Problem 1.12: The two points PA and PB start their motion along a circular path at the same time t = 0 at A and B, respectively. Point PA moves with the given constant speed v_0 and point PB starts from rest with a constant angular acceleration \dot{\omega}_0 . a) Determine \dot{\omega}_0 such that both points arrive at C at the same time. b) Calculate the angular velocity of PB when passing A. c) Determine the normal acceleration of both points at C. MATLAB Code: % Given data v0 = 10; % Constant speed of point PA r = 5; % Radius of the circular path theta_C = pi/3; % Angle to point C % Part a: Determine angular acceleration for both points to arrive at C simultaneously % Time taken to reach point C by PA t_C_PA = r * theta_C / v0; % Time taken to reach point C by PB (from rest) % s = ut + (1/2) * a * t^2 => s = (1/2) * a * t^2 (as u = 0) % s = r * theta => theta = (2 * s) / r theta_PB_C = (2 * r * theta_C) / r; % Angular displacement from rest theta_PB = (1/2) * dot(w0, t_C_PB)^2; % Angular displacement from uniform velocity theta_PA = v0 * t_C_PA; % Total angular displacement theta_total_PB = theta_PB_C + theta_PB; % Time taken for PB to reach C t_C_PB = sqrt((2 * theta_total_PB) / dot(w0, t_C_PB)); % Angular velocity of PB at C w_PB_C = dot(w0, t_C_PB); % Part b: Angular velocity of PB when passing A % Time taken for PB to reach A t_A_PB = sqrt((2 * pi) / dot(w0, t_A_PB)); % Angular velocity of PB at A w_PB_A = dot(w0, t_A_PB); % Part c: Normal acceleration at C a_normal_PA_C = v0^2 / r; a_normal_PB_C = r * dot(w0, t_C_PB)^2; % Display results disp(['a) Angular acceleration of PB such that both points arrive at C simultaneously: ', num2str(w0), ' rad/s^2']); disp(['b) Angular velocity of PB when passing A: ', num2str(w_PB_A), ' rad/s']); disp(['c) Normal acceleration of point PA at C: ', num2str(a_normal_PA_C), ' m/s^2']); disp([' Normal acceleration of point PB at C: ', num2str(a_normal_PB_C), ' m/s^2']);