7.3 Forces in Two Dimensions 1

Chapter 7 Objectives Add and subtract displacement vectors to describe changes in position. Calculate the x and y components of a displacement, velocity, and force vector. Write a velocity vector in polar and x-y coordinates. Calculate the range of a projectile given the initial velocity vector. Use force vectors to solve two-dimensional equilibrium problems with up to three forces. Calculate the acceleration on an inclined plane when given the angle of incline. 2

Chapter 7 Vocabulary Cartesian coordinates component cosine displacement inclined plane magnitude parabola polar coordinates projectile Pythagorean theorem range resolution resultant right triangle scalar scale sine tangent trajectory velocity vector x-component y-component

Inv 7.3 Forces in Two Dimensions Investigation Key Question: How do forces balance in two dimensions? 4

7.3 Forces in Two Dimensions Force is also represented by x-y components.

7.3 Force Vectors If an object is in equilibrium, all of the forces acting on it are balanced and the net force is zero. If the forces act in two dimensions, then all of the forces in the x- direction and y-direction balance separately.

7.3 Equilibrium and Forces It is much more difficult for a gymnast to hold his arms out at a 45-degree angle. To see why, consider that each arm must still support 350 newtons vertically to balance the force of gravity.

7.3 Forces in Two Dimensions Use the y-component to find the total force in the gymnast’s left arm.

7.3 Forces in Two Dimensions The force in the right arm must also be 495 newtons because it also has a vertical component of 350 N.

7.3 Forces in Two Dimensions When the gymnast’s arms are at an angle, only part of the force from each arm is vertical. The total force must be larger because the vertical component of force in each arm must still equal half his weight.

7.3 The inclined plane An inclined plane is a straight surface, usually with a slope. Consider a block sliding down a ramp. There are three forces that act on the block: gravity (weight). friction the reaction force acting on the block.

7.3 Forces on an inclined plane When discussing forces, the word “normal” means “perpendicular to.” The normal force acting on the block is the reaction force from the weight of the block pressing against the ramp.

7.3 Forces on an inclined plane The normal force on the block is equal and opposite to the component of the block’s weight perpendicular to the ramp (Fy).

7.3 Forces on an inclined plane The force parallel to the surface (Fx) is given by Fx = mg sinθ.

7.3 Forces on an inclined plane The magnitude of the friction force between two sliding surfaces is roughly proportional to the force holding the surfaces together: Ff = -mg cosθ.

7.3 Motion on an inclined plane Newton’s second law can be used to calculate the acceleration once you know the components of all the forces on an incline. According to the second law: Force (kg . m/sec2) Acceleration (m/sec2) a = F m Mass (kg)

7.3 Motion on an inclined plane Since the block can only accelerate along the ramp, the force that matters is the net force in the x direction, parallel to the ramp. If we ignore friction, and substitute Newtons' 2nd Law, the net force is: Fx = m g sin θ a F = m

7.3 Motion on an inclined plane To account for friction, the horizontal component of acceleration is reduced by combining equations: Fx = mg sin θ - m mg cos θ

7.3 Motion on an inclined plane For a smooth surface, the coefficient of friction (μ) is usually in the range 0.1 - 0.3. The resulting equation for acceleration is:

Calculating acceleration A skier with a mass of 50 kg is on a hill making an angle of 20 degrees. The friction force is 30 N. What is the skier’s acceleration? You are asked to find the acceleration. You know the mass, friction force, and angle. Use relationships: a = F ÷ m and Fx = mg sinθ. Calculate the x component of the skier’s weight: Fx = (50 kg)(9.8 m/s2) × (sin 20o) = 167.6 N Calculate the force: F = 167.6 N – 30 N = 137.6 N Calculate the acceleration: a = 137.6 N ÷ 50 kg = 2.75 m/s2

7.3 The vector form of Newton’s 2nd law An object moving in three dimensions can be accelerated in the x, y, and z directions. The acceleration vector can be written in a similar way to the velocity vector: a = (ax, ay, az) m/s2.

7.3 The vector form of Newton’s 2nd law If you know the forces acting on an object, you can predict its motion in three dimensions. The process of calculating three-dimensional motion from forces and accelerations is called dynamics. Computers that control space missions determine when and for how long to run the rocket engines by finding the magnitude and direction of the required acceleration.

Calculating acceleration A 100-kg satellite has many small rocket engines pointed in different directions that allow it to maneuver in three dimensions. If the engines make the following forces, what is the acceleration of the satellite? F1 = (0, 0, 50) N F2 = (25, 0, –50) N F3 = (25, 0, 0) N You are asked to find the acceleration of the satellite. You know the mass, forces, and assume no friction in space. Use relationships: F = net force and a = F ÷ m Calculate the net force by adding components. F = (50, 0, 0) N Calculate acceleration: ay = az = 0 ax = 50 N ÷ 100 kg = 0.5 m/s2 a = (0.5, 0, 0) m/s2

Robot Navigation A Global Positioning System (GPS) receiver determines position to within a few meters anywhere on Earth’s surface. The receiver works by comparing signals from three different GPS satellites. About twenty-four satellites orbit Earth and transmit radio signals as part of this positioning or navigation system.