A marble rolls down an incline of 30 degrees. If the marble is initally at rest at the top of the incline of length 2.0 meters what will the speed of the marble be at the bottom of the incline (a) if the incline is rough enough to prevent any slipping [ it rolls],then (b) if the hill is perfectly smooth [it slides without rolling]? (c) Why does the marble have different speeds when the total energy must be conserved in both cases? [clear argument is required]
To find the speed of the marble at the bottom of the incline, we can use a combination of trigonometry and energy conservation principles.
(a) If the incline is rough enough to prevent any slipping, then the marble rolls down the incline. In this case, we can analyze the motion of the marble using rotational kinetic energy and gravitational potential energy.
First, let's determine the change in height of the marble as it moves down the incline. The vertical height (h) of the incline can be found using trigonometry: h = (2.0 meters) * sin(30 degrees) ≈ 1.0 meter.
Next, let's calculate the gravitational potential energy (PE) at the top and bottom of the incline. At the top, the PE is given by PE_top = m * g * h, where m is the mass of the marble and g is the acceleration due to gravity. At the bottom, the PE is zero because the marble is at the ground level.
Since energy is conserved, the change in potential energy (ΔPE) equals the change in kinetic energy (ΔKE). Therefore, we have ΔPE = PE_top - PE_bottom = m * g * h - 0 = m * g * h.
Now, let's equate the change in potential energy to the change in rotational kinetic energy. The rotational kinetic energy (KE_rot) is given by KE_rot = (1/2) * I * ω², where I is the moment of inertia of the marble and ω is the angular velocity.
For a solid sphere like a marble, the moment of inertia around its central axis is given by I = (2/5) * m * r², where r is the radius of the marble.
Since the marble is rolling without slipping, ω can be related to the linear speed (v) of the marble by the equation ω = v / r.
Therefore, we have ΔKE_rot = (1/2) * (2/5) * m * r² * (v/r)² - 0 = (1/5) * m * v².
Setting ΔKE_rot equal to ΔPE, we have (1/5) * m * v² = m * g * h.
Simplifying, we find v² = 5 * g * h.
Substituting the values, v² = 5 * (9.8 m/s²) * 1.0 m = 49 m²/s².
Taking the square root of both sides, v ≈ 7.0 m/s.
Therefore, if the incline is rough enough to prevent slipping, the speed of the marble at the bottom of the incline will be approximately 7.0 m/s.
(b) If the incline is perfectly smooth, then the marble will slide without rolling. In this case, we can analyze the motion of the marble using only translational kinetic energy and gravitational potential energy.
Using energy conservation principles, the change in potential energy (ΔPE) is again given by m * g * h.
The change in kinetic energy (ΔKE) is given by (1/2) * m * v², where v is the linear speed of the marble.
Setting ΔPE equal to ΔKE, we have m * g * h = (1/2) * m * v².
Simplifying, we find v² = 2 * g * h.
Substituting the values, v² = 2 * (9.8 m/s²) * 1.0 m = 19.6 m²/s².
Taking the square root of both sides, v ≈ 4.4 m/s.
Therefore, if the incline is perfectly smooth, the speed of the marble at the bottom of the incline will be approximately 4.4 m/s.
(c) The marble has different speeds in the two cases because the distribution of energy changes between rotational and translational kinetic energy.
When the incline is rough and the marble rolls down without slipping, some of the energy is converted into rotational kinetic energy. The moment of inertia of the rolling marble allows it to gain rotational kinetic energy, resulting in a higher speed at the bottom compared to the case where the marble slides without rolling.
On the other hand, when the incline is perfectly smooth and the marble slides without rolling, all of the energy is converted into translational kinetic energy. There is no rotational motion to consider, so the marble has a lower speed at the bottom compared to the rolling case.
In both cases, the total energy of the marble is conserved. However, the distribution of energy between rotational and translational forms is different, resulting in different speeds at the bottom of the incline.