Show that the system of momenta for a rigid body in plane motion reduces to a single vector, and express the distance from the mass center G to the line of action of this vector in terms of the centroidal radius of gyration k of the body, the magnitude v of the velocity of G, and the angular velocity w.
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Show that the system of momenta for a rigid body in plane motion reduces to a single vector, and express the distance from the mass center G to the line of action of this vector in terms of the centroidal radius of gyration k of the body, the magnitude v of the velocity of G, and the
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- (a) The magnitude of the angular momentum about the origin of a particle of mass m moving with velocity v on a path that is a perpendicular distance d from the origin is given by m/v|d. Show that if r is the position of the particle then the vector J =r × mv represents the angular momentum. (b) Now consider a rigid collection of particles (or a solid body) rotating about an axis through the origin, the angular velocity of the collection being represented by w. (i) Show that the velocity of the ith particle is Vi = w X ri and that the total angular momentum J is J = Σm₁ [r}w - (r; · w)r;]. (ii) Show further that the component of J along the axis of rotation can be written as Iw, where I, the moment of inertia of the collection about the axis or rotation, is given by 1 = Σm₁p². Interpret pi geometrically. (iii) Prove that the total kinetic energy of the particles is 1².From a uniform disk of radius R, a circular hole of radius 2R/3 is cut out. The centre of the hole is at 2R/3 from the centre of the original disc.Locate the centre of gravity of the resulting flat body. Write the center of mass remainingThe specialty wrench shown in the figure is designed for access to the hold-down bolt on certain automobile distributors. For the configuration shown, where the wrench lies in a vertical plane and a horizontal 120-N force F is applied at A perpendicular to the handle, calculate the moment Mo applied to the bolt at O. For what value of the distance d would the z-component of Mo be zero? Assume a = 65 mm, d = 105 mm, h = 200 mm, 0 = 27°. F Answers Mo (i d = i A wanaosaRKEK wwwwwwww 1 0 a ₪12 d h Y_-_y i+ i mm j+ Mi k) N-m
- Engineering Dynamics need help from 4,5,6,7 thank you A ball of mass m is moving along a vertical semi-cylinder of radius R as it is guided by the arm OA. The arm moves in a clockwise direction with a constant angular velocity ω. Assume 0° ≤ Φ ≤ 90°. Neglect any friction. Neglect also the size of the ball and the thickness of the arm. Find the relationship between r, R and θ where r is the distance between O and the ball. Draw a free body diagram of the ball assuming that it is in contact with the cylinder and the arm OA. Write the equations of motion in the (r, θ) coordinate system. Find the normal force acting on the ball by the cylinder for Φ = Φ0. Find the normal force acting on the ball by the bar for Φ = Φ0. Determine the angle Φ at which the ball loses contact with the cylinder. Take m = 1 kg, R = 1.4 m, ω = 0.5 rad/s, and Φ = 60°A stick of length L and mass M1 is in free space (no gravity) and not rotating. A point mass m2 hasinitial velocity v heading in a trajectory perpendicular to the stick. The mass has a perfectly inelasticallycollision a distance b from the center of the stick. Find the velocity of the center of mass and the finalangular velocity.A particle of mass m is located at x = 1, y = 0,2 = 2. Find the tensor of inertia for the particle relative to the origin. The particle rotates about the z axis through a small angle a <<1 as shown below. Show that the moments of inertia are unchanged to second order in a but the products of inertia can change linearly with a.
- Problem 2: Equivalent force with no moment vector For an object shown in the figure below, a force and moment vector act at point A. The force is F= 20 [N]i + 15 [N]j and the moment is M, = 10 (Nm) k. Find a point on the object such that the equivalent system at that point consists of only a force (zero moment vector). Hint: Place the origin of you coordinate system at point A. The answer is going to be that the force can act on any point along a line.Two identical billiard balls can move freely on a horizontal table. Ball A has a velocity 10 m/s as shown and hits ball B, which is at rest, at an angle θ = 45°. Knowing that the coefficient of restitution between the two balls is e = 0.9 and assuming no friction, determine the velocity of each ball after impact.