Luke stands on a scale in an elevator that has a constant acceleration upward. The scale reads \(0.960 \mathrm{kN} .\) When Luke picks up a box of mass $20.0 \mathrm{kg},\( the scale reads \)1.200 \mathrm{kN} .$ (The acceleration remains the same.) (a) Find the acceleration of the elevator. (b) Find Luke's weight.

Since 1 kN = 1000 N, we'll first convert the scale readings to Newtons: For Luke alone: \(0.960 \mathrm{kN} \times 1000 = 960 \mathrm{N}\) For Luke with the box: \(1.200 \mathrm{kN} \times 1000 = 1200 \mathrm{N}\) Now we can work with these values to find the mass in each situation.

Since the only force acting on Luke and the box in the vertical direction is the net force, equal to the gravitational force and the force due to the acceleration of the elevator, we can use the following: \(F_{net} = m_{total} \times g = m_{total} \times 9.81\,\text{m/s}^2\) For Luke alone: \(m_L = 960\,\mathrm{N} / 9.81\,\text{m/s}^2 \approx 97.86\,\text{kg}\) For Luke and the box: \(m_{LB} = 1200\,\mathrm{N} / 9.81\,\text{m/s}^2 \approx 122.42\,\text{kg}\) We now have the total mass when Luke is alone and when he is holding the box.

According to Newton's second law, we have: \(F_{net} = m_{total} \times a\) For Luke alone: \(F_{net}^L = (97.86\,\text{kg})a\) For Luke and the box: \(F_{net}^{LB} = (122.42\,\text{kg})a\)

We can set up the following system of equations using the previous step: Equation 1: \((97.86\,\text{kg})a = 960\,\text{N}\) Equation 2: \((122.42\,\text{kg})a = 1200\,\text{N}\) We can solve this system of equations by dividing Equation 2 by Equation 1: \(\frac{(122.42\,\text{kg})a}{(97.86\,\text{kg})a} = \frac{1200\,\text{N}}{960\,\text{N}}\) It simplifies to: \(\frac {122.42\,\text{kg}}{97.86\,\text{kg}}= \frac{5}{4}\) Solving for the acceleration, we get: \(a=9.81 \,\text{m/s}^2 \times \frac{5}{4}\) \(a=12.26\,\text{m/s}^2\) Now we can find Luke's weight by using the acceleration value in Equation 1: \(W = m_L \times a\) \(W =(97.86 \,\text{kg})(12.26\,\text{m/s}^2)\) \(W \approx 1199.98\,\text{N}\) So, the acceleration of the elevator is \(12.26\,\text{m/s}^2\), and Luke's weight is approximately \(1200\,\text{N}\).