A Classical Approach of Newtonian Mechanics

7 CIRCULAR MOTION 7.6 The vertical pendulum v’ T B  r r cos  mg v (^) A mg cos     mg Figure 64: Motion in a vertical c ...

7 CIRCULAR MOTION 7.6 The vertical pendulum which reduces to vJ^2 = v^2 − 2 r g (1 − cos θ). (7.47) Let us now examine the radia ...

7 CIRCULAR MOTION 7.7 Motion on curved surfaces Note that this condition is independent of the mass of the object. Suppose that ...

7 CIRCULAR MOTION 7.7 Motion on curved surfaces v’ (^) R B mg cos  mg v  r cos (^)  r A mg Figure 65: Motion on the inside ...

7 CIRCULAR MOTION 7.7 Motion on curved surfaces the object will fly off the surface of the hoop, since it is no longer being pre ...

7 CIRCULAR MOTION 7.7 Motion on curved surfaces Figure 66: A skier on a hemispherical mountain. turning point lies in the lower ...

7 CIRCULAR MOTION 7.7 Motion on curved surfaces Suppose that the skier has reached angular coordinate θ. At this stage, the skie ...

7 CIRCULAR MOTION 7.7 Motion on curved surfaces R car^ of curvature  mg r (^). r g 2 / 2 centre banked curve Answer: Consider ...

7 CIRCULAR MOTION 7.7 Motion on curved surfaces √ Worked example 7.2: Circular race track Question: A car of mass m = 2000 kg tr ...

7 CIRCULAR MOTION 7.7 Motion on curved surfaces μ r ., reaction R due to the wall, which acts radially inwards. If the cylinder ...

7 CIRCULAR MOTION 7.7 Motion on curved surfaces R experiences an acceleration v^2 /r towards the centre of the loop, Newton’s se ...

7 CIRCULAR MOTION 7.7 Motion on curved surfaces Hence, vJ = m v . M + m Consider the case where the bob is suspended by a rigid ...

8 ROTATIONAL MOTION 8 Rotational motion 8.1 Introduction Up to now, we have only analyzed the dynamics of point masses (i.e., ob ...

8 ROTATIONAL MOTION 8.2 Rigid body rotation A rigid body P’ (^)  Q P       B (^) axis of rotation Figure 67: Rigid bod ...

8 ROTATIONAL MOTION 8.3 Is rotation a vector? velocities are conventionally measured in radians per second, whereas angular acce ...

8 ROTATIONAL MOTION 8.3 Is rotation a vector? direction of rotation vector Figure 68: The right-hand grip rule. The rotation “ve ...

8 ROTATIONAL MOTION 8.3 Is rotation a vector? x-axis z-axis z x y Figure 69: The addition of rotation is non-commutative. z-axis ...

8 ROTATIONAL MOTION 8.3 Is rotation a vector? longitude 90 ◦W could just as well be achieved by keeping the pointer fixed and ro ...

8 ROTATIONAL MOTION 8.4 The vector product rotation. We conclude that, although rotation can only be thought of as a vector quan ...

8 ROTATIONAL MOTION 8.4 The vector product × × × × × a x b (^) |a x b| = a b sin     b  a Figure 70: The vector product. I ...