Question

To understand two different techniques for computing the torque on an object due to an applied force. Imagine an object with a pivot point p at the origin of the coordinate system shown (Figure 1). The force vector F⃗ lies in the xy plane, and this force of magnitude F acts on the object at a point in the xy plane. The vector r⃗  is the position vector relative to the pivot point p to the point where F⃗  is applied. The torque on the object due to the forceF⃗  is equal to the cross product τ⃗ =r⃗ ×F⃗ . When, as in this problem, the force vector and lever arm both lie in the xy plane of the paper or computer screen, only the z component of torque is nonzero. When the torque vector is parallel to the z axis (τ⃗ =τk^), it is easiest to find the magnitude and sign of the torque, τ, in terms of the angleθ between the position and force vectors using one of two simple methods: the Tangential Component of the Forcemethod or the Moment Arm of the Force method. Note that in this problem, the positive z direction is perpendicular to the computer screen and points toward you (given by the right-hand rule i^×j^=k^), so a positive torque would cause counterclockwise rotation about thez axis.	Tangential component of the force Part A (Figure 2) Decompose the force vector F⃗  into radial (i.e., parallel to r⃗ ) and tangential (perpendicular tor⃗ ) components as shown. Find the magnitude of the radial and tangential components, Fr and Ft. You may assume that θ is between zero and 90 degrees. Enter your answer as an ordered pair. Express Ft and Fr in terms of F and θ. Hint 1. Magnitude of F⃗ ropened hint Use the given angle between the force vector F⃗ and its radial component F⃗ r to compute the magnitudeFr. (Fr,Ft)=nothing Part B Is the following statement true or false? The torque about point p is proportional to the length r of the position vector r⃗ . Is the following statement true or false? The torque about point p is proportional to the length of the position vector .truefalse Part C Is the following statement true or false? Both the radial and tangential components ofF⃗  generate torque about point p. Is the following statement true or false? Both the radial and tangential components of generate torque about point p.truefalse Part D Is the following statement true or false? In this problem, the tangential force vector would tend to turn an object clockwise around pivot point p. Is the following statement true or false? In this problem, the tangential force vector would tend to turn an object clockwise around pivot point p.truefalse Part E Find the torque τ about the pivot point p due to forceF⃗ . Your answer should correctly express both the magnitude and sign of τ. Express your answer in terms of Ft and r or in terms of F, θ, and r. τ=nothing Moment arm of the force In the figure, the dashed line extending from the force vector is called the line of action of F⃗ . The perpendicular distance rm from the pivot point p to the line of action is called the moment arm of the force. Part F (Figure 3) What is the length, rm, of the moment arm of the forceF⃗ about point p? Express your answer in terms of r and θ. rm=nothing Part G Find the torque τ about p due to F⃗ . Your answer should correctly express both the magnitude and sign ofτ. Express your answer in terms of rm and F or in terms of r, θ, and F. τ=nothing
A pivot point p at the origin of the xy plane is shown in the figure. The vector r is the position vector relative to the pivot point p to the point where force F is applied. Vector r lies in the first quadrant. Vector F starts from the end of r and is directed downward and to the right. The acute angle between F and straight line containing r is labeled theta. The perpendicular distance from point p to the line of action of F is labeled as r m. A positive torque would cause counterclockwise rotation about the z-axis perpendicular to the plane of the figure.

Answer 1