Lab Report: Determining the Optimal Angle to Drag a Box with
Minimal Applied Force
Course: AP Physics C: Mechanics
Date: August 31st, 2025
Lab Members: Jimmy T., Hanry L., Rithik K.
1.0 Introduction:
The objective of this lab was to determine and calculate for the optimal angle, , to
pull a box so that the applied for required to initiate some motion is minimized
while simultaneously maximizing horizontal acceleration. Pulling a box with an
angle, , involves both a horizontal component, which contribute to the acceleration
of the box, and a vertical component, in which it alters the normal force and
consequently the friction opposing motion
The static friction force is given by:
s = µsN
Where µs is the coefficient of static friction and N is the normal force. Optimizing
requires a balance: pulling too steeply increases the vertical component, which
causes it to reduce the normal force (and friction) but decreases the “horizontal
contribution” to acceleration; pulling to flat causes an increase in friction and will
require a greater force to get the box to move.
This lab incorporates three approaches to determine the optimal pulling
angle:
, 1. Calculus-Based Mathematical Optimization (DERIVATIVE)
By articulating the pulling force as a function of θ and taking the derivative,
we could find the mathematically optimal angle that minimizes the required force.
This method is “elegant”, “demure”, and provides understanding into the physics by
revealing how the horizontal and vertical components of the pulling force interact
through relationships.
2. Spreadsheet Modeling, Brute Force (MODEL)
Excel was used to simulate the applied force across a range of angles (0–90°)
using the derived equation. This brute-force modeling provides a visual
confirmation of the derivative solution and allows for practical verification without
solving derivatives manually.
3. Experimental Determination (EXPERIMENTAL)
A wooden box containing weights (total mass = 0.25 kg) was used for the
experiment. A string was attached to the box and connected to a Vernier Force
Sensor, in which we measured the applied force as the box was pulled. A protractor
was used to measure the pulling angle accurately. Trials were performed at the
predicted optimal angle (≈ 23°) and at angles ±5 ° and ± 10° around it. For each
trial, the box was pulled firmly until motion was visible, and the maximum force
recorded by the sensor was used as the applied force value.
The combination of these methods allows for a thorough comparison between
theoretical, modeled, and experimental results, highlighting the reliability of
physics principles in both idealized and real-world scenarios.
Given Values:
• Mass of the box, m = 0.25 kg
• Coefficient of static friction, µs = 0.4
• Gravitational acceleration, g = 9.81 m/s2
1.1 Procedure:
Finding the friction coefficient of our table setting:
For simplicity, we can just pull the box at 0 degrees, since the friction static
coefficient will be the same regardless of if with an angle for not. The work shown
below shows one of the calculations and graph for finding the table’s frictional
static coefficient. We did multiple trials for accuracy, so below will also have a table
of the different coefficients and the average. We want to do this for accuracy of the
Minimal Applied Force
Course: AP Physics C: Mechanics
Date: August 31st, 2025
Lab Members: Jimmy T., Hanry L., Rithik K.
1.0 Introduction:
The objective of this lab was to determine and calculate for the optimal angle, , to
pull a box so that the applied for required to initiate some motion is minimized
while simultaneously maximizing horizontal acceleration. Pulling a box with an
angle, , involves both a horizontal component, which contribute to the acceleration
of the box, and a vertical component, in which it alters the normal force and
consequently the friction opposing motion
The static friction force is given by:
s = µsN
Where µs is the coefficient of static friction and N is the normal force. Optimizing
requires a balance: pulling too steeply increases the vertical component, which
causes it to reduce the normal force (and friction) but decreases the “horizontal
contribution” to acceleration; pulling to flat causes an increase in friction and will
require a greater force to get the box to move.
This lab incorporates three approaches to determine the optimal pulling
angle:
, 1. Calculus-Based Mathematical Optimization (DERIVATIVE)
By articulating the pulling force as a function of θ and taking the derivative,
we could find the mathematically optimal angle that minimizes the required force.
This method is “elegant”, “demure”, and provides understanding into the physics by
revealing how the horizontal and vertical components of the pulling force interact
through relationships.
2. Spreadsheet Modeling, Brute Force (MODEL)
Excel was used to simulate the applied force across a range of angles (0–90°)
using the derived equation. This brute-force modeling provides a visual
confirmation of the derivative solution and allows for practical verification without
solving derivatives manually.
3. Experimental Determination (EXPERIMENTAL)
A wooden box containing weights (total mass = 0.25 kg) was used for the
experiment. A string was attached to the box and connected to a Vernier Force
Sensor, in which we measured the applied force as the box was pulled. A protractor
was used to measure the pulling angle accurately. Trials were performed at the
predicted optimal angle (≈ 23°) and at angles ±5 ° and ± 10° around it. For each
trial, the box was pulled firmly until motion was visible, and the maximum force
recorded by the sensor was used as the applied force value.
The combination of these methods allows for a thorough comparison between
theoretical, modeled, and experimental results, highlighting the reliability of
physics principles in both idealized and real-world scenarios.
Given Values:
• Mass of the box, m = 0.25 kg
• Coefficient of static friction, µs = 0.4
• Gravitational acceleration, g = 9.81 m/s2
1.1 Procedure:
Finding the friction coefficient of our table setting:
For simplicity, we can just pull the box at 0 degrees, since the friction static
coefficient will be the same regardless of if with an angle for not. The work shown
below shows one of the calculations and graph for finding the table’s frictional
static coefficient. We did multiple trials for accuracy, so below will also have a table
of the different coefficients and the average. We want to do this for accuracy of the
