Question

A point object P of mass m is sliding along a frictionless wall...

I want an intensive explanation for this problem.

3 A point object P of mass m is sliding along a fric- tionless wall of a radius of curvature, Ro, in a verti- cal plane as shown in Fig. 2. Assume that the point object is initially at rest at the point A and starts to move toward the bottom B. The angle between the normal (to the wall) at A and the vertical line is 8o radians, and assumed to be small. Fig. 2 (1) The elevation h between the points A and B is approximately (a) Robo (b) Ro0o2 (c Ro80 (d) Ro0o (2) The speed of the point object at B, Vax, is given by (b) 2gh (c) 2/gh where g is an acceleration constant due to gravity. (3) The arc length AB is given by (a) Robo (b) Robo (c) 2Ro0o (d) Ro0 (4) The time f which is defined as -AB/Vmix is given by (d) 2/ Rolg (5) The ratio of the actual time t reaching B from A to t,t/f, satisfies (b t/T-2 (e) t/t<1 (a) t/t> 2 (d) t/T-1

Answer 1

Elevation h = R_0 - R_0 cos theta = R_0 (1 - cos theta) h = R_0 (1 - (1 - 1/2 theta_0^2)) h = R_0 theta_0^2/2 (c) By energy conservation 1/2 m V_max^2 = mgh V_max = squareroot 2gh (b) Arc length is given by R_0 theta_0 (b) t^bar = AB/V_max = R_0 theta_0/ squareroot sgh = R_0 theta_0/ squareroot 2g R_0 theta_0^2/2 = squareroot R_0/g (b) t = AB/V_avg V_avg < V_max V_Avg < V_max for short theta_0 V_Avg = V_max/2 so t = 2t^bar t/t = 2 (b)