Braking Load A Toyota Sienna with a gross weight of 5300...

Braking Load A Toyota Sienna with a gross weight of 5300 pounds is parked on a street with an $8^{\circ}$ grade. See the figure. Find the magnitude of the force required to keep the Sienna from rolling down the hill. What is the magnitude of the force perpendicular to the hill? (IMAGE CANT COPY)

Key Concepts

Gravitational Force

This is the force that attracts the vehicle toward the Earth's center. It is directly proportional to the mass of the object and is often represented by the weight of the object on Earth. The gravitational force is the primary driver causing the vehicle to attempt to roll down the hill when it is parked on an incline.

Force Decomposition

Force decomposition involves breaking a force into its components along specific directions. In this context, the gravitational force acting on the vehicle can be resolved into components parallel and perpendicular to the angled road surface. This technique utilizes trigonometric functions—sine for the component parallel to the incline and cosine for the component perpendicular to the incline.

Inclined Plane Mechanics

This concept deals with the analysis of objects on slopes or ramps. It requires understanding how forces like gravity can be resolved into components that affect motion along the plane and the normal force exerted by the surface. Inclined plane mechanics is fundamental in solving problems related to objects at rest on an inclined surface, where friction or additional forces may be acting to prevent motion.

Static Equilibrium

Static equilibrium refers to the state of an object where all the forces acting upon it are balanced, resulting in no net acceleration. For a vehicle parked on a slope, it is crucial that the component of gravity attempting to pull it downhill is counteracted by friction, brakes, or another applied force. This balance of forces ensures that the vehicle remains stationary and is a key consideration in engineering and physics problems involving objects at rest.

Transcript

00:01 So we have a car weighing 5 ,300 pounds, and we have an 8 degree angle of incline.

00:09 We want to find the magnitude of the force required to keep the car from rolling down the hill, and then the magnitude of the force perpendicular to the hill.

00:19 So the force of gravity vector is 0, negative 5300, and w is going to be cosine 8, sine 8.

00:30 We're going to project f of g onto w.

00:33 So that is going to give us f of g dotted with w, dotted with cosine 8, sine 8, all over the magnitude of w, all of that squared, and then times w.

01:07 So if i keep going, if i do the dot product, 0 times cosine 8 is 0, and then negative 5300 times sine 8 is negative 5300 sign of 8 all over.

01:29 Now, sine squared plus cosine squared is always 1.

01:33 So the square root is 1, and then 1 squared is 1 times w.

01:39 So that's really just negative 5300 sine 8 times vector w.

01:52 Negative 5300 times sine of 8 is negative 737 .6.

02:04 And w was vector cossine 8, sine 8.

02:16 That's v.

02:19 And this magnitude is 1.

02:26 Okay.

02:27 So the magnitude of vector v is 737 .6.

02:39 It's got to be a length.

02:40 It's a positive amount.