Find the force required to keep a 75-lb sled from sliding down an incline that makes an angle of 27° with the horizontal. (Assume there is no friction.)

Verified step by step guidance

1. Identify the forces acting on the sled: the weight (gravity) acting vertically downward and the force required to keep the sled from sliding down the incline, which acts parallel to the incline surface.
2. Resolve the weight of the sled into two components: one perpendicular to the incline and one parallel to the incline. The component parallel to the incline causes the sled to slide down.
3. Use the formula for the component of weight parallel to the incline: \(W_{\parallel} = W \times \sin(\theta)\), where \(W\) is the weight (75 lb) and \(\theta\) is the angle of the incline (27°).
4. Since there is no friction, the force required to keep the sled from sliding is equal in magnitude and opposite in direction to the parallel component of the weight, so \(F = W_{\parallel}\).
5. Substitute the known values into the equation \(F = 75 \times \sin(27^\circ)\) to express the force required to keep the sled stationary on the incline.

Key Concepts

Resolving Forces on an Inclined Plane

When an object rests on an inclined plane, its weight can be resolved into two components: one perpendicular to the plane and one parallel to it. The parallel component causes the object to slide down, calculated as weight multiplied by the sine of the incline angle.

Force Required to Prevent Sliding

To keep the sled from sliding, an external force must counteract the component of weight pulling it down the slope. This force equals the parallel component of the weight, acting up the incline to maintain equilibrium.

Trigonometric Functions in Force Analysis

Trigonometric functions like sine and cosine relate the angle of the incline to the components of forces. Specifically, sine is used to find the component of weight parallel to the incline, essential for calculating the force needed to prevent sliding.