The systems shown below are in equilibrium (with m = 3.60 kg and θ = 33.0°). If the spring scales are calibrated in newtons, what do they read? Ignore the masses of the pulleys and strings and assume the pulleys and the incline are frictionless
To solve this problem, we need to analyze the forces acting on the system and apply Newton's laws of motion. Here are the steps to determine the readings on the spring scales:
Step 1: Identify the forces acting on the system.
The forces acting on the system are the weight (mg) acting vertically downwards, the normal force (N) exerted by the inclined plane perpendicular to it, and the tension forces (T) in each of the strings.
Step 2: Resolve the weight into its components.
Since the angle θ is given, we can resolve the weight (mg) into two components: mg*cos(θ) acting perpendicular to the inclined plane and mg*sin(θ) acting parallel to the inclined plane.
Step 3: Write the equations of motion for each object.
a) For the hanging mass:
In the vertical direction: T - mg*cos(θ) = 0 (Equation 1)
In the horizontal direction: T*sin(θ) = ma (Equation 2)
b) For the block on the incline:
In the vertical direction: N - mg*sin(θ) = 0 (Equation 3)
In the horizontal direction: N*sin(θ) = ma (Equation 4)
Step 4: Solve the equations.
First, we need to solve Equation 3 to find the normal force N:
N = mg*sin(θ)
Next, we substitute this value of N into Equation 4 and solve for a:
mg*sin(θ)*sin(θ) = ma
a = g*sin^2(θ)
Now, we substitute this value of a into Equation 2 to find the tension force T:
T*sin(θ) = ma
T*sin(θ) = m*g*sin^2(θ)
T = mg*sin(θ)
Finally, we can substitute the known values (m = 3.60 kg and θ = 33.0°) into the equation for T to find the tension force T:
T = (3.60 kg)(9.8 m/s^2)*sin(33.0°)
Step 5: Calculate the readings on the spring scales.
The readings on the spring scales are equal to the magnitudes of the tension forces (T) in each of the strings.
Therefore, the readings on the spring scales, calibrated in newtons, will be:
T = (3.60 kg)(9.8 m/s^2)*sin(33.0°) = ... (calculate the numerical value)