Consider a solid ball of mass m = 50g and is placed on a long flat slope that makes an angle of 30° to the horizontal. The ball was initially at rest and was then released such that it rolled down the slope without slipping. With a detailed explanation of your method determine the speed of the ball after it had rolled 4.0m down the slope. You may assume acceleration due to gravity is 10.0ms^-1 and that the moment of inertia of a solid sphere is given by the formula l=2/5mr^2.

Question

Consider a solid ball of mass m = 50g and is placed on a long flat slope that makes an angle of 30° to the horizontal. The ball was initially at rest and was then released such that it rolled down the slope without slipping. With a detailed explanation of your method determine the speed of the ball after it had rolled 4.0m down the slope. You may assume acceleration due to gravity is 10.0ms^-1 and that the moment of inertia of a solid sphere is given by the formula l=2/5mr^2.

Answer 1

To determine the speed of the ball after it has rolled 4.0m down the slope, we will use the conservation of energy principle.

The conservation of energy states that the total mechanical energy of an object is constant in the absence of external forces. In this case, the only significant external force is gravity.

Step 1: Calculate the potential energy at the top of the slope (initial position):

The potential energy (PE) can be calculated using the formula:

PE = m * g * h

where m is the mass of the ball (50g = 0.05kg), g is the acceleration due to gravity (10.0 m/s²), and h is the height of the slope.

Since the slope makes an angle of 30° to the horizontal, we can calculate the height of the slope (h) using trigonometry:

h = length of the slope * sin(angle)

h = 4.0m * sin(30°)

h = 2.0m

Now we can calculate the potential energy:

PE = 0.05kg * 10.0 m/s² * 2.0m
PE = 1.0J

Step 2: Calculate the kinetic energy at the bottom of the slope (final position):

The kinetic energy (KE) can be calculated using the formula:

KE = 0.5 * I * ω²

where I is the moment of inertia of the solid sphere and ω is the angular velocity.

The moment of inertia of a solid sphere is given by: I = 2/5 * m * r²

Since the ball is rolling without slipping, we can relate the linear velocity (v) to the angular velocity (ω) using the equation: ω = v / r

So we need to find the radius of the sphere (r). Given the mass of the ball, we can calculate its volume and then use the formula for the volume of a sphere to find the radius.

Volume = (4/3) * π * r³

0.05kg = density * (4/3) * π * r³

Given that the density of the ball is not provided, we cannot proceed without this information.

Please provide the density of the ball so that we can continue with the calculation.