Calculating Velocity and Angle - Normal & Tangential Accel.

[Cal Ripken]

Homework Statement

https://lh3.googleusercontent.com/7G78ypM5WmK-vnUpoo1yHc-u7cGRdLDiQV7akVJ8UWnuqtN9oJbY7_I6Yj4SJZCSRgUbxxZDCRv3fPX_fjJeWmFGgvv3dxz1rqz9WOrGnmviZ-YzGO5KoGSnbQVIDhNaX15IgY0mZ_1LLb-oEkZ-vdPX6JJi6unt8Bw7c4vsSUqJehsmmE7fQmyLUibE5ikeTcr7mSkhs_aQ8DKB2Oi8EXFF9iSKgufiawdSAR2MOVx0Nccn1yoFsKyhNkdqd0JPBKhzeMiPUES897hniRTTfvjMqUJBvi-sFS2SotX0VWrMoXk3s0b1dRnJTx1yBSkfKk8L8VztnjPe09B_1GMEMxL6-hlMIxMIKq4wz6rFkwalKLv3LXhr2kMZcubrzotG2HEmpnx718HV0Adn1Ife-3Zr6fTgR9nDJWa50vTCvyWrGVhdj8mC5_WXXXQgCTzk53XY9IAaz8K4TYO4VLalrkF3Tw0S4LlBaS3rICOYDFqQye1_uekZhv81JMPtCD4Z4lvVumVzDDAOxdBml-su0MaFupGoZzCnE8Sc0fWuMQ=w264-h195-no [/B]
A small 1 lb block is at rest at the top of a cylindrical surface. The block is given an initial velocity v₀ to the right which causes it to slide on the cylindrical surface. Assume smooth surface and no friction.

a. Use any computational software or Excel to calculate the velocity and angle theta at which the block leaves the surface if initial velocity v₀ is from 5 ft/s to 15 ft/s (every 1 ft/s).

b. If there is friction between the block and the surface what will be the difference between your equations? Set up the equations without solving. Assume the coefficient of kinetic friction μ_k between the block and the surface is given.

Homework Equations

∑F_n=ma_n
∑F_t=ma_t

The Attempt at a Solution

For part a, my attempt at a solution involved setting up the equations of motion as a function of theta. Recognizing that the block begins to leave the surface when the Normal force acting on the block is equal to zero, I set up an equation solving for theta when N=0, then plugged that value of theta into the equation established from the summation of forces in the tangential direction to solve for the final velocity of the block when it leaves the surface.

My equation setup:
https://lh3.googleusercontent.com/RI3-xHY-FdG0dczp2Pio9w4dmeMa3hfzZ2yfUt8jn9f-_qlWW66Ie_W5UqHJaLSS1Bzulu6CvIoM9hE8wUxjI-bbyIPlTguUALr7WbBuDC6xHCaFGp2Pkr5oVnMP5NDcjn_nQ0lZ2Obsp1vakar4OZQ0m-mtD6gHmhrEoBtqjsFlsJ5zOnuejAsTn9Wp6GWeTX_310ba3qbC1hyMFNPu_mZe1DJ3jdFA8OWoaHeBc4UFQX_T61tWpr00B9Yt3WAXEV6mOUPoYBK2bOT4r0PcODnGXuQW5s27lxpU2YPPqe1qcZd45LvJNYAdUv7mziqG9zf4IUucYL5qY3EeHw7HO4h1kM8pATbH3YF_yWtMb7A84Qu_W-t-8yMDbaV36iI1h_6uxzFP1dOybC3l8xdEWk5OVDYh52v5sUV5Z5i_bcAL-8ogRbdRibwUTQH6lMhFahRror6oBFwR2BLjpwAkwnKOFhry5W8f9VHmAdCFlCPHaauvNBMMc6mciDHp4JaFshKFUf2kzSYoZErmwzIa5p6oPfq3dXyUf8bov5FAsw=w526-h493-no

Values obtained for values of V₀ ranging from 5 ft/s to 15 ft/s:
https://lh3.googleusercontent.com/GnL529e_LtGTAKptTmmMMMbaT4Vm0-RSsrTJuOUetvwA58_DixtpxIWK2owVbTcvDmrKPSbxBJ4FmirdDJXOyuhPfap_HcPui525i4OQ0QpdNzlAszdOtfSlMfAuifPyRVoDhNO_aq-nHRwamouxn8Co393B8GjRDX0gBmDzI9IqPfEmJ3RyrjDLZc6jLhfty4KVhljUC3wWkbwiKL3I_b-qRRq5Ggd87Me_5e12w68rBJag0ioBXFty2kDuO7OFJS5e5ckRysNvLnfHzD5gnPpLWE92u7wCeVNfU4XmDHId4GXVReii_4IS5R73LA2VGMSqLsgaA0ZhPXWP6WvrHyjAH-7sI8g9Eb0wpryQ-jyG2Qf7qM2YUUc77LUw2zQY8AVoztfeGCgAjZX3aO3KEW4ZFsT98xxVQr_srcdpbYwGF-1ZZKELYh5zkl0-fS8K-sO5C7lPfz788bCdxUH4iXKrQUNwlurQNtmwrymNxq-OA_8XJdM1g91xs6bEiRPUOByxkuORAYGRwCbceQ2hu1UjzLOdAbyPxDvbSaHcFw=w287-h244-no

I would like to know if my equations are set up properly. Further, I would like some help interpreting the data. It makes sense that the velocity of the block would be greater when it leaves the surface because gravity is doing positive work. The angle theta is what's throwing me off, though. The values I obtained for theta when N=0 make sense if measured from the positive x axis, because lower values of V₀ should correspond to a greater change in theta. In setting up my second equation, my interval of integration was from 0 to theta. When I tried integrating from 90° to theta, I ended up with V_F =√ (- number) . My results make sense to me provided theta is measured from the x axis, but I intended to measure theta from the y axis. What am I missing here and what do I need to do to prevent making this mistake again?

Thanks for your considerations.

 

[Scott Redmond]

The values I obtained for theta when N=0 make sense if measured from the positive x axis, because lower values of V₀ should correspond to a greater change in theta.

My first thought upon seeing this was that your angle theta is measured from the y-axis. It seems to be measured from your weight vector, as drawn. So for v_o=5.00 ft/s, the block leaves the cylindrical surface when theta = 8.93 degrees -- as measured from the y-axis. I'll ask a question: what is the value of theta, according to your diagram, when the block is at the top of the cylinder? Then Δθ=θ_f-θ_i.

 

[haruspex]

Check your expression for the loss in gravitational PE.