular velocity `ω` plays the same role in rotation as the linear velocity `v` in translation. We wish to take this analogy further. In doing so we shall restrict the discussion only to rotation about fixed axis.
`●` This case of motion involves only one degree of freedom, i.e., needs only one independent variable to describe the motion. This in translation corresponds to linear motion. This section is limited only to kinematics. We shall turn to dynamics in later sections.
`●` We recall that for specifying the angular displacement of the rotating body we take any particle like P (Fig.7.33) of the body. Its angular displacement `θ` in the plane it moves is the angular displacement of the whole body; `θ` is measured from a fixed direction in the plane of motion of `P`, which we take to be the `x′` - axis, chosen parallel to the `x`-axis. Note, as shown, the axis of rotation is the `z` – axis and the plane of the motion of the particle is the `x - y` plane. Fig. 7.33 also shows `θ_o`, the angular displacement at `t = 0`.
`●` We also recall that the angular velocity is the time rate of change of angular displacement, `ω = dθ//dt`. Note since the axis of rotation is fixed, there is no need to treat angular velocity as a vector. Further, the angular acceleration, `alpha=(d omega//dt)`
`●` The kinematical quantities in rotational motion, angular displacement `(θ)`, angular velocity `(ω)` and angular acceleration `(α)` respectively correspond to kinematic quantities in linear motion, displacement `(x)`, velocity `(v)` and acceleration `(a)`. We know the kinematical equations of linear motion with uniform (i.e. constant) acceleration:
`color{blue}{v=v_0+at.......................(a)}`
`color{blue}{x=x_0+v_0t+1/2 at^2...............(b)}`
`color{blue}{v^2=v_0^2+2ax...................(c)}`
`=>` where `x_0 =` initial displacement and `v_0=` initial velocity. The word ‘initial’ refers to values of the quantities at `t = 0`
`●` The corresponding kinematic equations for rotational motion with uniform angular acceleration are:
`color{blue}{omega=omega_0 +alpha t..................(7.38)}`
`color{blue}{ theta=theta_0 +omega_0 t +1/2 alpha t^2.............(7.39)}`
and `color{blue}{omega^2=omega_0^2 + 2 alpha(theta-theta_0).............(7.40)}`
where `θ_0=` initial angular displacement of the rotating body, and `ω_0 =` initial angular velocity of the body.
`●` Let angular velocity `ω` plays the same role in rotation as the linear velocity `v` in translation. We wish to take this analogy further. In doing so we shall restrict the discussion only to rotation about fixed axis.
`●` This case of motion involves only one degree of freedom, i.e., needs only one independent variable to describe the motion. This in translation corresponds to linear motion. This section is limited only to kinematics. We shall turn to dynamics in later sections.
`●` We recall that for specifying the angular displacement of the rotating body we take any particle like P (Fig.7.33) of the body. Its angular displacement `θ` in the plane it moves is the angular displacement of the whole body; `θ` is measured from a fixed direction in the plane of motion of `P`, which we take to be the `x′` - axis, chosen parallel to the `x`-axis. Note, as shown, the axis of rotation is the `z` – axis and the plane of the motion of the particle is the `x - y` plane. Fig. 7.33 also shows `θ_o`, the angular displacement at `t = 0`.
`●` We also recall that the angular velocity is the time rate of change of angular displacement, `ω = dθ//dt`. Note since the axis of rotation is fixed, there is no need to treat angular velocity as a vector. Further, the angular acceleration, `alpha=(d omega//dt)`
`●` The kinematical quantities in rotational motion, angular displacement `(θ)`, angular velocity `(ω)` and angular acceleration `(α)` respectively correspond to kinematic quantities in linear motion, displacement `(x)`, velocity `(v)` and acceleration `(a)`. We know the kinematical equations of linear motion with uniform (i.e. constant) acceleration:
`color{blue}{v=v_0+at.......................(a)}`
`color{blue}{x=x_0+v_0t+1/2 at^2...............(b)}`
`color{blue}{v^2=v_0^2+2ax...................(c)}`
`=>` where `x_0 =` initial displacement and `v_0=` initial velocity. The word ‘initial’ refers to values of the quantities at `t = 0`
`●` The corresponding kinematic equations for rotational motion with uniform angular acceleration are:
`color{blue}{omega=omega_0 +alpha t..................(7.38)}`
`color{blue}{ theta=theta_0 +omega_0 t +1/2 alpha t^2.............(7.39)}`
and `color{blue}{omega^2=omega_0^2 + 2 alpha(theta-theta_0).............(7.40)}`
where `θ_0=` initial angular displacement of the rotating body, and `ω_0 =` initial angular velocity of the body.