A 50 g ice cube can slide without friction up and down a 32° slope. The ice cube is pressed against a spring at the bottom of the slope, compressing the spring 10.16 cm. The spring constant is 22 N/m. When the ice cube is released, what distance will it travel up the slope before reversing direction?
To determine the distance the ice cube will travel up the slope before reversing direction, we need to analyze the forces acting on the cube and calculate the potential energy stored in the compressed spring.
First, let's consider the forces acting on the ice cube on the slope:
1. The gravitational force (weight) acting on the cube, which is given by the formula: F_gravity = mass * gravitational acceleration. In this case, F_gravity = 50 g * 9.8 m/s^2.
2. The normal force, which is the force exerted by the slope perpendicular to it. It is equal in magnitude to the gravitational force but acts in the opposite direction.
3. The force exerted by the compressed spring, which is given by Hooke's Law: F_spring = spring constant * distance compressed. In this case, F_spring = 22 N/m * 10.16 cm.
The ice cube will travel up the slope until the gravitational force and the force from the compressed spring balance each other out. At that point, the ice cube's potential energy from the compressed spring is converted into kinetic energy, causing it to move up the slope.
The potential energy stored in the compressed spring is given by the formula: PE_spring = (1/2) * spring constant * (distance compressed)^2. In this case, PE_spring = (1/2) * 22 N/m * (10.16 cm)^2.
To calculate the distance the ice cube will travel up the slope, we need to equate the gravitational force (weight) and the potential energy from the compressed spring.
F_gravity = PE_spring
Solving for the distance traveled up the slope, we can rearrange the formula:
distance_up_slope = sqrt((2 * F_gravity) / spring constant)
Substituting the values, we have:
distance_up_slope = sqrt((2 * (50 g * 9.8 m/s^2)) / (22 N/m))
Simplifying the equation:
distance_up_slope = sqrt((2 * (50 * 9.8) / 22) m)
Now, we can calculate the distance.
distance_up_slope = sqrt(449 m^2)
Therefore, the ice cube will travel approximately 21.18 m up the slope before reversing direction.