A car is traveling up a hill that is inclined at an angle θabove the horizontal. Determine the r

step 1 In general, we have a car that is going up a hill with an inclination given by theta. So, what i

A car is traveling up a hill that is inclined at an angle...

A car is traveling up a hill that is inclined at an angle $\theta$ above the horizontal. Determine the ratio of the magnitude of the normal force to the weight of the car when (a) $\theta=15^{\circ}$ and $(\mathbf{b}) \theta=35^{\circ}$.

Key Concepts

Inclined Plane

An inclined plane is a flat surface tilted at an angle relative to the horizontal. In mechanics, it is used to analyze forces acting on objects such as gravitational forces, which must be decomposed into components parallel and perpendicular to the plane, making it a common setting for problems involving motion and force balance.

Normal Force

The normal force is the perpendicular reaction force exerted by a surface on an object in contact with it. For an object on an inclined plane, the normal force is less than the object’s weight because it only counteracts the component of weight perpendicular to the plane, typically calculated using the cosine of the inclination angle.

Weight

Weight is the force exerted on an object due to gravity, usually expressed as the product of mass and gravitational acceleration (mg). On an incline, weight is decomposed into two components: one perpendicular to the surface, which is balanced by the normal force, and one parallel to the surface, which influences the object’s tendency to slide.

Trigonometric Decomposition

Trigonometric decomposition involves breaking a vector into its perpendicular components using sine and cosine functions. In problems involving inclined planes, this method is crucial for determining the magnitude of the normal force (mg cos?) and the component of weight acting parallel to the incline (mg sin?), thereby facilitating the analysis of forces in different directions.

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