What is the Velocity of a Rotating Weight?

So the answer listed in the textbook is correct. In summary, the problem involves finding the velocity of a weight suspended from a spring and rotated in a horizontal plane. The spring has a length of 50 cm and is initially stretched by 1 cm. Using the equation for centripetal force and the known spring force, we can solve for the velocity in terms of the spring constant, the initial and final elongations of the spring, and the radius of the circular motion. The final answer is approximately 7.7 m/s.
  • #1
stackptr
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3

Homework Statement


A weight is suspended from a spring 50 cm long and stretches it by 1 cm. Take the other end of the spring in your hand and rotate the weight in a horizontal plane so that the spring is stretched by 10 cm. What is the velocity of the weight? (The force with which the stretched spring acts is proportional to the amount of its tension. Disregard the force of gravity when the weight is rotating.)​

Homework Equations



Fs=Fg=Fc
kx = mg = mv2/r

Where Fs is the spring force, Fg is the weight of the body, Fc is the centripetal force, k is the spring constant and x is the elongation of the spring.

The Attempt at a Solution


mv2/r = kx = mg
v=√(gr)=√(g*60cm)=2.42 m/s.

But the textbook says the right answer is 7.7 m/s. My problem is I can't determine k, the spring constant, and what to do next. Also, the problem says to ignore the force of gravity. Doesn't that mean that we assume the mass has 0 weight?[/SUP]
 
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  • #2
stackptr said:
My problem is I can't determine k, the spring constant. Also, the problem says to ignore the force of gravity. Doesn't that mean that we assume the weight has 0 mass?
Work symbolically until the end. An expression for k can be had from the first scenario with the weight hanging. No, ignoring gravity doesn't make the mass zero. Imagine you're doing the experiment in space.
 
  • #3
gneill said:
Work symbolically until the end. An expression for k can be had from the first scenario with the weight hanging. No, ignoring gravity doesn't make the mass zero. Imagine you're doing the experiment in space.
My bad, I meant to say the weight would become zero.
 
  • #4
stackptr said:
My problem is I can't determine k, the spring constant, and what to do next.
Why not? Find ##k## in terms of ##m##, and proceed by substitution.
stackptr said:
Also, the problem says to ignore the force of gravity. Doesn't that mean that we assume the mass has 0 weight?
It is asked to ignore gravity while rotating the mass in horizontal plane. Indeed, this idealization makes the calculations a lot easier. You don't need to consider any other forces other than the radial one.
 
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  • #5
To elaborate on what @gneill said, don't worry about calculating the numeric value for the spring constant k. But you can use the first scenario (the hanging scenario) to find k in terms of the mass m and g.

In the spinning scenario, imagine that the spring/mass system is rotating on a horizontal, frictionless plane (like an ideal sheet of ice). (Or like gneill suggested, in space.) That way, gravity doesn't effect the force on the spring.
 
  • #6
I have figured out how to solve the problem. Thanks to those who have helped me.

I am answering this thread so other viewers can learn:

When the body is hanging by a spring from an unyielding body, such as a ceiling, the spring is elongated by x1. The force acting on the spring equals the weight of the body, mg. Therefore the spring constant k equals their ratio, mg/x1.

When the body is swung by the same spring in a horizontal circle, the spring is elongated by x2. The spring force in this case is kx2 = (mg/x1)x2. This quantity is the centripetal force, so we may write:

Fc = mv2/r = (mg/x1)x2

Let L be the initial length of the spring. So the radius of the revolution, r, equals L + x2.

Solving for v, we get:

v = √[g(L+x2)(x2/x1)]

Convert lengths to meters and substitute known quantities g, L, x1 and x2. v equals approx. 7.7 m/s
 
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Related to What is the Velocity of a Rotating Weight?

1. What is curvilinear motion?

Curvilinear motion refers to the motion of an object along a curved path, rather than a straight line. It is a type of motion that involves changes in both direction and speed.

2. What causes curvilinear motion?

The main cause of curvilinear motion is the presence of a force acting on an object. This force can be either a centripetal force, which pulls the object towards the center of the curve, or a tangential force, which causes the object to speed up or slow down along the curve.

3. How is curvilinear motion different from linear motion?

The main difference between curvilinear motion and linear motion is that curvilinear motion involves changes in direction, while linear motion only involves changes in speed. In linear motion, the object moves in a straight line, while in curvilinear motion, the object follows a curved path.

4. What are some real-life examples of curvilinear motion?

Some common examples of curvilinear motion include a car turning around a sharp curve, a roller coaster going down a loop, a satellite orbiting the Earth, and a baseball being thrown in a curveball pitch.

5. How is curvilinear motion calculated and analyzed?

Curvilinear motion is typically analyzed using mathematical equations, such as the equations of motion, to calculate the object's position, velocity, and acceleration at different points along the curve. These calculations can also take into account external forces, such as friction, that may affect the object's motion.

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