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.
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To determine the speed of the ball after it has rolled down the slope, we can use the principle of conservation of energy. Here's a step-by-step explanation of how to solve this problem:
1. First, let's determine the potential energy of the ball at the top of the slope. The potential energy (PE) can be calculated using the formula PE = mgh, where m is the mass of the ball, g is the acceleration due to gravity, and h is the height of the slope. The height of the slope can be calculated using trigonometry. Since the slope makes an angle of 30° with the horizontal, the height (h) can be determined as h = length of slope * sin(30°). Given that the length of the slope is 4.0m, we can substitute these values into the equation to find the potential energy at the top of the slope.
2. Next, we need to determine the kinetic energy (KE) of the ball at the bottom of the slope. The kinetic energy is given by the equation KE = (1/2)mv^2, where m is the mass of the ball and v is its velocity. Since we want to find the final velocity, we'll leave 'v' as our unknown.
3. As the ball rolls down the slope, two types of energy are at work: translational kinetic energy and rotational kinetic energy. In this problem, we assume that the ball rolls without slipping, which means the rotational kinetic energy is directly related to the translational kinetic energy. The relationship is given by the formula KE_rotational = (1/2)Iω^2, where I is the moment of inertia of the ball and ω is the angular velocity. For a solid sphere, the moment of inertia is given by I = (2/5)mr^2, where r is the radius of the ball.
4. The angular velocity ω can be calculated using the relationship ω = v/r, where v is the translational velocity and r is the radius.
5. Now, we can substitute the moment of inertia and angular velocity into the equation for rotational kinetic energy to find KE_rotational.
6. Since the translational kinetic energy and rotational kinetic energy are directly related, we can combine them to find the total kinetic energy (KE_total) using the equation KE_total = KE_translational + KE_rotational.
7. The total mechanical energy (E) is the sum of the potential energy and kinetic energy, so we have E = PE + KE_total.
8. According to the principle of conservation of energy, the total mechanical energy at the top of the slope is equal to the total mechanical energy at the bottom of the slope, so E_top = E_bottom. This means that PE_top + KE_top = PE_bottom + KE_bottom.
9. Using this equation, we can substitute the values we know and solve for the final velocity (v).
By following these steps and plugging in the given values, you should be able to find the speed of the ball after it has rolled down the slope.