A child’s toy consists of a small wedge that has an acute angle θ(Fig. P6.42). The sloping side

step 1 Okay, so in this one we have a small wedge. I'm just going to draw the picture instead of being

A child’s toy consists of a small wedge that has an acute...

A child’s toy consists of a small wedge that has an acute angle $\theta$ (Fig. P6.42). The sloping side of the wedge is frictionless, and an object of mass $m$ on it remains at constant height if the wedge is spun at a certain constant speed. as an axis, a vertical rod that is as an axis, a vertical rod that is firmly attached to the wedge at the bottom end. Show that, when the boject sits at rest at a point at distance $L$ up along the wedge, the speed of the object must be $v=(g L \sin \theta)^{1 / 2}$

Key Concepts

Frictionless Dynamics

The assumption of a frictionless contact surface simplifies the analysis by eliminating the effects of frictional forces. This allows the focus to be solely on the interaction of gravitational and inertial forces, making it easier to derive the necessary conditions for equilibrium and constant height on the rotating inclined structure.

Static Equilibrium in a Rotating Frame

For an object to maintain a constant height on an inclined surface while the system rotates, the net force along the surface must balance out to zero. This involves equating the gravitational component along the incline to the effective forces arising from rotation, ensuring that the object remains stationary relative to the wedge.

Rotational Motion and Centripetal Force

This concept involves analyzing objects in circular motion where a centripetal force is required to keep moving in a curved path. In the context of spinning objects on a wedge, the necessary inward acceleration is provided by this force, which is fundamental in determining the conditions under which an object can maintain its position on a rotating inclined surface.

Force Decomposition on an Inclined Plane

When dealing with an inclined plane, the gravitational force acting on an object is split into two components: one parallel to the plane and one perpendicular to it. This decomposition is critical for understanding how the gravitational component along the plane interacts with other forces, such as the reaction force and any additional forces due to rotation.

Transcript

00:01 Okay, so in this one we have a small wedge.

00:05 I'm just going to draw the picture instead of being the problem.

00:09 So it's on a rod that is spinning.

00:15 And then here's the wedge.

00:18 Here's theta.

00:22 And the length is l.

00:27 And then the object on the top has a mass m.

00:32 Let's clean that up.

00:39 And let's see if there's any other information.

00:50 Okay.

00:51 And so when the object sits at rest, that the speed has to be whatever it is.

00:59 Okay, so basically we want, if it's going to sit at rest, then it's only going to have centripetal acceleration and it's not going to be, like it's only going to have an acceleration this way.

01:10 But, okay, so anyway, that'll become more clear as i draw the free body diagram for it.

01:14 So for the mass, we have the weight down and we have a normal force up.

01:24 And let's go ahead and actually draw this a bit more realistically.

01:35 So it has some kind of normal force from the wedge.

01:42 And we wanted to stay not going up and down the wedge.

01:48 So we basically want the acceleration in the vertical direction to be zero.

01:52 And then it's okay if we have some acceleration in this direction because that's going to give us, because it's moving in a circle.

01:59 So we'd actually expect it to be having some centripetal acceleration to the left.

02:06 So basically we want is this.

02:08 So if this normal force points like this and this is theta, yeah, let's find the, let's find a formula for the vertical component of this normal force.

02:23 So let's label everything.

02:26 I've done this enough times to know that this is theta and you can kind of draw out a bunch of triangles to sort of convince yourself.

02:34 So this is theta.

02:36 This is 90 degrees.

02:38 And then this whole thing is 180 degrees.

02:42 So 180 minus 90 minus theta...

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