Question

1. draw a free body diagram for a 12 kg ball rolling down a ramp. the ball is accelerating down the ramp at 5 m/s². are the forces balanced?

Explanation:

Free Body Diagram Explanation:

To draw the free - body diagram for the 12 - kg ball rolling down the ramp:

1. Gravitational Force (\(F_g\)): The gravitational force acts vertically downwards. The magnitude of the gravitational force is given by \(F_g=mg\), where \(m = 12\space kg\) and \(g=9.8\space m/s^{2}\). So, \(F_g=12\times9.8 = 117.6\space N\). We can resolve this force into two components: one parallel to the ramp (\(F_{g\parallel}\)) and one perpendicular to the ramp (\(F_{g\perp}\)).
2. Normal Force (\(F_N\)): The normal force acts perpendicular to the surface of the ramp, opposing the perpendicular component of the gravitational force.
3. Frictional Force (\(F_f\)): The frictional force acts parallel to the ramp, opposite to the direction of motion (up the ramp), opposing the parallel component of the gravitational force.

Force Balance Analysis:

According to Newton's second law, \(F_{net}=ma\). The ball is accelerating down the ramp with \(a = 5\space m/s^{2}\). If the forces were balanced, the net force \(F_{net}\) would be zero, and the acceleration \(a\) would also be zero. But since the ball has a non - zero acceleration (\(a = 5\space m/s^{2}\)), the net force \(F_{net}=ma=12\times5 = 60\space N\) (down the ramp). This means that the magnitude of the parallel component of the gravitational force (\(F_{g\parallel}\)) is greater than the magnitude of the frictional force (\(F_f\)), so the forces are not balanced.

Free Body Diagram Sketch Description:

• Draw a dot to represent the ball.
• Draw an arrow perpendicular to the ramp, pointing away from the ramp, to represent the normal force (\(F_N\)).
• Draw an arrow vertically downward to represent the gravitational force (\(F_g\)). Then draw two dashed lines to resolve \(F_g\) into \(F_{g\parallel}\) (parallel to the ramp, down the ramp) and \(F_{g\perp}\) (perpendicular to the ramp, towards the ramp).
• Draw an arrow parallel to the ramp, up the ramp, to represent the frictional force (\(F_f\)). The length of the \(F_{g\parallel}\) arrow should be longer than the length of the \(F_f\) arrow to show that there is a net force down the ramp.

Answer:

The forces are not balanced. The free - body diagram has the gravitational force (vertically down), normal force (perpendicular to ramp), frictional force (up ramp), and the resolved components of gravity (parallel and perpendicular to ramp), with the parallel component of gravity larger than friction.