AS Level foundations · Question 11

State State the independent variable, the dependent variable, and two quantities that must be kept constant in this investigation.

1. 1

   independent variable: the mass m hanging from the spring

2. 2

   dependent variable: the period T of the oscillations

3. 3

   constants: the same spring (same spring constant) and the same amplitude of oscillation — accept also the same support and no additional damping

Describe Describe how the student should measure T so that the uncertainty in each value is as small as possible.

1. 1

   time a number of complete oscillations — at least 10 or 20 — and divide by that number

2. 2

   start and stop the timing at the centre of the oscillation, where the mass moves fastest, using a fiducial marker

3. 3

   repeat the timing and take a mean

Explain Explain why a graph of lg T against lg m tests the suggested relationship, and state what the gradient and the y-intercept of that graph represent.

1. 1

   taking logarithms of T = km^p gives lg T = p lg m + lg k

2. 2

   this has the form y = mx + c, so if the relationship holds the points lie on a straight line

3. 3

   the gradient of the line is p

4. 4

   the y-intercept is lg k, so k = 10^(intercept)

Determine Use the first and last data points to determine values for p and for k. Give k to two significant figures.

1. 1

   gradient p = (0.051 − (−0.401)) / (−0.097 − (−1.000)) = 0.452 / 0.903

2. 2

   The logarithmic graph gradient gives the exponent p = 0.50.

3. 3

   intercept: lg k = −0.401 − 0.50 × (−1.000) = 0.099

4. 4

   k = 10^0.099 = 1.3 (accept 1.26)

Deduce Theory predicts T = 2π√(m/k_s), where k_s is the spring constant. Deduce a value for k_s and comment on whether the student's results support the theory.

1. 1

   the theory predicts p = 0.5, which matches the gradient found, so the form of the relationship is supported

2. 2

   comparing constants, k = 2π/√k_s, so k_s = (2π/k)² = (2π/1.26)²

3. 3

   k_s = 25 N m⁻¹