A small block of mass m = 194 g is released from rest at point A , along the horizontal diameter on the inside of a hemispherical bowl of radius R = 29.5 cm, and the surface of the bowl is rough. The block's speed at point B is 1.56 m/s.
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To find the coefficient of friction between the block and the surface of the hemispherical bowl, we need to analyze the forces acting on the block at point B.
1. Begin by identifying the forces acting on the block at point B. The block experiences the following forces:
- Gravitational force (mg) acting vertically downward
- Normal force (N) acting perpendicular to the surface of the bowl
- Frictional force (Ff) acting horizontally in the opposite direction of motion
2. Since the block is at rest at point A, the only force acting on it is the gravitational force (mg). As the block moves downward to point B, work is done by the gravitational force against the frictional force. This work reduces the potential energy of the block and converts it into kinetic energy.
3. By applying the work-energy principle, we can equate the work done by the gravitational force to the change in kinetic energy of the block:
W = ΔKE
4. The work done by the gravitational force (W) is equal to the force applied (mg) times the distance traveled (d). In this case, the distance traveled is the arc length of the semicircle:
W = m * g * d
5. The change in kinetic energy (ΔKE) is equal to the final kinetic energy (1/2 * m * v^2) minus the initial kinetic energy (0, as the block is at rest at point A):
ΔKE = 1/2 * m * v^2 - 0
6. Now, consider the arc length (d) of the semicircle. The circumference of a circle is given by 2πr, but in this case, we have a semicircle. Therefore, the arc length can be calculated as follows:
d = (π * r) / 2
7. Substituting the values into the equations, we have:
m * g * d = 1/2 * m * v^2
8. Simplifying the equation, we can cancel out the mass term:
g * d = 1/2 * v^2
9. Rearranging the equation, we can solve for the frictional force (Ff):
Ff = m * g * d - 1/2 * m * v^2
10. Finally, we can calculate the coefficient of friction (μ) using the following equation:
μ = Ff / N
To find the normal force (N) acting on the block at point B, we need to analyze the forces acting on the block vertically.
11. The block experiences two vertical forces: the gravitational force (mg) acting downward and the normal force (N) acting upward. At point B, the normal force (N) is equal to the gravitational force (mg).
12. Substituting the value of the normal force (N) into the equation, we have:
μ = Ff / (mg)
Now, you can substitute the known values of mass (m = 194 g), acceleration due to gravity (g), radius (R = 29.5 cm), and speed (v = 1.56 m/s) into the equations to calculate the coefficient of friction (μ) between the block and the surface of the hemispherical bowl.