Physics 170 - Mechanics Lecture 25 Torque Torque From experience, we know that the same force will be much more effective at rotating an object such as a nut or a door if our hand is not too close to the axis. This is why we have long-handled wrenches, and why the doorknobs are not next to the hinges. Torque We define a quantity called torque: The torque increases as the force increases and also as the distance increases. Torque The ability of a force to cause a rotation or twisting motion depends on three factors: 1. The magnitude F of the force; 2. The distance r from the point of application to the pivot; 3. The angle φ at which the force is applied. = r x F Torque This leads to a more general definition of torque: Two Interpretations of Torque Torque Only the tangential component of force causes a torque: Sign of Torque If the torque causes a counterclockwise angular acceleration, it is positive; if it causes a clockwise angular acceleration, it is negative. Sign of Torque Question Five different forces are applied to the same rod, which pivots around the black dot. Which force produces the smallest torque about the pivot? (a) (b) (c) (d) (e) Gravitational Torque An object fixed on a pivot (taken as the origin) will experience gravitational forces that will produce torques. The torque about the pivot from the ith particle will be τi=−ximig. The minus sign is because particles to the right of the origin (x positive) will produce clockwise (negative) torques. Example: The Gravitational Torque on a Beam The 4.0 m long 500 kg steel beam is supported 1.20 m from the right end. What is the gravitational torque about the support? Torque & Angular Acceleration Newton’s Second Law: If we consider a mass m rotating around an axis a distance r away, we can reformat Newton’s Second Law to read: Or equivalently, Torque & Angular Acceleration Once again, we have analogies between linear and angular motion: F = m a τ = I α Problem Solving Strategy Picture: Angular acceleration for rigid objects can be found by using free-body diagrams and Newton’s 2nd Law for rotation, τnet ext = Στext = Iα. If τnet ext is constant, then the constant angular acceleration equations apply. Time intervals and angular positions, angular velocities, and angular accelerations can be determined using these equations. Solve: 1. Draw a free-body diagram with the object shown as a likeness of the object (not as a dot). 2. Draw each force vector along the line of action of that force. 3. On the diagram, indicate the positive direction (cw or ccw) for rotations. Check: Make sure that the signs of your results are consistent with your choice of positive directions of rotation. Example: A Stationary Bike To get some exercise without going anywhere, you set your bike on a stand with the rear wheel free to turn. As you pedal, the chain applies a force of F = 18 N to the rear sprocket wheel at a distance of rs = 7.0 cm from the rotation axis of the wheel. Consider the wheel to be a hoop of radius R = 35 cm and mass M = 2.4 kg. What is the angular velocity ω of the wheel after 5.0 s? Dynamic Applications of Torque When dealing with systems that have both rotating parts and translating parts, we must be careful to account for all forces and torques correctly and to keep the signs straight. Example: Pulley and Block A 2.54-kg block is attached to a disk-shaped pulley of radius 0.108 m and mass 0.816 kg. (a) If the block is allowed to fall, what is its linear acceleration? (b) What is the angular acceleration of the pulley? (c) How far does the block drop in 1.80 s? Example: A Uniform Rod Pivoted at an End A uniform thin rod of length L and mass M is pivoted at one end. It is held horizontal and released. Neglect friction and air drag. (a) Find the angular acceleration α of the rod immediately after its release. (b) Find the magnitude of the force FA exerted by the rod on the pivot at that instant. Example: A Rolling Sphere A solid sphere with radius r=0.10 m and mass m=0.50 kg rolls down an inclined plane that makes an angle θ=30o with the horizontal. What is the angular acceleration α of the sphere? What is the linear acceleration a of the sphere? 2 ( = ) (The static friction exerts a torque about the center of mass).