PHY 131 Chapter 8-Part 1

Objectives Calculate the torque created by a force. Define the center of mass of an object. Describe a technique for finding the center of mass of an irregularly shaped object. Calculate the moment of inertia for a mass rotating on the end of a rod. Describe the relationship between torque, angular acceleration, and rotational inertia.

Torque Key Question: How does force create rotation?

Torque A torque is an action that causes objects to rotate. Torque is not the same thing as force. For rotational motion, the torque is what is most directly related to the motion, not the force.

Torque Motion in which an entire object moves is called translation. Motion in which an object spins is called rotation. The point or line about which an object turns is its center of rotation. An object can rotate and translate.

Torque Torque is created when the line of action of a force does not pass through the center of rotation. The line of action is an imaginary line that follows the direction of a force and passes though its point of application.

Torque To get the maximum torque, the force should be applied in a direction that creates the greatest lever arm. The lever arm is the perpendicular distance between the line of action of the force and the center of rotation

Torque-Formula Lever arm length (m) t = r x F Torque (N.m) Force (N)

Calculate a torque A force of 50 newtons is applied to a wrench that is 30 centimeters long. 1) You are asked to find the torque. 2) You are given the force and lever arm. 3) The formula that applies is τ = rF. 4) Solve: τ = (-50 N)(0.3 m) = -15 N.m Calculate the torque if the force is applied perpendicular to the wrench so the lever arm is 30 cm.

Rotational Equilibrium When an object is in rotational equilibrium, the net torque applied to it is zero. Rotational equilibrium is often used to determine unknown forces. What are the forces (FA, FB) holding the bridge up at either end?

Rotational Equilibrium

When the force and lever arm are NOT perpendicular

Calculate a torque A 20-centimeter wrench is used to loosen a bolt. The force is applied 0.20 m from the bolt. 1) You are asked to find the force. 2) You are given the force and lever arm for one condition. 3) The formula that applies is τ = rF. 4) Solve: The torque required to loosen the bolt τ = (50 N)(0.2 m) = 10 N.m To get the same torque with a force applied at 30 degrees: 10 N.m = F × (0.2 m)cos30o = 0.173 F F = 10 N.m ÷ 0.173 = 58 N. It takes a larger force. It takes 50 newtons to loosen the bolt when the force is applied perpendicular to the wrench. How much force would it take if the force was applied at a 30-degree angle from perpendicular?

Center of Mass Key Question: How do objects balance?

Center of Mass There are three different axes about which an object will naturally spin. The point at which the three axes intersect is called the center of mass.

Finding the center of mass If an object is irregularly shaped, the center of mass can be found by spinning the object and finding the intersection of the three spin axes. There is not always material at an object’s center of mass.

Finding the center of gravity The center of gravity of an irregularly shaped object can be found by suspending it from two or more points. For very tall objects, such as skyscrapers, the acceleration due to gravity may be slightly different at points throughout the object.

Balance and center of mass For an object to remain upright, its center of gravity must be above its area of support. The area of support includes the entire region surrounded by the actual supports. An object will topple over if its center of mass is not above its area of support.

Rotational Inertia Key Question: Does mass resist rotation the way it resists acceleration?

Rotational Inertia Inertia is the name for an object’s resistance to a change in its motion (or lack of motion). Rotational inertia is the term used to describe an object’s resistance to a change in its rotational motion. An object’s rotational inertia depends not only on the total mass, but also on the way mass is distributed.

Linear and Angular Acceleration-Formula Angular acceleration (kg) Linear acceleration (m/sec2) a = a r Radius of motion (m)

Rotational Inertia To put the equation into rotational motion variables, the force is replaced by the torque about the center of rotation. The linear acceleration is replaced by the angular acceleration.

Rotational Inertia A rotating mass on a rod can be described with variables from linear or rotational motion.

Rotational Inertia The product of mass × radius squared (mr2) is the rotational inertia for a point mass where r is measured from the axis of rotation.

Rotation and Newton's 2nd Law If you apply a torque to a wheel, it will spin in the direction of the torque. The greater the torque, the greater the angular acceleration.