The relationship between mass (m) and the square of the period (T2) for a single spring is given by the formula for the period of a mass-spring system, T = 2pi sqrt((m)/(k1)), where $k1

Here are the solutions to the questions: (xvii) Determine the slope, s_1, of the graph. Step 1: The relationship between mass (m) and the square of the period (T^2) for a single spring is given by the formula for the period of a mass-spring system, T = 2 sqrt((m)/(k_1)), where k_1 is the spring constant of the single spring. Squaring both sides gives: T^2 = 4^2 (m)/(k_1) Step 2: Rearranging the equation to match the graph axes (m on the vertical axis and T^2 on the horizontal axis), we get: m = ((k_1)/(4^2)) T^2 Step 3: Comparing this to the equation of a straight line y = s_1 x, where y=m and x=T^2, the slope s_1 is: s_1 = (k_1)/(4^2) Since no experimental data is provided, the numerical value of s_1 cannot be determined. It represents the ratio of the spring constant to 4^2. (xviii) On the same axes, plot m on the vertical axis and T_0^2 on the horizontal axis. This step was explained in the previous response (Step xvi). You would plot the data points for the parallel spring system (m vs. T_0^2) on the same graph as the single spring system, using different symbols or colors to distinguish them. A best-fit straight line passing through the origin would then be drawn for these points. (xix) Determine the slope, s_2, of the graph. Step 1: For two identical springs in parallel, the effective spring constant k_eff is the sum of the individual spring constants. If each spring has a constant k_1, then k_eff = k_1 + k_1 = 2k_1. Step 2: The period of oscillation T_0 for the parallel spring system is T_0 = 2 sqrt((m)/(k_eff)) = 2 sqrt((m)/(2k_1)). Squaring both sides gives: T_0^2 = 4^2 (m)/(2k_1) Step 3: Rearranging the equation to match the graph axes (m on the vertical axis and T_0^2 on the horizontal axis), we get: m = ((2k_1)/(4^2)) T_0^2 Step 4: Comparing this to the equation of a straight line y = s_2 x, where y=m and x=T_0^2, the slope s_2 is: s_2 = (2k_1)/(4^2) Since no experimental data is provided, the numerical value of s_2 cannot be determined. It represents the ratio of twice the spring constant to 4^2. (xx) Evaluate = (s_2)/(s_1). Step 1: Substitute the expressions for s_1 and s_2: = (s_2)/(s_1) = (2k_1)/(4^2)(k_1)/(4^2) Step 2: Simplify the expression: = (2k_1)/(4^2) × (4^2)/(k_1) = 2 The value of is 2. (xxi) State two precautions taken to ensure accurate results when performing this experiment. • Ensure oscillations are purely vertical and not swinging sideways to avoid energy loss and inaccurate period measurements. • Avoid parallax error when reading the stopwatch or measuring the extension of the spring. How does change in the number of springs affects the frequency of the system? The frequency (f) of a mass-spring system is inversely proportional to the period (T), and directly proportional to the square root of the spring constant (k). When the number of springs in parallel increases, the effective spring constant (k_eff) of the system increases. Since f = (1)/(2) sqrt(k_eff)m, an increase in k_eff leads to an increase in the frequency of the system. A spring has a stiffness of 125 \, Nm^-1. If a weight of 8.0 \, N is suspended on it, calculate the extension, e of the spring. Step 1: Apply Hooke's Law, which states that the force (F) applied to a spring is directly proportional to its extension (e), where k is the spring constant (stiffness): F = ke Step 2: Substitute the given values: F = 8.0 \, N and k = 125 \, Nm^-1. 8.0 \, N = (125 \, Nm^-1) × e Step 3: Solve for e: e = 8.0 \, N125 \, Nm^-1 e = 0.064 \, m The extension of the spring is 0.064 m. That's 2 down. 3 left today — send the next one.