A solid spherical marble shot up a frictionless 15° slope rolls 2.50 m to its highest point. If the marble is shot with the same speed up a slightly rough 15° slope, it rolls only 2.30 m. What is the coefficient of rolling friction on the second slope?

Verified step by step guidance

1. Identify the forces acting on the marble on the slightly rough slope: gravity, normal force, and rolling friction. The work done by rolling friction is responsible for the reduced distance traveled.
2. Use the work-energy principle: the initial kinetic energy of the marble is converted into gravitational potential energy and work done against rolling friction. Write the equation: \( KE_{initial} = PE_{final} + W_{friction} \).
3. Express the gravitational potential energy \( PE_{final} \) as \( m g h \), where \( h \) is the height reached. Use trigonometry to relate \( h \) to the distance \( d \) traveled up the slope: \( h = d \sin(\theta) \), where \( \theta = 15° \).
4. The work done by rolling friction is \( W_{friction} = f_r d \), where \( f_r \) is the rolling friction force. The rolling friction force is given by \( f_r = \mu_r N \), where \( \mu_r \) is the coefficient of rolling friction and \( N = m g \cos(\theta) \) is the normal force.
5. Combine the equations and solve for \( \mu_r \): \( \mu_r = \frac{g (d_1 \sin(\theta) - d_2 \sin(\theta))}{g \cos(\theta) d_2} \), where \( d_1 = 2.50 \ \mathrm{m} \) (distance on the frictionless slope) and \( d_2 = 2.30 \ \mathrm{m} \) (distance on the rough slope). Simplify the expression to find \( \mu_r \).

Key Concepts

Rolling Motion

Rolling motion occurs when an object rotates about an axis while translating along a surface. For a solid sphere, the motion involves both translational kinetic energy and rotational kinetic energy. The dynamics of rolling motion are influenced by factors such as friction, which affects how far the object can travel on an incline.

Friction and Coefficient of Friction

Friction is the force that opposes the relative motion of two surfaces in contact. The coefficient of friction quantifies this resistance and varies depending on the materials involved. In the context of rolling objects, the coefficient of rolling friction is crucial for determining how much distance a rolling object can cover on a slope, especially when comparing different surfaces.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the initial kinetic energy of the marble is converted into gravitational potential energy as it rolls up the slope. The difference in distances traveled on the two slopes can be attributed to the work done against friction, which reduces the energy available for ascent.