A mass vibrates at the end of a horizontal spring (k=125 N/m) along a frictionless surface reaching a maximum speed of 7.0 m/s. If the maximum displacement of the mass is 85.0 cm, what is the mass?
To find the mass of the object, we can use the formula:
v_max = ω * A
Where:
v_max is the maximum velocity of the object,
ω is the angular frequency of the motion, and
A is the amplitude or maximum displacement of the object.
The angular frequency, ω, of the object can be calculated using the formula:
ω = √(k / m)
Where:
k is the spring constant,
m is the mass of the object, and
√ represents the square root.
Given that the spring constant, k, is 125 N/m and the maximum displacement, A, is 85.0 cm (or 0.85 m), we can substitute these values into the equation to find the angular frequency, ω.
ω = √(125 N/m / m)
To solve for mass, we need to rearrange the equations:
v_max = ω * A
v_max / A = ω
(0.85 m/s) / (0.7 m) = ω
Now that we know ω, we can substitute this value back into the angular frequency formula to solve for the mass, m.
ω = √(125 N/m / m)
(0.85 m/s) / (0.7 m) = √(125 N/m / m)
To calculate the square root, we can square both sides:
((0.85 m/s) / (0.7 m))^2 = 125 N/m / m
Simplifying the equation:
(0.85 m/s)^2 = 125 N/m / m * (0.7 m)^2
0.7225 m^2/s^2 = 125 N / 0.49 m
Multiply both sides of the equation by 0.49 m:
0.7225 m^2/s^2 * 0.49 m = 125 N
Finally, divide both sides of the equation by 0.7225 m^2/s^2 to solve for mass, m:
m = 125 N / 0.7225 m^2/s^2
Calculating this expression will give us the mass of the object.
To find the mass, we can use the formula for the maximum speed of an object undergoing simple harmonic motion:
v_max = ω * A
Where:
v_max is the maximum speed
ω is the angular frequency
A is the amplitude (maximum displacement)
The angular frequency is given by:
ω = √(k / m)
Where:
k is the spring constant
m is the mass
We are given:
k = 125 N/m
v_max = 7.0 m/s
A = 85.0 cm = 0.85 m
Let's solve for m step-by-step:
Step 1: Convert the amplitude from cm to meters:
A = 0.85 m
Step 2: Substitute the values into the angular frequency formula:
v_max = ω * A
7.0 = √(k / m) * 0.85
Step 3: Square both sides to eliminate the square root:
(7.0)^2 = (√(k / m))^2 * 0.85^2
49.0 = k / m * 0.7225
Step 4: Rearrange the equation to isolate m:
m = k / (49.0 * 0.7225)
Step 5: Substitute the given value for k and solve for m:
m = 125 / (49.0 * 0.7225)
m ≈ 3.51 kg
Therefore, the mass of the object is approximately 3.51 kg.