Problem 74 - Algebra and Trigonometry Real Mathematics, Real People

Question

A sport utility vehicle with a gross weight of 5400 pounds is parked on a slope of \(10^{\circ} .\) Assume that the only force to overcome is the force of gravity. Find the force required to keep the vehicle from rolling down the hill. Find the force perpendicular to the hill.

Step-by-Step Solution

Answer

The force required to keep the vehicle from rolling down the hill is given by \(F_d = F_w \cdot \sin(10^{\circ})\) and the force perpendicular to the hill is given by \(F_p = F_w \cdot \cos(10^{\circ})\) where \(F_w = 5400 \times 4.448 = 24000 \, N\). Substitute \(F_w\) into these expressions to find the required forces.

Step 1: Convert weight to force

Start by converting the weight of the vehicle from pounds to force. We know that 1 pound = 4.448 Newtons, so we multiply the gross weight of the vehicle by 4.448 to convert it into force. So, \(F_w = 5400 \times 4.448 = 24000 \, N\)

Step 2: Resolve the weight into components

Next, let's break this force into two components: one that acts perpendicularly to the hill (which we will call \(F_p\)), and the one that will try to cause the vehicle to move down the hill (which we will call \(F_d\)). The component of the weight that is perpendicular to the surface of the slope is given by \(F_p = F_w \cdot \cos(10^{\circ})\). And the component of the weight that acts parallel to the slope and tries to roll the vehicle down the slope is given by \(F_d = F_w \cdot \sin(10^{\circ})\)

Step 3: Compute the components

One needs to substitute \(F_w\) from step 1 into the expressions in step 2 and compute \(F_p\) and \(F_d\). The force that needs to be overcome to prevent the vehicle from rolling down the hill is equal to \(F_d\) and the force perpendicular to the hill is \(F_p\).