Math Problem Statement

A 74.8 kg man pushes his daughter in a stroller (combined mass of the stoller and the daughter is 19.2 kg) on a flat surface, by exerting a force of 90 N at 18degrees below the horizontal. If the coefficient of friction is 0.235, determine the net acceleration of the stoller if it moves horizontally along the level surface

Solution

We can solve this problem step by step. Here's how to calculate the net acceleration of the stroller:

Step 1: Resolve the applied force into horizontal and vertical components

The applied force (FappliedF_{\text{applied}}) is 90 N at 18° below the horizontal:

  • Horizontal component: Fx=Fappliedcos(θ)=90cos(18°)F_x = F_{\text{applied}} \cdot \cos(\theta) = 90 \cdot \cos(18°)
  • Vertical component: Fy=Fappliedsin(θ)=90sin(18°)F_y = F_{\text{applied}} \cdot \sin(\theta) = 90 \cdot \sin(18°)

Step 2: Calculate the normal force

The normal force (FNF_N) is influenced by the vertical component of the applied force and the weight of the stroller: FN=mstrollergFyF_N = m_{\text{stroller}} \cdot g - F_y where mstroller=19.2kgm_{\text{stroller}} = 19.2 \, \text{kg} and g=9.8m/s2g = 9.8 \, \text{m/s}^2.

Step 3: Calculate the force of friction

The frictional force (FfF_f) is given by: Ff=μFNF_f = \mu \cdot F_N where μ=0.235\mu = 0.235.

Step 4: Find the net horizontal force

The net horizontal force (FnetF_{\text{net}}) is the horizontal applied force minus the frictional force: Fnet=FxFfF_{\text{net}} = F_x - F_f

Step 5: Calculate the net acceleration

The acceleration of the stroller (aa) can be calculated using Newton's second law: a=Fnetmstrollera = \frac{F_{\text{net}}}{m_{\text{stroller}}}

Let me calculate this for you.import math

Given data

m_stroller = 19.2 # mass of stroller in kg m_man = 74.8 # mass of man in kg (not directly needed here) F_applied = 90 # applied force in N theta = 18 # angle in degrees mu = 0.235 # coefficient of friction g = 9.8 # acceleration due to gravity in m/s^2

Step 1: Resolve applied force into components

F_x = F_applied * math.cos(math.radians(theta)) # horizontal component F_y = F_applied * math.sin(math.radians(theta)) # vertical component

Step 2: Calculate normal force

F_N = (m_stroller * g) - F_y

Step 3: Calculate frictional force

F_f = mu * F_N

Step 4: Net horizontal force

F_net = F_x - F_f

Step 5: Calculate net acceleration

a = F_net / m_stroller

a2.49547895299968532.4954789529996853successThe net acceleration of the stroller is approximately 2.50 m/s².

Do you want me to delve deeper into any specific part of the solution? Here are some related questions:

  1. How does the angle of the force affect the acceleration?
  2. What happens if the coefficient of friction changes?
  3. How would adding mass to the stroller influence the result?
  4. What if the force were applied horizontally instead of at an angle?
  5. How do we calculate the normal force when friction is negligible?

Tip: Always resolve forces into components when dealing with forces at an angle to simplify the calculations.

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Math Problem Analysis

Mathematical Concepts

Force resolution
Newton's second law of motion
Frictional forces
Normal force calculation

Formulas

F_x = F_applied * cos(θ)
F_y = F_applied * sin(θ)
F_N = m * g - F_y
F_f = μ * F_N
a = F_net / m

Theorems

Newton's second law of motion

Suitable Grade Level

Grades 9-12