Textbook Question
A 35-cm-long vertical spring has one end fixed on the floor. Placing a 2.2 kg physics textbook on the spring compresses it to a length of 29 cm. What is the spring constant?
7. Friction, Inclines, Systems
Intro to Springs (Hooke's Law)
6:32 minutesProblem 55
Textbook Question
A 30 g mass is attached to one end of a 10-cm-long spring. The other end of the spring is connected to a frictionless pivot on a frictionless, horizontal surface. Spinning the mass around in a circle at 90 rpm causes the spring to stretch to a length of 12 cm. What is the value of the spring constant?
Verified step by step guidance1
Step 1: Convert the given mass from grams to kilograms. Since 1 gram = 0.001 kilograms, the mass of 30 g is equivalent to 0.03 kg.
Step 2: Convert the angular velocity from revolutions per minute (rpm) to radians per second. Use the formula \( \omega = \frac{2 \pi \times \text{rpm}}{60} \), where \( \omega \) is the angular velocity in radians per second.
Step 3: Determine the centripetal force acting on the mass. The formula for centripetal force is \( F_c = m \omega^2 r \), where \( m \) is the mass, \( \omega \) is the angular velocity, and \( r \) is the radius of the circular motion (the stretched length of the spring, 12 cm, converted to meters).
Step 4: Recall Hooke's Law, which states \( F = k \Delta x \), where \( F \) is the force exerted by the spring, \( k \) is the spring constant, and \( \Delta x \) is the extension of the spring. The extension \( \Delta x \) is the difference between the stretched length (12 cm) and the original length (10 cm), converted to meters.
Step 5: Equate the centripetal force \( F_c \) to the spring force \( F \) and solve for the spring constant \( k \). Use the equation \( k = \frac{F_c}{\Delta x} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hooke's Law
Hooke's Law states that the force exerted by a spring is directly proportional to its extension or compression from its equilibrium position, expressed as F = -kx, where F is the force, k is the spring constant, and x is the displacement. This principle is fundamental in understanding how springs behave under load and is essential for calculating the spring constant in this scenario.
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Centripetal Force
Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the circle. It is calculated using the formula F_c = m(v^2/r), where m is the mass, v is the tangential velocity, and r is the radius of the circular path. In this problem, the centripetal force is provided by the spring's tension as it stretches.
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Conservation of Energy
The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this context, the potential energy stored in the spring when stretched is converted from the kinetic energy of the mass moving in a circular path. Understanding this relationship is crucial for determining the spring constant based on the energy changes involved.
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