Math / Geometry

QuestionFigure 1 A sled of mass $m$ slides down a rough ramp with a constant speed $v_{0}$ . The angle between the ramp and the horizontal is $\theta$ , as shown in Figure 1. The ramp smoothly transitions to a horizontal surface. The coefficients of static and kinetic friction between the sled and the ramp are $\mu_{s}$ and $\mu_{b}$ respectively. The ramp and the horizontal surface are made of identical materials. (a) The dot in Figure 2 represents the sled when the sled is sliding down the ramp at a constant speed. Draw and label arrows that represent the forces (not components) that are exerted on the sled. Each force in your free-body diagram must be represented by a distinct arrow starting on, and pointing away from the dot.

Studdy Solution

STEP 1

1. The sled is moving at a constant speed, indicating that the net force acting on it is zero.
2. The ramp is inclined at an angle $\theta$ with respect to the horizontal.
3. The forces acting on the sled include gravitational force, normal force, and frictional force.
4. The frictional force is kinetic since the sled is moving.

STEP 2

1. Identify all forces acting on the sled.
2. Draw the free-body diagram with labeled forces.

STEP 3

Identify all forces acting on the sled:
- Gravitational Force ( $F_g$ ): Acts vertically downward. Its magnitude is $mg$ , where $m$ is the mass of the sled and $g$ is the acceleration due to gravity. - Normal Force ( $N$ ): Acts perpendicular to the surface of the ramp. It balances the component of gravitational force perpendicular to the ramp. - Frictional Force ( $F_f$ ): Acts parallel to the surface of the ramp, opposing the motion of the sled. Its magnitude is $\mu_k N$ , where $\mu_k$ is the coefficient of kinetic friction.

STEP 4

Draw the free-body diagram with labeled forces:
- Draw a dot to represent the sled. - Draw an arrow starting from the dot pointing downward to represent the gravitational force ( $F_g$ ). - Draw an arrow starting from the dot perpendicular to the inclined surface to represent the normal force ( $N$ ). - Draw an arrow starting from the dot parallel to the inclined surface, pointing up the ramp, to represent the frictional force ( $F_f$ ).
Label each arrow with the corresponding force name.
The free-body diagram should include: - A downward arrow labeled $F_g = mg$ . - A perpendicular arrow to the ramp labeled $N$ . - A parallel arrow to the ramp labeled $F_f = \mu_k N$ .

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