Question

A bowling ball of mass \(M_{1}=6.0 \mathrm{~kg}\) is initially at rest on the sloped side of a wedge of mass \(M_{2}=9.0 \mathrm{~kg}\) that is on a frictionless horizontal floor. The side of the wedge is sloped at an angle of \(\theta=36.9^{\circ}\) above the horizontal. a) With what magnitude of horizontal force should the wedge be pushed to keep the bowling ball at a constant height on the slope? b) What is the magnitude of the acceleration of the wedge, if no external force is applied?

Short answer

#Answer# a) The magnitude of the horizontal force required to keep the bowling ball at a constant height on the slope is 47.1 N. b) The magnitude of the acceleration of the wedge when no external force is applied is 5.23 m/s².

Step-by-step solution

Bowling ball free body diagram

Draw a free body diagram for the bowling ball on the slope. The forces acting on the bowling ball are: 1. Gravity acting downward, \(M_{1}g\) 2. Normal force acting perpendicular to the slope, \(N\) 3. Friction force acting along the slope, \(F_{friction}\) The friction force acts in the same direction and equal to the horizontal force acting on the wedge.

Wedge free body diagram

Draw a free body diagram for the wedge on the horizontal floor. The forces acting on the wedge are: 1. Gravity acting downward, \(M_{2}g\) 2. Normal force acting upward, \(N'\) 3. Friction force acting horizontally, \(F_{friction}\) Step 2: Write equations of motion

Newton's second law for the bowling ball:

Applying Newton's second law and focusing on the vertical direction, we have: $$ N - M_{1}g\sin{(\theta)} =0 $$ On the horizontal direction: $$ F_{friction} - M_1 g \cos{(\theta)} = 0 $$

Newton's second law for the wedge:

Applying Newton's second law on the horizontal direction, we have : $$ F_{h} = M_{2} a_{w} + F_{friction} $$ Step 3: Solve equations to get the required values #a)# For part (a), the force required to keep the ball at a constant height on the slope is equal to the friction force and can be calculated using: $$ F_{friction} = M_1 g\cos(\theta) $$

Calculation for the force required to keep the bowling ball at constant height:

$$ F_{friction} = (6.0 \ \text{kg}) (9.81 \ \text{m/s}^2) \cos(36.9^\circ) = 47.1 \ \text{N} $$ The magnitude of the horizontal force required to keep the bowling ball at a constant height on the slope is 47.1 N. #b)# For part (b), we want to find the acceleration of the wedge when there is no external force, which means \(F_{h} = 0\). Hence we have: $$ 0 = M_{2}a_{w} + F_{friction} $$ Divide both sides by \(M_{2}\): $$ a_{w} = - \frac{F_{friction}}{M_{2}} $$

Calculation for the acceleration of the wedge:

$$ a_{w} = -\frac{47.1 \ \text{N}}{9.0 \ \text{kg}} = -5.23 \ \text{m/s}^2 $$ The magnitude of the acceleration of the wedge when no external force is applied is 5.23 m/s².

Key concepts

Free Body Diagram

In classical mechanics physics, a free body diagram (FBD) is used to visualize the forces acting upon an object. To create a FBD, one must isolate the object and represent all external forces, moments, and reactions affecting that object.

When dealing with problems involving objects at rest or moving on a plane, it is crucial to accurately represent both the magnitude and direction of forces. This approach facilitates the application of Newton's laws of motion. For example, in the case of the bowling ball, we included gravity, normal force, and friction. For the wedge, we included its weight and the reaction from the floor, depicted as normal force. The friction force on the wedge is shown as equal in magnitude but opposite in direction to the friction force on the ball, adhering to Newton's third law.

Newton's Second Law

Newton's second law is the core principle underlying the dynamics of motion in classical mechanics. It states that the acceleration of an object is dependent on two variables - the net force acting on the object, and the mass of the object, as shown in the equation \[ F = ma \].

In terms of our exercise, we applied Newton's second law separately to each object in the system. For the bowling ball, we examined the forces in both the vertical and horizontal directions to determine the normal and frictional forces. Similarly, for the wedge, we used the law to relate the horizontal force to the frictional force and the wedge's acceleration.

Static Equilibrium

A state of static equilibrium exists when an object is at rest or moving with a constant velocity, and all the forces acting upon it are balanced, resulting in no net force and no acceleration. The condition for static equilibrium in terms of forces is \[ \sum F = 0 \].

This concept is crucial when determining the force necessary to keep the bowling ball at a constant height on the slope. By ensuring the sum of the vertical forces are zero, we deduced that the frictional force is what's needed to maintain static equilibrium, allowing us to use the known variables to find the required horizontal force.

Kinematics Equations

The kinematics equations describe motion without considering the forces that cause it. They relate the variables of motion — position, velocity, acceleration, and time — without reference to the nature of the force applied.

While our exercise focuses primarily on forces and equilibrium, kinematics enters the picture as part of the broader set of tools needed in mechanics problems. If, for instance, we knew the acceleration and wanted to determine how far the wedge moved over a period of time without an external force, kinematics equations would be the ideal tool for that calculation.