If you stand on a bathroom scale, the spring inside the scale compresses 0.60 mm, and it tells you your weight is 760 N. Now if you jump on the scale from a height of 1.0 m, what does the scale read at its peak? Assume Hooke’s law holds.

Verified step by step guidance

1. Determine the spring constant (k) of the scale using Hooke's law: \( F = k \Delta x \). Rearrange to solve for \( k \): \( k = \frac{F}{\Delta x} \). Substitute \( F = 760 \ \text{N} \) and \( \Delta x = 0.60 \ \text{mm} = 0.0006 \ \text{m} \).
2. Calculate the velocity just before impact using the conservation of energy principle. The potential energy at the height \( h \) is converted into kinetic energy at the point of impact: \( mgh = \frac{1}{2}mv^2 \). Rearrange to solve for \( v \): \( v = \sqrt{2gh} \). Substitute \( g = 9.8 \ \text{m/s}^2 \) and \( h = 1.0 \ \text{m} \).
3. Determine the additional force exerted on the scale at the peak compression due to the impact. Use the relationship \( F = ma \), where \( a \) is the deceleration caused by the spring. The deceleration can be found using \( a = \frac{v^2}{2\Delta x} \), where \( \Delta x \) is the additional compression of the spring.
4. Combine the forces acting on the scale at the peak. The total force is the sum of the static weight (760 N) and the additional force due to the impact. Use \( F_{\text{total}} = F_{\text{static}} + F_{\text{impact}} \).
5. Substitute all known values into the equations to find the total force \( F_{\text{total}} \) that the scale reads at its peak. Ensure all units are consistent throughout the calculations.

Key Concepts

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position, as long as the elastic limit is not exceeded. Mathematically, it is expressed as F = kx, where F is the force, k is the spring constant, and x is the displacement. This principle is crucial for understanding how the bathroom scale measures weight and how it will respond to additional forces when jumping.

Weight and Gravitational Force

Weight is the force exerted on an object due to gravity and is calculated as the product of mass and gravitational acceleration (W = mg). In this scenario, the weight of the person is given as 760 N, which indicates the gravitational force acting on them. Understanding weight is essential for determining how the scale will react when additional forces are applied, such as during a jump.

Kinetic and Potential Energy

Kinetic energy is the energy of an object due to its motion, while potential energy is the stored energy based on an object's position in a gravitational field. When the person jumps from a height of 1.0 m, they convert potential energy into kinetic energy as they fall. At the peak of the jump, the scale will read a maximum force due to the combined effects of the person's weight and the additional force from the jump, which can be analyzed using energy conservation principles.