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 body that is in a pure translational motion can be represented by a particle of same mass placed at its center of mass.

- Rotational motion of a body about a fixed axis has to be represented by the angular variables with respect to axis of rotation.

- When the axis of rotation of a body is in translational motion, the motion of the body becomes complicated.

- For this case, the motion of the body can be represented as a combination of translational motion of center of mass and the rotation of body with respect to center of mass.

A Translational circular motion, rotational motion

B Rotational motion, Rotational motion

C Straight line translational motion, Rotational motion

D Rotational Motion, Translational motion

- Velocity of a point is the vector sum of \(v_{CM}\) and velocity (\(v\)) of a point rotating with respect to Center of Mass

- The motion of wheel can be seen as :

Motion of center of mass + Motion with respect to center of mass

- Velocity of any point is the vector sum of velocity of center of mass and velocity with respect to center of mass.
- Velocity at point P, v
_{P}:

\(|\vec v_P|=\sqrt {(v_{CM})^2+(\omega r)^2}\)

- Velocity at point Q, v
_{Q}:

\(|\vec v_Q|=v_{CM}+\omega r\)

- Velocity at point R, v
_{R}:

\(|\vec v_R|=\sqrt {(v_{CM})^2+(\omega r)^2}\)

- Velocity at point S, v
_{S}:

\(|\vec v_S|=|v_{CM}-\omega r|\)

A 2 m/sec

B 0 m/sec

C 4 m/sec

D 6 m/sec

- Rolling is a combination motion in which an object rotates about an axis that is itself moving, such that there is no slipping between the surfaces in contact.
- Consider a ball, rolling on surface B.

- Motion of Ball A can be expressed as

- The velocity and tangential acceleration of contact points of two surfaces are equal.
- Relative velocity of the contact point is zero.

\(v_P=v-\omega R=v_B\)

**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.

- Relative tangential acceleration of contact point is zero.

\((a_P)_{\text { tangential}}=a-r\alpha=a_B\)

- If relative tangential acceleration is non-zero, the relative velocity at next instant would be non-zero which would result in slipping. Hence, for pure rolling, relative tangential acceleration a
_{t}is also zero.

**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 \(a=a_B+r\alpha ,\ v=v_B+r\omega\)

- Consider a string, wrapped on a cylinder is pulled by a force and the cylinder moves. Let the cylinder be in pure rolling motion on ground (rolling without slipping)

- Acceleration of each point on the body has two components :

- Net acceleration at different points

- For pure rolling

\((a_P)_{tangential}=a_{ground}\)

\(\therefore a-r\alpha=0\)

\(a=r\alpha\)

- Net acceleration at point T

\(a_T=\sqrt {a^2+a^2+2a^2\ cos(\pi-\theta)}\)

\(=\sqrt {2a^2-2a^2\;cos\,\theta}\)

\(a_T=a\sqrt {2-2\;cos\,\theta}\)

- Acceleration at a point on a body is the vector sum of \(a_{CM}\) and \(a\) of point with respect to center of mass.
- A disk is in motion, as shown in figure. To calculate acceleration at any point at time t = 0 (\(\omega=0\))

- The motion of disk can be seen as :

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, \(\vec a_T\) :

\(|\vec a_T|=\sqrt {a^2+(r\alpha)^2+2ar\alpha\;cos(\pi-\theta)}\)

- Rolling is a combination of translational motion, which is same as the translational motion of center of mass and rotational motion of the body with respect to Center of Mass.

The rolling constraint requires the motion without slipping.

- The velocity of contact point is equal.

- The tangential acceleration of contact point is equal.

\((a_P)_{\text {tangential}}=a_{Q}\)

- The velocity of contact point is same not only at the instant shown but also for any time that follows.

A \(v_C = v_P\)

B \(v_B = v_P\)

C \(v_C =r\ {\omega}\)

D \(v_A = v_P\)