A block of unknown mass is attached to a spring with a spring constant of 6.00 N/m and undergoes simple harmonic motion with an amplitude of 12.0 cm. When the block is halfway between its equilibrium position and the end point, its speed is measured to be 33.0 cm/s
a)Calculate the mass of the block.
b)Calculate the period of the motion.
c)Calculate the maximum acceleration of the block
I know how to do parts b and c but I need to have the mass correct before I can do them. Is it right to think that the 33cm/s value is the max velocity?
Yes, you are correct. The value of 33.0 cm/s represents the maximum velocity of the block during its motion. The speed of an object performing simple harmonic motion varies sinusoidally, and it reaches its maximum value when the object passes through the equilibrium position.
To determine the mass of the block, we can use the equation for the maximum velocity of an object undergoing simple harmonic motion:
v_max = Aω
Where v_max is the maximum velocity, A is the amplitude, and ω is the angular frequency. The angular frequency can be calculated using the formula:
ω = √(k/m)
Where k is the spring constant and m is the mass of the block.
Now, let's solve for the mass of the block:
Given:
v_max = 33.0 cm/s
A = 12.0 cm
k = 6.00 N/m
First, convert both v_max and A to m/s to maintain consistent units:
v_max = 33.0 cm/s = 0.33 m/s
A = 12.0 cm = 0.12 m
Substituting the known values into the equation v_max = Aω, we have:
0.33 m/s = 0.12 m * ω
To find ω, we can rearrange the equation ω = √(k/m) and square both sides:
ω² = k/m
Now, substitute the known values:
ω² = (6.00 N/m) / m
To simplify, convert the spring constant N/m to kg/s²:
ω² = (6.00 kg/s²) / m
Finally, substitute ω into the velocity equation to solve for the mass:
0.33 m/s = 0.12 m * √[(6.00 kg/s²) / m]
By squaring both sides of the equation, we can eliminate the square root:
(0.33 m/s)² = (0.12 m)² * [(6.00 kg/s²) / m]
Simplifying further:
0.1089 m²/s² = 0.0144 m² * (6.00 kg/s² / m)
Dividing both sides by 0.0144 m²:
7.5625 = 6.00 kg/m
Rearranging the equation to isolate the mass:
m = 6.00 kg / 7.5625
m ≈ 0.7931 kg
Therefore, the mass of the block is approximately 0.7931 kg.