A Solid Sphere Release From Rest at the top of a Ramp.

In summary, the acceleration of the solid sphere down the ramp can be calculated using the formula a = g sin θ, where g is the acceleration due to gravity (9.8 m/s^2) and θ is the angle of the ramp. The mass of the solid sphere does not affect its acceleration down the ramp, as long as there is no air resistance or friction. The speed of the solid sphere as it rolls down the ramp can be influenced by the angle of the ramp, the mass of the sphere, and the presence of friction or air resistance. As the solid sphere rolls down the ramp, its potential energy is converted into kinetic energy, resulting in an increase in its kinetic energy. The solid sphere will continue to roll
  • #1
aZenki
1
0
Please help me with this question. Thank you.

A 2.0 kg solid sphere (radius = 0.10m) is released from rest at the top of a ramp and allow to roll without slipping. The ramp is 0.75m high and 5.3 m long. Find Ktotal, Krot, and Ktranslation.
 
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  • #2
Pure rolling takes place, work done by friction is 0. Energy is conserved. Ktotal=mgh, Krot/Ktranslation=2/5 ie in ratio of moment of inertia. Ktranslation=5/7 mgh etc
 
  • #3


I would approach this question by first identifying the known and unknown variables. We know that the solid sphere has a mass of 2.0 kg and a radius of 0.10m, and that it is released from rest at the top of a ramp with a height of 0.75m and a length of 5.3m. The unknown variables are the total kinetic energy (Ktotal), the rotational kinetic energy (Krot), and the translational kinetic energy (Ktranslation).

To find Ktotal, we can use the formula Ktotal = 1/2 * mv^2, where m is the mass of the sphere and v is its velocity. Since the sphere is released from rest, its initial velocity is 0. We can calculate its final velocity using the conservation of energy principle, where the potential energy at the top of the ramp is converted into kinetic energy at the bottom. This gives us v = √(2gh), where g is the acceleration due to gravity (9.8 m/s^2) and h is the height of the ramp (0.75m). Plugging in the values, we get v = 3.4 m/s. Therefore, Ktotal = 1/2 * 2.0 kg * (3.4 m/s)^2 = 11.6 J.

To find Krot, we can use the formula Krot = 1/2 * I * ω^2, where I is the moment of inertia and ω is the angular velocity. Since the sphere is rolling without slipping, its angular velocity is equal to its linear velocity divided by its radius, ω = v/r. The moment of inertia for a solid sphere is given by I = 2/5 * mr^2. Plugging in the values, we get Krot = 1/2 * (2/5 * 2.0 kg * (0.10m)^2) * (3.4 m/s / 0.10m)^2 = 0.46 J.

To find Ktranslation, we can simply subtract Krot from Ktotal, since Ktranslation represents the remaining kinetic energy. Therefore, Ktranslation = Ktotal - Krot = 11.6 J - 0.46 J = 11.14 J.

In conclusion, the total kinetic energy of the solid sphere released from rest at the top of the ramp is
 

Related to A Solid Sphere Release From Rest at the top of a Ramp.

1. What is the acceleration of the solid sphere down the ramp?

The acceleration of the solid sphere down the ramp can be calculated using the formula a = g sin θ, where g is the acceleration due to gravity (9.8 m/s^2) and θ is the angle of the ramp.

2. How does the mass of the solid sphere affect its acceleration down the ramp?

The mass of the solid sphere does not affect its acceleration down the ramp, as long as there is no air resistance or friction. This is because the acceleration due to gravity is constant for all objects, regardless of their mass.

3. What factors can influence the speed of the solid sphere as it rolls down the ramp?

The speed of the solid sphere as it rolls down the ramp can be influenced by the angle of the ramp, the mass of the sphere, and the presence of friction or air resistance. These factors can affect the acceleration and therefore the final speed of the sphere.

4. What happens to the kinetic energy of the solid sphere as it rolls down the ramp?

As the solid sphere rolls down the ramp, its potential energy is converted into kinetic energy. This means that the kinetic energy of the sphere increases as it moves down the ramp. At the bottom of the ramp, all of the potential energy will have been converted into kinetic energy.

5. Can the solid sphere reach a point where it stops rolling on the ramp?

No, the solid sphere will continue to roll down the ramp until it reaches the bottom, unless there is a force acting on it that causes it to stop. This is because there is no friction or air resistance present to slow down the sphere's motion.

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