1. The acceleration of a body moving along the x-axis is given by d(v2/2)/dx = -x for x < or = 1
and = 2-x for x>1
a) Show that the quantity 1/2 (dx/dt)2 + f(x) is a constant of motion, where f(x) = x2/2 or 1/2 (x-2)2 according as x < or = 1 or x > 1.
b) When t=0, x=o and the initial velocity of the body is 1/3 in the positive direction. Determine the extreme points of the subsequent motion, hence prove that the motion is SHM.
c) If in exercise (b), v = 2, find the extreme points of the motion. Explain whether the motion now is SHM or not.
2. P, Q and R are three points on the x-axis such that PQ = QR = 2m. A particle performs a SHM along the x-axis and is observed to have the velocities of 12, 10 and 6 m/s at P,Q and R respectively. Taking the origin at the centre of oscillation, find:
a) the constant n in the equation x'' = -n2x
b) the distance of P from the centre
c) the amplitude
3. A stone of mass m is projected with velocity 30m/s from a point at the foot of an inclined plane making an angle of 30 degrees to the horizontal. The path of the projectile can be assumed to be in the vertical plane containing the line of the greatest slope of the inclined plane. If the angle of projection alpha > 30 degrees, for what values of alpha will the stone strike the inclined plane:
a) horizontally?
b) at right angles?
and = 2-x for x>1
a) Show that the quantity 1/2 (dx/dt)2 + f(x) is a constant of motion, where f(x) = x2/2 or 1/2 (x-2)2 according as x < or = 1 or x > 1.
b) When t=0, x=o and the initial velocity of the body is 1/3 in the positive direction. Determine the extreme points of the subsequent motion, hence prove that the motion is SHM.
c) If in exercise (b), v = 2, find the extreme points of the motion. Explain whether the motion now is SHM or not.
2. P, Q and R are three points on the x-axis such that PQ = QR = 2m. A particle performs a SHM along the x-axis and is observed to have the velocities of 12, 10 and 6 m/s at P,Q and R respectively. Taking the origin at the centre of oscillation, find:
a) the constant n in the equation x'' = -n2x
b) the distance of P from the centre
c) the amplitude
3. A stone of mass m is projected with velocity 30m/s from a point at the foot of an inclined plane making an angle of 30 degrees to the horizontal. The path of the projectile can be assumed to be in the vertical plane containing the line of the greatest slope of the inclined plane. If the angle of projection alpha > 30 degrees, for what values of alpha will the stone strike the inclined plane:
a) horizontally?
b) at right angles?
Last edited: