Question

Suppose you are a space traveler, and you land on the surface of a strange new planet. As stated in your Starfleet manual, your first job is to measure the acceleration due to gravity on this planet, g. You construct an 'Atwood machine', which consists of two masses connected to a pulley by a string of negligible mass. One of the masses is m1 = 62.0 kg and the other is m2 = 50.0 kg. The pulley is a uniform disk of mass m = 117.0 kg and radius r = 65.0 cm which is mounted on frictionless bearings. When released from rest, the heavier mass moves down 95.9 cm in 1.05 s, and there is no slippage of the string on the pulley. What is the acceleration due to gravity on this planet?

Answer 1

The acceleration due to gravity on the planet is approximately 8.85 m/s².

Step-by-step explanation:

To determine the acceleration due to gravity (g) on the planet using the Atwood machine setup, we can start by analyzing the forces acting on the system.

The equation governing the motion of an Atwood machine is:

`\[ g = (2(m_1 - m_2)h)/(m_1 + m_2 + m) \]`

Where:

`m_1` = 62.0 kg (mass of one object)\(
`m_2` = 50.0 kg) (mass of the other object)\( m = 117.0 kg) (mass of the pulley)\( h = 95.9 cm) = 0.959 m (distance moved by the heavier mass)The time taken, t = 1.05 seconds.

First, let's calculate the acceleration of the system using the distance traveled by the heavier mass:

Distance = Initial velocity × time + 1\2 × acceleration × time²

`0.959 \, \text{m} = 0 * 1.05 \, \text{s} + (1)/(2) * a * (1.05 \, \text{s})^2 \]\[ a = \frac{2 * 0.959 \, \text{m}}{(1.05 \, \text{s})^2} \]\[ a = 1.784 \, \text{m/s}^2 \]`

Now, using the derived acceleration in the Atwood machine formula:

`\[ g = (2(m_1 - m_2)h)/(m_1 + m_2 + m) \]\[ g = \frac{2 * (62.0 \, \text{kg} - 50.0 \, \text{kg}) * 0.959 \, \text{m}}{62.0 \, \text{kg} + 50.0 \, \text{kg} + 117.0 \, \text{kg}} \]\[ g = \frac{2 * 12.0 \, \text{kg} * 0.959 \, \text{m}}{229.0 \, \text{kg}} \]\[ g \approx 8.85 \, \text{m/s}^2 \]`

Therefore, the acceleration due to gravity on this planet, as determined by the Atwood machine setup, is approximately
`\( 8.85 \, \text{m/s}^2 \)`.

Answer 2

To determine the planet's acceleration due to gravity using an Atwood machine, one must apply the principles of force, motion, and rotational inertia, followed by kinematic equations to solve for acceleration, and subsequently for the acceleration due to gravity.

Step-by-step explanation:

To calculate the acceleration due to gravity on this new planet using an Atwood machine, we first must analyze the motion of the two masses and the pulley. Since the string does not slip on the pulley, the tension in the string provides the force that accelerates both masses and rotates the pulley. For mass m1 (62.0 kg), the downward force is m1g, and for mass m2 (50.0 kg), the upward force is m2g. The pulley has a rotational inertia, I = 0.5mr², where m is the mass of the pulley and r is its radius.

Using Newton's second law for rotation, τ = Iα where τ is the net torque and α is the angular acceleration of the pulley. The torque caused by each tension force is Torque = Tr, and the angular acceleration of the pulley is related to the linear acceleration a of the masses by the equation a = rα. We can set up equations for forces on masses and for rotational motion of the pulley and solve for the acceleration, a. Lastly, we use the kinematic equation s = ut + 0.5at² to relate the acceleration to the displacement s and the time t.

From the given displacement (95.9 cm) and time (1.05 s), we could calculate the expected acceleration. Then by equating the net force on the system, which is (m2 - m1)g, to the total mass of the system multiplied by the acceleration, (m1 + m2)a, we can solve for g, the acceleration due to gravity on the planet.