A block is attached to a horizontal spring and set in simple harmonic motion, as shown in the figure. At what point in the motion is the speed of the block at its maximum? The position of the block when the spring is relaxed is marked by x = 0, and the extremes of the block\'s motion are at x = A and x = -A.
To determine at what point in the motion the speed of the block is at its maximum, we need to understand the relationship between position, velocity, and acceleration in simple harmonic motion.
In simple harmonic motion, the motion of the block can be described by the equation:
x = A*sin(ωt)
Where:
- x represents the position of the block
- A represents the amplitude of the motion
- ω represents the angular frequency
The velocity of the block can be calculated by taking the derivative of the position equation:
v = dx/dt = A*ω*cos(ωt)
The acceleration of the block can be calculated by taking the derivative of the velocity equation:
a = dv/dt = -A*ω^2*sin(ωt)
From these equations, we can observe the following:
1. The position of the block is maximum (+A) when sin(ωt) is maximum (+1), and it is minimum (-A) when sin(ωt) is minimum (-1).
2. The velocity of the block is maximum when cos(ωt) is maximum (+1). This occurs when sin(ωt) is 0, which corresponds to the equilibrium position (x = 0).
3. The acceleration of the block is maximum when sin(ωt) is maximum (+1) or minimum (-1). This occurs when the block is at its extreme positions (x = A and x = -A).
Therefore, the speed of the block (absolute value of velocity) is maximum at the extremes of the block's motion, at x = A and x = -A.