Figure 6-78 shows a multiple-exposure photograph of a ball...

Figure $6-78$ shows a multiple-exposure photograph of a ball rolling up an inclined plane. (The ball is rolling in the dark, the camera lens is held open, and a brief flash occurs every $3 / 4$ sec, four times in total.) The leftmost ball corresponds to an instant just after the ball was released. The rightmost ball is at the highest point the ball reaches. (a) Copy this picture on your paper. Draw an arrow at each of the four ball locations to indicate the velocity of the ball at that instant. Make the relative lengths of the arrows indicate the relative magnitudes of the velocities. Explain what is happening ("tell the story" of the picture). (b) For the instant of time when the ball is at the second position shown from the left, draw a free-body diagram for the ball and indicate all forces acting on it. (c) If your force diagram doesn't include an arrow pointing up the ramp, explain why the ball keeps rolling up the ramp. (d) If the mass of the ball is $m$, what is its acceleration? (e) If the angle $\theta$ is equal to $30^{\circ}$, how long is the distance $s$ ?

Figure $6-78$ shows a multiple-exposure photograph of a ball rolling up an inclined plane. (The ball is rolling in the dark, the camera lens is held open, and a brief flash occurs every $3 / 4$ sec, four times in total.) The leftmost ball corresponds to an instant just after the ball was released. The rightmost ball is at the highest point the ball reaches.
(a) Copy this picture on your paper. Draw an arrow at each of the four ball locations to indicate the velocity of the ball at that instant. Make the relative lengths of the arrows indicate the relative magnitudes of the velocities. Explain what is happening ("tell the story" of the picture).
(b) For the instant of time when the ball is at the second position shown from the left, draw a free-body diagram for the ball and indicate all forces acting on it.
(c) If your force diagram doesn't include an arrow pointing up the ramp, explain why the ball keeps rolling up the ramp.
(d) If the mass of the ball is $m$, what is its acceleration?
(e) If the angle $\theta$ is equal to $30^{\circ}$, how long is the distance $s$ ?

Key Concepts

Uniform Acceleration on an Inclined Plane

When analysing motion on an inclined plane without friction, the object experiences constant acceleration determined solely by the component of gravitational force along the ramp. This acceleration is given by g sin ?, which is a direct consequence of decomposing gravity into parallel and perpendicular components relative to the incline.

Newton's Laws and Inertia

Newton's first law states that an object in motion continues in its state of motion unless acted upon by a net external force. In the context of this problem, even though no applied force is pushing the ball up the ramp, its inertia keeps it moving upward until gravity (through its parallel component) decelerates it to a stop and then accelerates it in the opposite direction.

Vector Representation of Velocity

Vector representation involves depicting physical quantities like velocity with arrows, where the direction indicates the direction of the velocity and the arrow's length corresponds to its magnitude. In problems like this, drawing velocity vectors at different time intervals helps visualize the changes in speed and the effects of acceleration due to gravity along the inclined plane.

Free-Body Diagram

A free-body diagram is a simplified representation of an object that isolates it from its environment and shows all of the external forces acting on it. It helps in visualizing and analyzing the forces and their directions, making it easier to apply Newton's laws of motion to determine the object's acceleration or other dynamics.

Force Decomposition on an Inclined Plane

This concept involves breaking down the gravitational force into two components when an object is on an incline: one perpendicular to the surface and the other parallel to it. The perpendicular component is balanced by the normal force of the surface, while the parallel component (given by mg sin ?) is responsible for the acceleration or deceleration of the object along the plane.

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