Physics 1301: Lecture 13 - Springs and Weight

Physics 1301: Lecture 13

Springs and Weight

Date: February 19th
Speaker: Jason Alicea (Caltech)
Title: "Measurement-altered quantum matter"
Location: Tate B50
Note: Cookies available after colloquium

Introduction to Springs

• Springs are typically constructed from a coil of wire.
• When a spring is deformed from its natural length, it tries to return to that length by applying a restoring force (denoted as F).

Force Exerted by Springs

• The amount of force F exerted by a spring depends on the change in length from its natural length.
• If x0 is the natural length and x is the new length, the spring applies a force which can be quantified as: $F(x - x_0)$
  • This implies that the force is a function of the difference between the current length and the natural length, which can be expressed as:
    $F(x - x_0)$

Taylor Series and Spring Behavior

Taylor's Theorem

• All reasonably smooth functions can be approximated by a polynomial sum known as Taylor series.
• For a function f(x), it can be constructed from simpler functions as follows:
  $f(x) ext{ can be expressed using } 1, (x - x<em>0), (x - x</em>0)^2, (x - x_0)^3, ext{ etc.}$
• Mathematicians refer to this approximation method as Taylor's Theorem.

Small Change Approximation

• When the change in x (denoted as Δx) is small, the higher-order terms of the Taylor series become less significant:
  • For example, if x is close to x0, then $(x - x<em>0)$ is small, $(x - x</em>0)^2$ is even smaller, leading to very good approximations using only the first few terms:
    $f(x) ext{ approximated by lower order terms when } |x - x_0| ext{ is small.}$

Hooke's Law

Definition

• Hooke's Law states that the force exerted by a spring is proportional to the change in length from the natural length and acts in the opposite direction: $F = -k(x - x_0)$
  • Here, k is the spring constant, representing the stiffness of the spring.
  • In this formula, Δx represents the change in length defined as Δx = x - x0.
• Hooke's Law is only valid within certain limits; although it can apply to larger deformations in practice, it primarily holds for small displacements.

Assumptions of Ideal Springs

• For the purposes of analysis, we assume that springs are ideal, which means they:
  • Obey Hooke's Law.
  • Are considered massless.
  • Maintain uniform tension throughout the length of the spring.
  • Perfectly transmit forces applied to them.

Examples of Spring Forces

Example Scenario

• Identical Springs Under Load:
  • If two identical springs are used, one with a 1 kg weight and the other with a 2 kg weight, we can measure how far each weight displaces the spring from its initial position.
  • This provides a basis for understanding the calculated difference in displacement (noted as Δx).

Stacking Springs

• When stacking springs end-to-end, relation between the total length L' and the individual spring lengths can be established.
  • For example, if weight m and additional effects are considered, the equations would arise from balancing the forces acting on the combined spring system.
  • The relation can be represented mathematically as:
    $L' = 2L$
  • This equation illustrates how stacking affects the total spring length under load.

Applications of Springs

Springs as Scales

Measuring Weight with Springs

• Springs can be utilized to measure force due to the linear relationship established by Hooke’s Law.
• Since weight is a force (defined as gravitational force), springs offer an effective means of measuring weight changes:
  • Calibration Process:
  • To calibrate a spring scale, measure the spring length under known weights. From this, determine the spring constant k along with the baseline spring length L0.
  • Using calibrated measurements, the weight of an unknown object can be inferred from the change in spring length during the test.

Example Calibration Measurements

• Given a set of weights:
  • For weight w1 = 5N, water level is measured as L1 = 5m.
  • For weight w2 = 3N, water level is L2 = 4m.
  • For an unknown weight w3, measured against L3 = 6m.

Springs in Non-Standard Conditions

Example of Weight Measurement in an Elevator

• Scenario: In an upward-accelerating elevator, check suitcase weight with a spring scale.
  • If the suitcase weighs 10 kg and the elevator accelerates upwards at 5 m/s², the scale reading can be calculated as follows:
  • Using F = ma, where net force applied = T - mg, we can find the tension in the spring scale under acceleration:
    $T = mg + ma$
  • Hence, replacing with given values, $(m = 10 ext{ kg}, a = 5 ext{ m/s}^2)$ provides the reading of:
    $T = 10kg imes (10 ext{ m/s}^2 + 5 ext{ m/s}^2) = 150N$

Free Fall Situation

• Scenario: If the elevator cable is cut while a person is holding the spring scale, the reading would drop to zero due to free fall conditions:
  • This can be expressed as: $T - mg = -mg$
  • Hence, the scale reads zero as free fall means no tension acting on the spring.