Conservational of Energy

translational rotlltional ΙΙ)ΙΙΙΙ 2 translalional

Example 1: Disc Rolling Down an Incline

  • Find the speed of a disc of radius R which starts at rest and rolls down an incline of height H

    ء «U إ د للاجها - مر) تي (بردل١ه ا :لا8 ا دء "نم -لا4 لادلا ت لاه آـ تلا ولاءا ا باو

Rotational Dynamics

F' ma

Example 2: Strings with Massive Pulleys

  • Two blocks are connected by a light string over a pulley of mass mp</sub and radius R.

  • Find the acceleration of mass m2</sub if m1</sub stis on a frictionless surface

    C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image235.png

Example 3: Rolling Without Slipping

  • A disc of radius R rolls down an incline of angle θ without slipping.

  • Find the force of friction on the disc

    H

Example 4: Rolling with Slipping

  • A bowling ball of mass M and radius R skids horizontally down the alley with an initial velocity of v0. Find the distance the ball skids before rilling given a coefficient of kenetic friction μk

    72 6

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Example 5: Amusement Park Swing

  • An amusement park ride of radius x allows children to sit in a spinning swing held by a cable of length L.

  • At maximum angular speed, the cable makes an angle of θ with the vertical as shown in the diagram below

  • Determine the maximum angular speed of the rider in terms of g, θ, x and L.

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    7 亻 ㄨ

2002 Free Response Question 2

m/4 Mech 2. m Bumper The cart shown above is made of a block of mass m and four solid rubber tires each of mass m/4 and radius r. Each tire may be considered to be a disk. (A disk has rotational inertia 1 ML2 , where M is the mass and L is the radius of the disk.) The cart is released from rest and rolls without slipping from the top of an inclined plane of height h. Express all algebraic answers in terms of the given quantities and fundamental constants. (a) Determine the total rotational inertia of all four tires. (b) Determine the speed of the cart when it reaches the bottom of the incline. (c) After rolling down the incline and across the horizontal surface, the cart collides with a bumper of negligible mass attached to an ideal spring, which has a spring constant k. Determine the distance the spring is compressed before the cart and bumper come to rest. (d) Now assume that the bumper has a non-neglible mass. After the collision with the bumper, the spring is compressed to a maximum distance of about 90% of the value of Xm in part (c). Give a reasonable explanation for this decrease.

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2006 Free Response Question 3

0 Mech 3. A thin hoop of mass M, radius R, and rotational inertia MR2 is released from rest from the top of the ramp of length L above. The ramp makes an angle 9 with respect to a horizontal tabletop to which the ramp is fixed. The table is a height H above the floor. Assume that the hoop rolls without slipping down the ramp and across the table. Express all algebraic answers in terms of given quantities and fundamental constants. (a) Derive an expression for the acceleration of the center of mass of the hoop as it rolls down the ramp. (b) Derive an expression for the speed of the center of mass of the hoop when it reaches the bottom of the ramp. (c) Derive an expression for the horizontal distance from the edge of the table to where the hoop lands on the floor. (d) Suppose that the hoop is now replaced by a disk having the same mass M and radius R. How will the distance from the edge of the table to where the disk lands on the floor compare with the distance determined m part (c) for the hoop? Less than The same as Greater than Briefly justify your response.

C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image244.png

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2010 Free Response Question 2

4.0 m 300 3.0 m Note: Figure not drawn to scale. Mech. 2. A bowling ball of mass 6.0 kg is released from rest from the top of a slanted roof that is 4.0 m long and angled at 300 , as shown above. The ball rolls along the roof without slipping. The rotational inertia of a sphere of mass M and radius R about its center of mass is —MR (a) On the figure below, draw and label the forces (not components) acting on the ball at their points of application as it rolls along the roof. (b) Calculate the force due to friction acting on the ball as it rolls along the roof. If you need to draw anything other than what you have shown in part (a) to assist in your solution, use the space below. Do NOT add anything to the figure in part (a). (c) Calculate the linear speed of the center of mass of the ball when it reaches the bottom edge of the roof. (d) A wagon containing a box is at rest on the ground below the roof so that the ball falls a vertical distance of 3.0 m and lands and sticks in the center of the box. The total mass of the wagon and the box is 12 kg. Calculate the horizontal speed of the wagon immediately after the ball lands in it.

C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image247.png

ZSS 7

2013 Free Response Question 3

Note: Figure not drawn to scale. Mech 3. A disk of mass M = 2.0 kg and radius R = 0.10 m is supported by a rope of negligible mass, as shown above. The rope is attached to the ceiling at one end and passes under the disk. The other end of the rope is pulled upward with a force FA . The rotational inertia of the disk around its center is MR2 2 . (a) Calculate the magnitude of the force FA necessary to hold the disk at rest. At time t = 0, the force FA is increased to 12 N, causing the disk to accelerate upward. The rope does not slip on the disk as the disk rotates. (b) Calculate the linear acceleration of the disk. (c) Calculate the angular speed of the disk at t = 3.0 s. (d) Calculate the increase in total mechanical energy of the disk from t = O to t = 3.0 s. (e) The disk is replaced by a hoop of the same mass and radius. Indicate whether the linear acceleration of the hoop is greater than, less than, or the same as the linear acceleration of the disk. Greater than Less than The same as Justify your answer.

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