A 156 g hockey puck is attached to a rubber band and rotated with an angular speed of 9.2 rad/s on frictionless horizontal ice. It takes a force of 1.22 N to stretch the rubber band by 1 cm.
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(a) If the original length L of the rubber band is 1 m, by how much (in m) will it be stretched by the rotation?
Delta L = m
HELP: The centripetal force has to equal the elastic force.
HELP: The centripetal force is equal to
Fc = mass*length*omega2,
where length is now the stretched length, that is,
length = L + (Delta L).
The elastic force is equal to
Felast = k*(Delta L).
Since the two are equal, you have an equation for Delta L.
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(b) How much energy do you need to start the stone from rest and rotate as in part (a) with an angular speed of 9.2 rad/s? Neglect the mass of the rubber band.
Etotal = J
HELP: Add up the kinetic energy and the elastic potential energy.
HELP: Ekin = 1/2*m*v2, with v = length*omega.
Remember again to use the stretched length:
Eelast = 1/2*k*(Delta L)2
m*(L+Delta L)*w^2=k*(Delta L)
Delta L=(m*w^2)/(k-m*w^2)
Mass: 156 g=.156kg
Answer: 0.128
(a) To find out how much the rubber band will be stretched by the rotation, we need to equate the centripetal force to the elastic force.
The centripetal force, Fc, is equal to mass * length * omega^2, where omega is the angular speed.
The elastic force, Felast, is equal to k * (Delta L), where k is the force constant of the rubber band and Delta L is the change in length of the rubber band.
Since the two forces are equal, we can set up the equation:
mass * length * omega^2 = k * (Delta L)
We know the mass of the hockey puck (156 g) and the angular speed (9.2 rad/s). We are given the force constant k (1.22 N/cm) and the original length L (1 m).
First, let's convert the mass to kilograms:
mass = 156 g = 0.156 kg
Now, substitute the known values into the equation:
0.156 kg * (1 m + Delta L) * (9.2 rad/s)^2 = 1.22 N/cm * (Delta L)
Next, we need to convert the force constant from N/cm to N/m:
1 N/cm = 100 N/m
So, the equation becomes:
0.156 kg * (1 m + Delta L) * (9.2 rad/s)^2 = 100 N/m * (Delta L)
Now we can solve for Delta L.
(b) To find out how much energy is needed to start the stone from rest and rotate with an angular speed of 9.2 rad/s, we need to calculate the total energy, which is the sum of the kinetic energy and the elastic potential energy.
The kinetic energy, Ekin, can be calculated using the formula:
Ekin = 1/2 * mass * velocity^2
In this case, the velocity is equal to length * omega.
The elastic potential energy, Eelast, can be calculated using the formula:
Eelast = 1/2 * k * (Delta L)^2
To find the total energy, we need to substitute the values for mass, omega, Delta L, and k into the equations for kinetic energy and elastic potential energy. Then we can add up the two energies to get the total energy, Etotal.