a) To determine The acceleration of the punch when θ = 0 . Explanation Write the equation for the harmonic motion of the pivoted sector. θ = θ 0 sin 2 π t (I) Here, amplitude is θ 0 and time is t . Substitute π 12 rad for θ 0 in Equation (I). θ = ( π 12 rad ) sin 2 π t = π 12 sin 2 π t (II) Differentiate Equation (II) with respect to time. d d t ( θ ) = d d t ( π 12 sin 2 π t ) θ ˙ = π 2 6 cos 2 π t (III) Here, angular velocity is θ ˙ . Differentiate Equation (III) with respect to time. d d t ( θ ˙ ) = d d t ( π 2 6 cos 2 π t ) θ ¨ = − π 3 3 sin 2 π t (IV) Here, angular acceleration is θ ¨ . Draw the schematic diagram of the system as shown in Figure (I). For Figure (I), apply law of cosine rule. l 2 = y 2 + b 2 − 2 y b cos θ (V) Here, distance between points A and B is l , distance between points O and A is y and distance between points O and B is b . Differentiate Equation (V) with respect to time. d d t ( l 2 ) = d d t ( y 2 + b 2 − 2 y b cos θ ) 0 = y y ˙ − y ˙ b cos θ + y b θ ˙ sin θ (VI) Here, velocity of the punch is y ˙ . Differentiate Equation (VI) with respect to time. 0 = d d t ( y y ˙ − y ˙ b cos θ + y b θ ˙ sin θ ) 0 = y ¨ ( y − b cos θ ) + y ˙ 2 + 2 b y ˙ θ ˙ sin θ + b y θ ˙ 2 cos θ + b y θ ¨ sin θ (VII) Here, acceleration of the punch is y ¨ . Conclusion: Substitute 0 for θ in Equation (II). 0 = π 12 sin 2 π t 0 = sin 2 π t sin π = sin 2 π t π = 2 π t t = 1 2 Substitute 1 2 for t in Equation (III). θ ˙ = π 2 6 cos 2 π ( 1 2 ) = − π 2 6 Substitute 1 2 for t in Equation (IVI) b) To determine The acceleration of the punch when θ = π 12 .

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ISBN: 9781119047315

Author: Bolton

Publisher: VST

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Chapter 5.3, Problem 58P

To determine

The acceleration of the punch when θ=0.

To determine

The acceleration of the punch when θ=π12.

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Chapter 5.3, Problem 57P

Chapter 5.4, Problem 59P

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