Handout 6: Rotational motion and moment of inertia. Angular velocity and angular acceleration

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1 1 Handout 6: Rotational motion and moment of inertia Angular velocity and angular acceleration In Figure 1, a particle b is rotating about an axis along a circular path with radius r. The radius sweeps out and angle θ and the particle moves a distance s. The size of tangential velocity v and the size of the angular velocity ω are r s ω = dθ dt v = ds dt, dθ ω = dt. Figure 1: Angular velocity The direction of angular velocity is perpendicular to the plane of rotation according to right hand rule in Figure 2. The relationship between the tangential velocity and the angular velocity is that v = ω r, v = ωr sin θ, where θ is angle between ω and r. Note that vector v must be perpendicular to both ω and r. Right hand rule (Figure 2) can be used to find direction of ω. If ω and r are making right angle, the above expression is reduce to v = ωr. Figure 2: Right hand rule If the size of tangential velocity is not constant, there is the tangential acceleration a and hence the angular acceleration α defined by a = dv dt, dω α = dt. It can be shown that, for a constant r, a = αr. It must be noted that, for a rotating extended object, r is the distance of a point to the rotation axis, not the radius of the object. Example 1 A particle moves on a plane. When its position vector is r = i + 4j k, the angular velocity is ω = i + 2j + k. Figure 3: Angular velocity and tangential velocity a) Find the velocity of the particle at this point. b) Find the speed of the particle at this point.

2 Example 2 Points A and B are on a circular disc of radius 0.30 m. The disc is rotating with period T = 2π/3 s about a fixed axis passing through the center of the disc and perpendicular to the plane of the disc. Points A and B are at distances 0.18 m and 0.27 m, respectively, from the axis. Determine the tangential speed of A and B. 2 Example 3 A plane equilateral triangle ABC is rotating with angular speed ω = 12 rad s -1 about an axis perpendicular to the plane at point B. Each side of the triangle is 0.25 m long. Find the velocity of point A and point C. Example 4 A circular disc or radius 0.5 m is rotating about an axis perpendicular to the plane at the center. At time t seconds, the swept angle is given by θ t = 2t 2 + 3t radians. Evaluate the net acceleration of the edge of the disc at time t = 0.1.

3 3 Moment of inertia Consider a body rotating about a fixed axis. It is sometimes harder to rotate some bodies than the others. This is because of the moment of inertia (I) of the body. The greater the moment of inertia of a body about an axis is, the harder it is to rotate that body. In Figure 4, a particle (point mass) of mass m is rotating about an axis at perpendicular distance r. The moment of inertia of the particle is defined as Figure 4: Point pass rotating about an axis. I = mr 2. For a collection of particles rotating about the same axis in the same direction, the total moment of inertia is the sum of individuals. In Figure 5, the total moment of inertia is therefore I = m i r i 2 i = m 1 x m 2 x m 3 x 3 2. Figure 5: A collection of masses rotating in the same direction about an axis For a rigid object such as that in Figure 6, the moment of inertia is calculated by integration: I = r 2 dm, where dm is the mass element at distance r from axis of rotation. The integral runs over all volume of the object. Note that the axis of rotation does not need to pass through the body of the object. The moment of inertia depends on the distribution of mass around the axis of rotation. Consider a rotation of rod in Figure 7. When the rotation axis is along the length of the rod, the distribution of mass about the rotation axis is very small. This makes the moment of inertia small and hence it is to rotate the rod about this axis. However, if the rotation axis is perpendicular to the rod at one end, the distribution of mass around the rotation axis is very large. As a result, the moment of inertia is large and it is very hard to rotate the rod about this rotation axis. The formulae for the moments of inertia of rigid bodies are given in Appendix. Figure 6: Mass element dm at distance r from rotation axis (z-axis) Figure 7: Comparison of rotation of rod about different axes

4 Example 5 Three point masses are joined by light rods in shape of triangle. Determine the moment of inertia about axis through B and perpendicular to x-y plane. 4 Example 6 A rod of mass M and length L is rotating about axis perpendicular to the rod at one end. Show that the moment of inertia I = 1 3 ML2. Example 7 Show that a ring of mass m and radius r has moment of inertia about axis through the center of the ring perpendicular to the plane of the ring is given by I = mr 2. Example 8 Show that a disc of mass M and radius R rotating about an axis through the center perpendicular to the disc is given by I = 1 2 MR2.

5 Example 9 A rigid body consists of a solid cylinder of mass 4 kg and radius 0.2 m and a hemisphere of mass 2.5 kg and the same radius attached at one end of the cylinder. The rotation axis passes through the center along the length of the body. Evaluate the moment of inertia of the body. 5 Parallel axis theorem and perpendicular In Figure 8, a rigid body of mass M rotating about an axis through the center of mass has the moment of inertia I cm. If the axis of rotation is moved to a new axis parallel to the original by distance d, the moment of inertia about this new axis is given by I = I cm + Md 2. This is known as parallel axis theorem. Figure 8: Parallel axis theorem Consider a planar object in Figure 9. If the moments of inertia about x-axis and y-axis are I x and I y respectively, perpendicular axis theorem states that the moment of inertia about z-axis, perpendicular to x- axis and y-axis, is given by I z = I x + I y. Example 10 Find the expression for the moment of inertia of a disc, mass m and radius R, about an axis passing through the edge of the disc perpendicular to the plane. Figure 9: Perpendicular axis theorem

6 Example 11 Consider a circular disc of mass m and radius r in the figure. Determine the expression for moments of inertia about a) axis PQ b) axis SR. 6 Example 12 Find the formula for the moment of inertia of a rigid body consisting of a rod and a sphere as shown in the diagram. Example 13 Two rods, each of mass m and length L are joined to form a T-shaped rigid body. Point A is at the end of the rod shown in the figure. Find the expression for the moment of inertia about axis passing through point A perpendicular to the plane of the rods. A L L

7 Appendix: Formulae of moment of inertia of rigid bodies 7