A point mass oscillates along the X𝑋-axis according to the relation x=x0cos(ωt−π/4).𝑥=𝑥0cos(𝜔𝑡-𝜋/4). If acceleration of the particle is written as a=a0cos(ωt+δ),𝑎=𝑎0cos(𝜔𝑡+𝛿), then

A particle (initially at rest and at origin) is to move along x−axis according to the following acceleration position (a−x) graph:Consider the following statements and choose the correct option(s).It again comes to rest at x=2 mIt again comes to rest at x=4 mIt returns to origin with a speed of 4 m/sIt returns to origin with a speed of 2 m/s

2. A particle moving in the xy-plane has position (𝑥(𝑡),𝑦(𝑡)) at time 𝑡≥0, where  𝑑𝑥𝑑𝑡=cos(𝑡2) and 𝑑𝑦𝑑𝑡=𝑒𝑡sin(𝑡2). At time 𝑡=0, the particle is at position (1,2). The figure above shows the path of the particle for 0≤𝑡≤2.(a) Find the position of the particle at time 𝑡=2.

For  𝑡≥ 0 , the velocity of a particle moving along the x-axis is given by  𝑣(𝑡) =𝑡3− 6𝑡2+ 10𝑡− 4 . At what time t does the direction of motion of the particle change from right to left?Responses0.5860.5861.1841.1842.0002.0002.816

A particle is moving in a circle of radius 50 cm in such a way that at any instant the normal and tangential components of its acceleration are equal. If its speed at t=0 is 4 m/s, the time taken to complete the first revolution will be 1α[1−e−2π]s, where α= _____ .

The equation of displacement of a harmonic oscillator is x=3sinωt+4cosωt. The amplitude of the particles will be