(a) x(t) M ᎾᎥ I 1 m
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A mass M is free to move without friction on a horizontal track. A pendulum of length l and mass m is connected to mass M, as illustrated in figure (a). Consider that x is the coordinate that
represents the position of M, and let θ be the angle between the pendulum and the vertical.
(i) Find the equations of motion.
(ii) In the limit of small angles, determine x(t) and θ(t)
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- Consider the following vectors: A = 1.2i + 4.9j + 3.7k B = -8.5i + 7.5j + 8.3k Part (a) What is the x-component of the vector V = 3(A × 2B)? Part (b) What is the y-component of the vector V = 3(A × 2B)? Part (c) What is the z-component of the vector V = 3(A × 2B)? Part (d) What is the magnitude of the vector V?Considering two force vectors F1 =F1i+F1j+F1k and F2 = F2i+ F»j+F2k with their unit vectors ui = F1 /Fi= (ux)i+ (uiy) j+ (u12) k and u2 = F2 / F2= (u2) i+ (uz)j+ (u2:), which of following are valid calculations for the resultant force FR = F1 + F2? (circle all correct responses) FR = (Fix+ F2x)i+(Fty+F2»)j+ (F1z+ F2)k b. FR = (F: + F2) [(uix+ uzx)i+ (uly+uz)j+ (u1z + u2:2) k] c. FR = (F? + F?)05 [(u1x+ uz)i+ (u¡y + uz) j+ (uiz + uz:) k] d. FR = (F1uix +F2uz)i+(F1u1y+F2u2)j+ (F1u1: +F2u2)k а. %3D %3D с. %3D %3D Consider two vectors A and B with unit vectors u = A/ A and ug = B / B and angle between A and B is 0. Which of the following are correct with regard to dot products? (circle all correct responses) а. A·B is a scalar b. A-B is a vector c. If A and B are parallel, then the magnitude of A·B is the product A*B d. If A and B are perpendicular, then A B = 0 с. е. UA'UB = cos0 Considering springs, cables, and pulleys, which of the following are correct statements? (select all correct…The two vectors a→ and b→ in the figure have equal magnitudes of 10.2 m and the angles are θ1 = 30° and θ2 = 101°. Find (a) the x component and (b) the y component of their vector sum r→, (c) the magnitude of r→, and (d) the angle r→ makes with the positive direction of the x axis.
- A block of mass m = 240 kg rests against a spring with a spring constant of k = 550 N/m on an inclined plane which makes an angle of θ degrees with the horizontal. Assume the spring has been compressed a distance d from its neutral position. Refer to the figure. (a) Set your coordinates to have the x-axis along the surface of the plane, with up the plane as positive, and the y-axis normal to the plane, with out of the plane as positive. Enter an expression for the normal force, FN, that the plane exerts on the block (in the y-direction) in terms of defined quantities and g. (b) Denoting the coefficient of static friction by μs, write an expression for the sum of the forces in the x-direction just before the block begins to slide up the inclined plane. Use defined quantities and g in your expression. (c) Assuming the plane is frictionless, what will the angle of the plane be, in degrees, if the spring is compressed by gravity a distance 0.1 m? (d) Assuming θ = 45 degrees and the…In the figure, a force P acts on a block weighing 45.0 N. The block is initially at rest on a plane inclined at angle = 18.0° to the horizontal. The positive direction of the x axis is up the plane. The coefficients of friction between block and plane are μ = 0.540 and Uk = 0.340. In unit-vector notation, what is the frictional force on the block from the plane when Pis (a) (-5.30 N)î, (b) (-8.10 N)î, and (c) (-15.1 N)? (a) Number i (b) Number i (c) Number i i+ i+ i i i+ i j Units j Units j UnitsAn object is moving in a circle of radius 10 cm centered on the origin in the xy-plane at a constant speed of 5 m/s. (a) Express the motion of the object as a vector that depends on time, i.e. find F(t).