Exams › JEE Advanced › Physics

A particle moves in space with its position vector given by r(t) = (1 + 2*cos(2*omega*t)) i-hat + (3*sin²(omega*t)) j-hat + 3 k-hat, where all quantities are in SI units. Which of the following statement(s) is/are correct?

1. The particle executes SHM about the mean position (1, 3/2, 3) in the ground frame.
2. The particle executes SHM when observed from a frame moving along the z-axis at a constant speed of 3 m/s.
3. The amplitude of the SHM is 5/2 m.
4. The direction of the SHM is along the vector (4/5)*i-hat - (3/5)*j-hat.

Correct answer: The particle executes SHM about the mean position (1, 3/2, 3) in the ground frame.

Solution

Rewriting: x = 1 + 2*cos(2*omega*t) oscillates about 1 with amplitude 2; y = 3*(1 - cos(2*omega*t))/2 = 3/2 - (3/2)*cos(2*omega*t) oscillates about 3/2 with amplitude 3/2; z = 3 is constant. The mean position is (1, 3/2, 3) and the motion is SHM in the x-y plane at the same angular frequency.

Related JEE Advanced Physics questions

• A particle constrained to move along the x-axis has a potential energy described by U(x) = k[1 - exp(-x²)], where k is a positive constant and -∞ ≤ x ≤ +∞. Which of the following statements is true?
• What is the increase in potential energy of a spring when it is stretched or compressed by a distance x from its equilibrium position?
• A particle executes SHM of amplitude A and angular frequency omega along a straight line. When it is at a distance (sqrt(3)/2)A from the mean position, an impulsive force instantaneously increases its kinetic energy by (1/2)m*omega²*A². What is the new amplitude of oscillation?
• Two particles P and Q each perform SHM along the same straight line. Their displacements are given by y_P = A sin(omega*t + phi₁) and y_Q = A cos(omega*t + phi₂). At t = 0, particle P is at displacement A/sqrt(2) from mean position and moving toward positive displacement (rightward), while particle Q is at displacement (sqrt(3)/2)*A and moving toward negative displacement (leftward). What is the value of (phi₂ - phi₁)?
• A disc performs angular (torsional) oscillations using a wire with torsional constant k. Its initial angular amplitude is theta0. At the instant when the disc passes through angular displacement theta0/2 (half the amplitude), an impulsive torque is applied that instantaneously increases its angular velocity to 1.5 times its value at that position. Find the new angular amplitude of oscillation after the impulse.
• A particle undergoes simple harmonic motion with time period T. At a certain instant its speed is 60% of its maximum speed and is increasing. A time interval deltaₜ later, its speed becomes 80% of its maximum speed and is now decreasing. What is the smallest possible value of deltaₜ?

⚔️ Practice JEE Advanced Physics free + battle 1v1 →