Learn combination of translational and rotational motion, rolling motion without Slipping. Practice acceleration of a point on a body and the rolling constraint requires the motion without slipping.
A Translational circular motion, rotational motion
B Rotational motion, Rotational motion
C Straight line translational motion, Rotational motion
D Rotational Motion, Translational motion
Motion of center of mass + Motion with respect to center of mass
\(|\vec v_P|=\sqrt {(v_{CM})^2+(\omega r)^2}\)
\(|\vec v_Q|=v_{CM}+\omega r\)
\(|\vec v_R|=\sqrt {(v_{CM})^2+(\omega r)^2}\)
\(|\vec v_S|=|v_{CM}-\omega r|\)
A 2 m/sec
B 0 m/sec
C 4 m/sec
D 6 m/sec
\(v_P=v-\omega R=v_B\)
Case 1 : Rolling on ground when Velocity of Ground is Zero
\(v_{\text { Ground}}=0\) [ As ground is stationary ]
In that case, \(v-r\omega=0\)
or, \(v=r\omega\)
The velocities of different points of ball are
as \(v=R\ \omega\)
Hence, point P is at rest instantaneously.
\((a_P)_{\text { tangential}}=a-r\alpha=a_B\)
Case 2 : Rolling on Ground when acceleration of Ground is Zero
As acceleration of ground is zero
\((a_P)_t=0\)
or, \(a-r\;\alpha=0\)
\(a=r\;\alpha\)
The acceleration of different points of ball.
As \(a=r\;\alpha\)
A \(a=r\alpha,\ v=r\omega\)
B \(a=r\alpha ,\ v=r\omega+v_B\)
C \(\alpha=ar ,\ \omega=vr\)
D \(a=a_B+r\alpha ,\ v=v_B+r\omega\)
\((a_P)_{tangential}=a_{ground}\)
\(\therefore a-r\alpha=0\)
\(a=r\alpha\)
\(a_T=\sqrt {a^2+a^2+2a^2\ cos(\pi-\theta)}\)
\(=\sqrt {2a^2-2a^2\;cos\,\theta}\)
\(a_T=a\sqrt {2-2\;cos\,\theta}\)
Motion of center of mass + Motion with respect to center of mass
Acceleration at P, \(\vec a_P\) :
\(|\vec a_P|=\sqrt {a^2+(r\alpha)^2}\)
Acceleration at Q, \(\vec a_Q\) :
\(|\vec a_Q|=a+r\alpha\)
Acceleration at R, \(\vec a_R\) :
\(|\vec a_R|=\sqrt {a^2+(r\alpha)^2}\)
Acceleration at S, \(\vec a_S\) :
\(|\vec a_S|=|a-r\alpha|\)
Acceleration at point T, \(\vec a_T\) :
\(|\vec a_T|=\sqrt {a^2+(r\alpha)^2+2ar\alpha\;cos(\pi-\theta)}\)
The rolling constraint requires the motion without slipping.
\((a_P)_{\text {tangential}}=a_{Q}\)
A \(v_C = v_P\)
B \(v_B = v_P\)
C \(v_C =r\ {\omega}\)
D \(v_A = v_P\)