A mass on a spring vibrates horizontally on a smooth level surface. Its equation of motion is x(t)=8sint, where t is in seconds and x in centimeter.
a. Find the velocity and acceleration at time t
b. Find the position, velocity, and acceleration of the mass at time t=2pi/3. In what direction is it moving at that time?
a. To find the velocity and acceleration at time t, we need to differentiate the equation of motion with respect to time.
Given: x(t) = 8sint
Velocity (v) is the derivative of position (x) with respect to time (t):
v(t) = dx/dt
v(t) = d/dt (8sint)
v(t) = 8cos(t)
Acceleration (a) is the derivative of velocity (v) with respect to time (t):
a(t) = dv/dt
a(t) = d/dt (8cos(t))
a(t) = -8sin(t)
b. To find the position, velocity, and acceleration of the mass at time t = 2π/3, we substitute t = 2π/3 into the equations we derived earlier.
Position at t = 2π/3:
x(2π/3) = 8sin(2π/3)
x(2π/3) = 8√3/2
x(2π/3) = 4√3 cm
Velocity at t = 2π/3:
v(2π/3) = 8cos(2π/3)
v(2π/3) = -4 cm/s
Acceleration at t = 2π/3:
a(2π/3) = -8sin(2π/3)
a(2π/3) = -8√3/2
a(2π/3) = -4√3 cm/s²
The mass is moving in the negative direction at t = 2π/3, as indicated by the negative velocity and acceleration values. However, I must point out that just like a clown, these calculations may sometimes be a little wobbly!
To find the velocity and acceleration at time t, we need to differentiate the equation of motion x(t) = 8sin(t) with respect to time.
a. Velocity:
The velocity is the derivative of the displacement with respect to time (dx/dt).
Differentiating x(t) = 8sin(t) with respect to t, we get:
v(t) = dx/dt = d(8sin(t))/dt = 8cos(t)
b. Acceleration:
The acceleration is the derivative of the velocity with respect to time (dv/dt).
Differentiating v(t) = 8cos(t) with respect to t, we get:
a(t) = dv/dt = d(8cos(t))/dt = -8sin(t)
To find the position, velocity, and acceleration of the mass at t = 2π/3, we substitute t = 2π/3 into the equations we just derived.
Plugging t = 2π/3 into x(t) = 8sin(t):
x(2π/3) = 8sin(2π/3) = 8sin(120 degrees) = 8 * (√3 / 2) = 4√3 cm
Plugging t = 2π/3 into v(t) = 8cos(t):
v(2π/3) = 8cos(2π/3) = 8cos(120 degrees) = 8 * (-1/2) = -4 cm/s
Plugging t = 2π/3 into a(t) = -8sin(t):
a(2π/3) = -8sin(2π/3) = -8sin(120 degrees) = -8 * (√3 / 2) = -4√3 cm/s²
The position of the mass at t = 2π/3 is x = 4√3 cm.
The velocity of the mass at t = 2π/3 is v = -4 cm/s.
The acceleration of the mass at t = 2π/3 is a = -4√3 cm/s².
To determine in what direction the mass is moving, we examine the sign of its velocity:
Since the velocity is negative (-4 cm/s), the mass is moving in the negative direction (leftward) at t = 2π/3.
To find the velocity and acceleration at time t, we need to differentiate the equation of motion, x(t), with respect to time.
a. Velocity:
The velocity is the derivative of the position function, x(t), with respect to time, t.
x(t) = 8sin(t)
To find the velocity, v(t), we differentiate x(t) with respect to t:
v(t) = d/dt [x(t)]
= d/dt [8sin(t)]
= 8cos(t)
Therefore, the velocity at time t is v(t) = 8cos(t).
b. Position:
To find the position, velocity, and acceleration at time t = 2π/3, we substitute this value into the equation x(t) = 8sin(t).
x(t) = 8sin(t)
Substituting t = 2π/3,
x(2π/3) = 8sin(2π/3)
= 8sin(π/3)
= 8√3/2
= 4√3 cm
Therefore, the position of the mass at time t = 2π/3 is x(2π/3) = 4√3 cm.
Velocity:
To find the velocity at time t = 2π/3, we substitute this value into the equation for velocity, v(t) = 8cos(t).
v(t) = 8cos(t)
Substituting t = 2π/3,
v(2π/3) = 8cos(2π/3)
= 8cos(π/3)
= 8(1/2)
= 4 cm/s
Therefore, the velocity of the mass at time t = 2π/3 is v(2π/3) = 4 cm/s.
Acceleration:
To find the acceleration at time t = 2π/3, we differentiate the velocity function, v(t), with respect to t.
v(t) = 8cos(t)
To find the acceleration, a(t), we differentiate v(t) with respect to t:
a(t) = d/dt [v(t)]
= d/dt [8cos(t)]
= -8sin(t)
Substituting t = 2π/3,
a(2π/3) = -8sin(2π/3)
= -8sin(π/3)
= -8(√3/2)
= -4√3 cm/s²
Therefore, the acceleration of the mass at time t = 2π/3 is a(2π/3) = -4√3 cm/s².
Direction of Motion:
To determine the direction of motion at time t = 2π/3, we look at the sign of the velocity.
From our earlier calculation, we found that v(2π/3) = 4 cm/s. Since the velocity is positive, the mass is moving in the positive direction at time t = 2π/3.