A helicopter flies over the arctic ice pack at a constant altitude, towing an airborne 99-kg laser sensor which measures the thickness of the ice (see the drawing). The helicopter and the sensor move only in the horizontal direction and have a horizontal acceleration of magnitude 4.18 m/s2. Ignoring air resistance, find the tension in the cable towing the sensor.
To find the tension in the cable towing the sensor, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
In this case, the helicopter and the sensor are moving in the horizontal direction only, so we can focus on the horizontal forces.
Let's assume the tension in the cable is T.
The only force acting in the horizontal direction is the tension in the cable. Therefore, we can equate the net force to the mass of the sensor multiplied by its horizontal acceleration:
T = m * a
where T is the tension in the cable, m is the mass of the sensor (99 kg), and a is the horizontal acceleration (4.18 m/s^2).
Plugging in these values:
T = 99 kg * 4.18 m/s^2
Calculating the tension:
T = 413.82 N
Therefore, the tension in the cable towing the sensor is approximately 413.82 N.
To find the tension in the cable towing the sensor, we need to consider the forces acting on the sensor in the horizontal direction.
Given:
Mass of the sensor (m) = 99 kg
Horizontal acceleration (a) = 4.18 m/s^2
When a body is accelerated horizontally, there are two main forces acting on it: the gravitational force (mg) and the tension force in the cable (T). In this case, we can ignore air resistance, so there is no drag force.
Considering Newton's second law in the horizontal direction, the net force (F_net) acting on the sensor can be written as:
F_net = T - mg
The gravitational force is given by:
mg = (mass of the sensor) x (acceleration due to gravity)
= (99 kg) x (9.8 m/s^2)
≈ 970.2 N
Now, substituting the known values into the equation for F_net, we have:
F_net = T - mg
= T - 970.2 N
Since the horizontal acceleration of the sensor is given as 4.18 m/s^2, the net force is also equal to the mass of the sensor multiplied by the acceleration:
F_net = (mass of the sensor) x (horizontal acceleration)
= (99 kg) x (4.18 m/s^2)
≈ 413.82 N
Now we can equate the expressions for F_net:
F_net = T - 970.2 N
413.82 N = T - 970.2 N
To find Tension (T), rearrange the equation:
T = F_net + 970.2 N
T = 413.82 N + 970.2 N
T ≈ 1384.02 N
Therefore, the tension in the cable towing the sensor is approximately 1384.02 N.