7.19 A hoop of radius 2 m weighs 100 kg. It rolls along a horizontal floor so that its centre of mass has a speed of 20 cm/s. How much work has to be done to stop it?
Radius of the hoop, r = 2 m, mass of the hoop, m = 100 kg, velocity of the hoop, v = 20 cm /s = 0.2 m/s
Total energy of the hoop = Translational KE + Rotational KE = m +
Moment of inertia about the centre, I = mr2
So the total energy = m +
Since v = r we get
Total energy = m + =
Required work to be done = 100 x 0.2 x 0.2 J = 4 J
<p>Radius of the hoop, r = 2 m, mass of the hoop, m = 100 kg, velocity of the hoop, v = 20 cm /s = 0.2 m/s</p><p>Total energy of the hoop = Translational KE + Rotational KE = <span title="Click to copy mathml"><math><msub><mrow><mrow><mi>θ</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math></span> m <span title="Click to copy mathml"><math><mi>α</mi><mi></mi></math></span> + <span title="Click to copy mathml"><math><mi>α</mi><mi></mi></math></span></p><p>Moment of inertia about the centre, I = mr2</p><p>So the total energy = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac></math></span> m <span title="Click to copy mathml"><math><msup><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></math></span> + <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac><mi>I</mi><msup><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></math></span></p><p>Since v = r <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac></math></span> we get</p><p>Total energy = <span title="Click to copy mathml"><math><msup><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></math></span> m <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac><mi>m</mi><msup><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><msup><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></math></span> + <span title="Click to copy mathml"><math><mi>ω</mi></math></span> = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac></math></span></p><p>Required work to be done = 100 x 0.2 x 0.2 J = 4 J</p>
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