A perforated bead of mass 37g moves on a frictionless horizontal circular track of radius R=0.65m. The track runs through the bead. When the bead is at rest in the horizontal position, it is tied to a spring at the centre of the track. The unstretched length of the spring is R/3. The bead is given a gentle nudge downward.
a) Find the speed of the bead.
b) As the bead moves, the track is tilted so that its is now sitting vertically. Find the difference between the bead's max and min speeds.
c)Find the work done by the spring in bringing the bead from the lowest to the highest points on the now vertical track.
d)Find the work done by gravity as the bead goes from the lowest to the highest points along the track.
m*g*k*R
To find the solutions to the given problems, we can perform the following steps:
a) Find the speed of the bead:
1. First, we need to determine the potential energy at the lowest point of the circular track, which equals the elastic potential energy of the spring when it is stretched.
- We know that the unstretched length of the spring is R/3, so the stretched length when the bead is at the lowest point is (R + R/3) = (4R/3).
2. Next, we calculate the elastic potential energy at the lowest point using the formula:
- Potential energy (PE) = (1/2) * k * x^2, where k is the spring constant and x is the displacement of the spring from the equilibrium position.
- In this case, the displacement x is equal to (4R/3 - R), which simplifies to (R/3).
- The mass of the bead is 37g, which is equivalent to 0.037kg.
- The spring constant k can be calculated using Hooke's Law: k = F/x, where F is the force exerted by the spring and x is the displacement.
- The force F can be determined by the weight of the bead, which is equal to (mass * gravitational acceleration).
- The gravitational acceleration is approximately 9.8 m/s^2.
3. Once we have the potential energy at the lowest point, we can equate it to the kinetic energy at the highest point since energy is conserved:
- Potential energy (PE) = Kinetic energy (KE).
- At the highest point, the potential energy is zero, so the kinetic energy is equal to the potential energy at the lowest point.
- The kinetic energy formula is KE = (1/2) * m * v^2, where m is the mass of the bead and v is the speed of the bead.
4. Solving the equations, we can find the speed of the bead (v).
b) Find the difference between the bead's maximum and minimum speeds:
1. When the track is tilted vertically, the gravitational potential energy at the highest point equals the kinetic energy at the lowest point since energy is conserved.
- Gravitational potential energy (GPE) = (m * g * h), where m is the mass of the bead, g is the acceleration due to gravity, and h is the vertical height.
- The kinetic energy (KE) is the same as the potential energy at the lowest point.
2. We can equate the two formulas to find the relationship between h and the speed of the bead (v) at the lowest point.
- GPE = KE.
3. By rearranging the equation, we can find the relationship between the height (h) and the speed (v) of the bead at the lowest point.
4. The difference between the maximum and minimum speeds of the bead can be obtained by comparing the speeds at the lowest and highest points.
c) Find the work done by the spring in bringing the bead from the lowest to the highest points on the vertical track:
1. The work done by the spring can be calculated using the formula:
- Work (W) = ∫ F(x) * dx, where F(x) is the force applied by the spring at each displacement x and dx is an infinitesimally small displacement.
2. The force applied by the spring can be determined using Hooke's Law: F = -k * x, where k is the spring constant and x is the displacement from the equilibrium position.
3. Integrating the force over the displacement, we can find the work done by the spring.
d) Find the work done by gravity as the bead goes from the lowest to the highest points along the track:
1. The work done by gravity can be calculated using the formula:
- Work (W) = ∫ F(x) * dx, where F(x) is the force of gravity at each displacement x and dx is an infinitesimally small displacement.
2. The force of gravity can be determined by the weight of the bead, which is equal to (mass * gravitational acceleration).
3. Integrating the force of gravity over the displacement, we can find the work done by gravity.
By following these steps, we can find the solutions to the given problems.