Problem 35 - Algebra and Trigonometry

Question

Ramp Angle Billy and Timmy are using a ramp to load furniture into a truck. While rolling a 250 -pound piano up the ramp, they discover that the truck is too full of other furniture for the piano to fit. Timmy holds the piano in place on the ramp while Billy repositions other items to make room for it in the truck. If the angle of inclination of the ramp is \(20^{\circ}\), how many pounds of force must Timmy exert to hold the piano in position?

Step-by-Step Solution

Answer

About 85.5 pounds.

Step 1: Understand the Problem

Identify that Timmy needs to exert a force to keep the piano stationary by counteracting the component of the piano's weight parallel to the ramp. Use the given weight (250 pounds) and angle (20°) to find this component.

Step 2: Identify the Relevant Physics Formula

The force parallel to the ramp can be determined using the formula for the component of the gravitational force: \[\text{{Force}}_{\parallel} = W \times \text{{sin}}(\theta)\]where \(W = 250\) pounds (weight of the piano) and \(\theta = 20^{\circ}\) (angle of the ramp).

Step 3: Plug in the Values

Substitute the known values into the formula: \[\text{{Force}}_{\parallel} = 250 \times \text{{sin}}(20^{\text{\circ}})\]

Step 4: Calculate the Sine Value

Find the sine of 20 degrees using a calculator: \[\text{{sin}}(20^{\text{\circ}}) \approx 0.342\]

Step 5: Multiply to Find the Force

Multiply the weight by the sine of the angle to find the parallel component: \[250 \times 0.342 \approx 85.5\] pounds.

Key Concepts

Ramp Angle

The ramp angle is crucial in this exercise. It refers to the angle between the ramp and the ground. This angle influences how much of the object’s weight pushes down the ramp. In physics, we often denote this angle using the Greek letter \( \theta \). In Billy and Timmy's case, the ramp angle is given as \(20^{\text{°}}\). A steeper ramp angle means that the component of the gravitational force acting along the ramp will be greater. This is because the force is split into two components: one parallel to the ramp and one perpendicular to it.

Force Decomposition

Force decomposition is a method used to break down a single force into two or more component forces. For our ramp problem, the gravitational force (weight) of the piano is decomposed into two components: parallel and perpendicular to the ramp. Timmy needs to counteract the parallel component to keep the piano from sliding down. The perpendicular component presses the piano against the ramp but does not contribute to its motion along the ramp. This decomposition simplifies problem-solving by allowing us to focus only on the parallel component, which is directly responsible for the movement or tendency of the piano to slide.

Sine Function

The sine function is a fundamental trigonometric function often used in physics. It helps to relate the ramp angle to the forces acting on the objects. In our example, we use the sine of the ramp angle (\theta) to find the component of the weight acting parallel to the ramp. The sine function is defined as the ratio of the length of the side opposite the angle to the hypotenuse, in a right-angled triangle. For the given ramp angle of \(20^{\text{°}}\), we calculate \(\text{sin}(20^{\text{°}}) \approx 0.342.\)

Weight Component

In this scenario, the weight component is the part of the piano's weight that acts parallel to the ramp. To calculate this, we use the formula: \(\text{Force}_{\text{parallel}} = W \times \text{sin}( \theta )\). Here, the weight (\text{W}) of the piano is 250 pounds, and \( \theta \) (the ramp angle) is \(20^{\text{°}}\). By substituting these values, we get: \(\text{Force}_{\text{parallel}} = 250 \times 0.342 \approx 85.5 \text{pounds}\). Thus, Timmy needs to exert approximately 85.5 pounds of force to hold the piano in place on the ramp.