7 The acceleration is exactly what one expects for a mass m 1 +m 2
8 Using Newton’s Laws General Method Determine the object, or objects, of interest. Determine real forces acting on each object. For each object, find the net force. Insert the net force into the 2 nd law and solve:
19 Circular Motion is a unit vector that points from the center of the circle to the object. r is the velocity of the object. As the object moves around the circle, the direction of the velocity changes as does the direction of the unit vector.
20 Circular Motion The acceleration is directed towards the center of the circle, that is, it is centripetal, and its magnitude is r
21 Example 5.7 – Loop-the-Loop! What is the minimum speed needed to guarantee that a roller-coaster car stays on the track at the top of the loop? Identify forces on car Gravity Normal force from track
22 Example 5.7(2) We have just seen that to move in a circle, an object must have a centripetal acceleration. According to the 2 nd law, the acceleration is caused by a net force.
23 Example 5.7(3) Since the acceleration is in the same direction as the net force, it follows that the net force must be centripetal, that is, directed towards the center of the loop. What are these forces?
24 Example 5.7(4) Presumably, they must be the two forces we have identified: the weight and the normal force. As usual, we need to set up a coordinate system.
25 Example 5.7(5) Coordinate system Take +y to be downwards Take +x to the right
26 Example 5.7(6) At the top of the loop, the normal force and the gravitational force point downwards and towards the center of the circle. Therefore, in the y direction
27 Example 5.7(7) Solving for v we get At the minimum speed the car is on the verge of leaving the tracks at the top of the loop. This occurs when the normal force, n, is zero!
29 The Nature of Friction Friction is an electrical force between the molecules of surfaces in contact. Unlike gravity, however, friction is a very complicated force to describe accurately.
30 The Nature of Friction But, for many everyday situations, such as dragging an object along a floor, we can describe frictional forces using simple, approximate, expressions.
31 Frictional Forces Static Friction – This is the frictional force between surfaces that are at rest relative to each other. The maximum static frictional force is found to be f s = μ s n where n is the magnitude of the normal force. μ s is called the coefficient of static friction.
32 Frictional Forces Kinetic Friction – This is the frictional force between surfaces that are moving relative to each other. Its value is found to be f k = k n where n is the magnitude of the normal force. k is called the coefficient of kinetic friction.
33 Frictional Forces It is found that as the applied force increases so does the opposing frictional force until a maximum value is reached. When the applied force exceeds the maximum frictional force the object accelerates. During acceleration the frictional force decreases and remains constant when the motion is constant.
34 Frictional Forces Constant speed Time Frictional force Maximum frictional force Accelerating At rest Frictional force remains equal to and opposite the applied force. s ns n k nk n
35 Friction in Action Without friction it would be impossible to walk or make a vehicle move. As you push against the ground, the ground pushes you forwards!
36 Example – Dragging a Box What rope tension is needed to move the box at constant velocity, assuming a coefficient of kinetic friction k between box and floor?
37 Example – Dragging a Box Draw free-body diagram for box. The magnitude of the kinetic friction force is f k = μ k n y x
38 Example – Dragging a Box The motion is constant, so the forces cancel: – f k + T cos = 0(x-dir.) –mg + n + T sin = 0(y-dir.) y x
39 Example – Dragging a Box The magnitude of the tension is therefore: y x
40 Summary Big Idea: A net force causes changes in motion. How to Apply: Find all real forces on a body, sum them, and apply Newton’s 2 nd and 3 rd laws. Frictional force Increases until object moves, then reduces and remains constant when motion is constant.