Question

A uniform solid sphere of mass \(M\) and radius \(R\) is rolling without sliding along a level plane with a speed \(v=3.00 \mathrm{~m} / \mathrm{s}\) when it encounters a ramp that is at an angle \(\theta=23.0^{\circ}\) above the horizontal. Find the maximum distance that the sphere travels up the ramp in each case: a) The ramp is frictionless, so the sphere continues to rotate with its initial angular speed until it reaches its maximum height. b) The ramp provides enough friction to prevent the sphere from sliding, so both the linear and rotational motion stop when the object reaches its maximum height.

Short answer

Answer: In the case of a frictionless ramp, the maximum distance the sphere travels up the ramp can be calculated using the height (h) and the angle (θ) of the ramp. Based on the conservation of energy principle, we found h = ((7 / 5) * 3^2) / (2 * 9.81) (approx.). Then, we calculated the distance (d) as d = h / sin(23.0°).

Step-by-step solution

Kinetic Energy and Potential Energy Formulae

In both cases, the total mechanical energy must be conserved. Let's first recall the formulae for kinetic energy (KE) and gravitational potential energy (PE): 1. Kinetic Energy: KE = 0.5 * m * v^2 (for linear motion) + 0.5 * I * ω^2 (for rotational motion) 2. Gravitational Potential Energy: PE = m * g * h Here, I is the moment of inertia of the sphere which is (2/5) * m * R^2 for a solid sphere, and ω is the angular speed related to the linear speed by ω = v / R.

Case (a): Frictionless Ramp

Since the sphere continues to rotate with its initial angular speed, there is no loss in rotational kinetic energy. Only linear kinetic energy transforms into potential energy due to the change in height. Step 1: Calculate the initial rotational KE: KE_rot_initial = 0.5 * I * ω^2 = 0.5 * (2/5) * m * R^2 * (v/R)^2 = 0.5 * (2/5) * m * v^2 Step 2: Calculate the initial linear KE: KE_lin_initial = 0.5 * m * v^2 Step 3: Equate the total KE to PE: KE_rot_initial + KE_lin_initial = m * g * h Step 4: Solve for h: h = (0.5 * (2/5) * m * v^2 + 0.5 * m * v^2) / (m * g) = ((7 / 5) * v^2) / (2 * g) = ((7 / 5) * 3^2) / (2 * 9.81) (approx.) Step 5: Calculate the distance traveled up the ramp using the height and the angle θ: d = h / sin(θ) = h / sin(23.0°)

Key concepts

Conservation of Mechanical Energy

The concept of the Conservation of Mechanical Energy states that in an isolated system not subject to any non-conservative forces (like friction), the sum of the potential energy (PE) and kinetic energy (KE), which together comprise the mechanical energy, remains constant throughout any process.

This principle is particularly crucial when dealing with physics problems involving objects moving under the influence of gravity, like a rolling sphere on a ramp. When the sphere moves upwards, its KE is gradually converted into PE. Conversely, as the sphere descends, the PE is transformed back into KE. It's this energy conversion that allows us to solve for unknown quantities, such as how high the sphere travels up a ramp, by setting the total mechanical energy at the beginning and end of the motion equal to one another.

Kinetic Energy Formula

The Kinetic Energy Formula is used to calculate the energy possessed by a moving object due to its motion. For linear motion, the formula is expressed as \(KE_{lin} = 0.5 \times m \times v^2\), where 'm' stands for the mass of the object and 'v' is its velocity.

However, when an object is rolling, such as a sphere, it not only translates but also rotates. Accordingly, we must consider the rotational kinetic energy, given by \(KE_{rot} = 0.5 \times I \times \omega^2\), where 'I' is the moment of inertia, and 'omega' (\(\omega\)) is the angular speed. This addition is essential in problems where the object's rotation plays a role, like a sphere rolling up a ramp.

Potential Energy Calculation

The calculation of Potential Energy (PE) is straightforward in situations involving height changes due to gravity. The formula \(PE = m \times g \times h\) describes this energy, where 'g' is the acceleration due to gravity, and 'h' is the height that the object is raised to.

The potential energy is directly proportional to the height; an object higher above a reference point (usually the ground) will have a greater potential energy. When solving the physics problem of a rolling sphere ascending a ramp, the sphere's mechanical energy shifts from kinetic to potential as it moves against the force of gravity, increasing its height and thus its PE.

Moment of Inertia

The Moment of Inertia (I) is an object's resistance to change in its rotational motion. It's analog to mass in linear motion but for rotation. The moment of inertia depends on the object's shape and the distribution of its mass.

For a solid sphere, like the one in our physics problem, the moment of inertia is \(I = \frac{2}{5} m R^2\), where 'm' is the mass and 'R' is the radius of the sphere. This value is incorporated when calculating the rotational kinetic energy. A larger moment of inertia implies that it's harder for the sphere to be brought into rotation or be stopped from rotating, which heavily influences the sphere's behavior when rolling up a ramp.

Angular Speed

The term Angular Speed (\(\omega\)) refers to the rate at which an object rotates or spins. It's the angular equivalent of linear velocity. The angular speed is related to the linear velocity through the formula \(\omega = \frac{v}{R}\), where 'v' is the linear speed and 'R' is the radius of the object.

In the context of a sphere rolling without slipping, angular speed can be directly derived from the linear speed, since every point on the sphere completes one rotation in the same time it takes to traverse a path equal to its circumference. Understanding angular speed is essential for correctly calculating both the kinetic energy related to an object's rotation and predicting the behavior of the object on surfaces with different frictional properties.