Chapter 15: Dynamics Problem Set

Question 1: The 200-lb cabinet is subjected to the force F = 20(t + 1) lb where t is in seconds. If the cabinet is initially moving to the left with a velocity of 20 ft/s, determine its speed when t = 5 s. Neglect the size of the rollers. Matlab % Defining known values mass = 200/32.2; % Mass in slugs v_initial = -20; % Initial velocity in ft/s time = 5; % Time in seconds % Defining force function force = @(t) 20*(t+1)*cosd(30); % Force in lbf % Calculating integral of force*dt dt = 0.001; % Small time step t = 0:dt:time; % Time vector F_dt = 0; % Initializing integral for i = 1:length(t) F_dt = F_dt + force(t(i))*dt; % Numerical integration using Riemann sum end % Calculating final velocity v_final = (F_dt + mass*v_initial)/mass; % Printing result fprintf('The speed of the cabinet at t = %d s is %.2f ft/s\n',time,abs(v_final)) Question 2: If it takes 35 s for the 50-Mg tugboat to increase its speed uniformly to 25 km/h starting from rest, determine the force of the rope on the tugboat. The propeller provides the propulsion force F which gives the tugboat forward motion, whereas the barge moves freely. Also, determine F acting on the tugboat. The barge has a mass of 75 Mg. Matlab % Defining known values m_tug = 50e3; % Mass of tugboat in kg m_barge = 75e3; % Mass of barge in kg time = 35; % Time taken in seconds v_final = 25*1000/3600; % Final velocity in m/s % Calculating force on tugboat F_tug = (m_tug + m_barge)*v_final/time; % Force in N fprintf('The force on the tugboat is %.2f kN\n',F_tug/1e3) % Calculating tension in rope T = m_barge*v_final; % Tension in N fprintf('The tension in the rope is %.2f kN\n',T/1e3) Question 3: The thrust on the 4-Mg rocket sled is shown in the graph. Determine the sled's maximum velocity and the distance the sled travels when t = 35 s. Neglect friction. Matlab % Defining known values mass = 4e3; % Mass of sled in kg % Defining thrust function T = @(t) 4*sqrt(t).*(t<=25) + 20e3.*(1-(t>25)).*(t<=35); % Calculating velocity dt = 0.01; % Small time step t = 0:dt:35; % Time vector v = zeros(size(t)); % Initializing velocity vector for i = 2:length(t) v(i) = v(i-1) + (T(t(i))/mass)*dt; % Numerical integration using Euler method end v_max = max(v); % Maximum velocity in m/s fprintf('The maximum velocity is %.2f m/s\n',v_max) % Calculating displacement s = zeros(size(t)); % Initializing displacement vector for i = 2:length(t) s(i) = s(i-1) + v(i)*dt; % Numerical integration using Euler method end disp_35 = s(end); % Displacement at t = 35 s in m fprintf('The displacement at t = 35 s is %.0f m\n',disp_35) Question 4: The motor pulls on the cable at A with a force F = (30 + t^2) lb, where t is in seconds. If the 34-lb crate is originally on the ground at t = 0, determine its speed at t = 4 s. Neglect the mass of the cable and pulleys. Matlab % Defining known values mass = 34/32.2; % Mass of crate in slugs g = 32.2; % Acceleration due to gravity in ft/s^2 % Defining force function force = @(t) 30 + t^2; % Force function in lbf % Finding time when crate starts lifting t_start = sqrt(34); % Time when force = crate weight % Calculating integral of (force - weight)*dt dt = 0.001; % Small time step t = t_start:dt:4; % Time vector F_W_dt = 0; % Initializing integral for i = 1:length(t) F_W_dt = F_W_dt + (force(t(i)) - 34)*dt; % Numerical integration using Riemann sum end % Calculating final velocity v_final = sqrt(2*F_W_dt/mass); % Printing result fprintf('The speed of the crate at t = 4 s is %.2f ft/s\n',v_final)