A lift of mass 1500 kg is raised by means of a rope passing round a drum of diameter 1.50 m. Given that the lift is accelerating upwards at 0.255 m s-2, draw a free body diagram of the lift, calculate the tension in the rope, the moment applied to the drum and the angular acceleration of the drum.
To draw a free body diagram of the lift, we will consider the forces acting on it:
1. Weight (mg): This force acts vertically downward and is given by the formula F = mg, where m is the mass of the lift and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Tension in the rope (T): This force acts vertically upwards and is responsible for lifting the lift. It is the force we want to calculate.
3. Normal force (N): This force acts perpendicular to the surface of the lift and counteracts the weight of the lift. Since the lift is accelerating upwards, N will be slightly less than mg.
Now, let's calculate the tension in the rope:
Using Newton's second law of motion, F = ma, we can calculate the net force acting on the lift in the vertical direction.
Net force = T - mg
Given that the lift is accelerating upwards at 0.255 m/s^2 and the mass of the lift is 1500 kg, we substitute these values into the equation:
T - mg = ma
T - (1500 kg)(9.8 m/s^2) = (1500 kg)(0.255 m/s^2)
Rearranging the equation:
T = (1500 kg)(0.255 m/s^2) + (1500 kg)(9.8 m/s^2)
T = 382.5 N + 14700 N
T = 15082.5 N
Therefore, the tension in the rope is 15082.5 N.
To calculate the moment applied to the drum, we will consider the torque exerted by the tension force on the drum. The torque is given by the formula:
Torque = Force × distance
In this case, the force is the tension in the rope (15082.5 N), and the distance is the radius of the drum (diameter/2 = 1.50 m/2 = 0.75 m).
Torque = 15082.5 N × 0.75 m
Torque = 11311.875 Nm (or 1.1312 × 10^4 Nm)
Therefore, the moment applied to the drum is 11311.875 Nm.
To calculate the angular acceleration of the drum, we can use the formula:
Angular acceleration = Torque / Moment of inertia
The moment of inertia of a solid cylinder is given by the formula:
Moment of inertia = (1/2) × mass × radius^2
Given that the mass of the drum is unknown, we cannot directly calculate the moment of inertia. However, assuming the mass of the drum is negligible compared to the mass of the lift, we can ignore it in our calculations.
Angular acceleration = Torque / Moment of inertia
Angular acceleration = 11311.875 Nm / ((1/2) × 1500 kg × (0.75 m)^2)
Calculating the value gives:
Angular acceleration = 11311.875 Nm / (0.75 kgm^2)
Angular acceleration = 15082.5 N / kgm
Therefore, the angular acceleration of the drum is 15082.5 N/kgm.