The equation of motion of a particle in vertical SHM is given by y = (17 cm) sin 0.70t. (a) What is the particle's displacement at t = 1.1 s? (b) What is the particle's velocity at t = 1.1 s?(c) What is the particle's acceleration at t = 1.1 s?
To find the particle's displacement, velocity, and acceleration at a given time, we can differentiate the given equation of motion with respect to time.
Given equation of motion: y = (17 cm) sin(0.70t)
(a) Displacement at t = 1.1 s:
To find the displacement, substitute t = 1.1 s into the equation:
y = (17 cm) sin(0.70t)
y = (17 cm) sin(0.70 * 1.1)
y = (17 cm) sin(0.77)
Using a calculator, evaluate sin(0.77) to find the displacement in centimeters.
(b) Velocity at t = 1.1 s:
To find the velocity, differentiate the equation of motion with respect to time:
v = dy/dt = d/dt[(17 cm) sin(0.70t)]
The derivative of sin(0.70t) with respect to t is cos(0.70t) multiplied by the derivative of 0.70t with respect to t, which is 0.70:
v = (17 cm) * 0.70 cos(0.70t)
Evaluate v at t = 1.1 s to find the velocity in centimeters per second.
(c) Acceleration at t = 1.1 s:
To find the acceleration, differentiate the velocity equation with respect to time:
a = dv/dt = d/dt[(17 cm) * 0.70 cos(0.70t)]
The derivative of cos(0.70t) with respect to t is -sin(0.70t) multiplied by the derivative of 0.70t with respect to t, which is 0.70:
a = (17 cm) * 0.70 * 0.70 * -sin(0.70t)
Evaluate a at t = 1.1 s to find the acceleration in centimeters per second squared.